Final Report Summary - ACMAC (Archimedes Center for Modeling, Analysis and Computation in Crete)
Executive Summary:
An executive summary
ACMAC
The Project and its objectives
The objective of the ACMAC project was to strengthen the research profile of the Applied Mathematics Department and to enable its members to enhance their contacts with major research centers in Europe. To this end certain measures were taken that reinforced the Department’s research capability in areas where it had recognized expertise. These measures were organized around a Center, named Archimedes Center for Modeling, Analysis, and Computation (ACMAC), aiming to promote academic excellence in Applied Mathematics and fostering the interface of modeling, analysis and computation.
The main objectives of the ACMAC project were:
• Exchange of know-how and expertise between ACMAC and established Centers of Excellence and other Collaborating Institutions with which ACMAC members had ongoing projects. The Centers of Excellence were:
• University of Bonn, Institute for Applied Mathematics, University of Bonn,
• Cambridge University, DAMTP, Cambridge University,
• Politecnico di Milano, MOX, Milan,
• University of Paris VI, Laboratoire J.L. Lions, Paris,
• University of Oxford, Computing Laboratory, Oxford.
• Recruitment of incoming experienced researchers
• Organization of thematic programs taking place in the “Archimedes Center”, Crete, which included research visits and the organization of workshops in targeted themes. This was a principal activity envisaged to bring in Crete and involved in our project several internationally leading Applied Mathematicians.
• Dissemination of research results through participation of ACMAC members in conferences, workshops and research visits.
• Upgrading the existing computational infrastructure and enhancing the computational capacity of the group.
• Evaluation of the project at the end of the reporting periods.
ACMAC had a sizable impact to the evolution of Applied Mathematics in Crete as an internationally recognized centre in modern computational and applied mathematics.
Project Context and Objectives:
The main objectives of the ACMAC project, listed in Annex I of the contract, is to strengthen the research profile of the Applied Mathematics Department and to enable its members to enhance their contacts with major research centers in Europe.
This is achieved by a series of actions, detailed below, and organized in a research center called “Archimedes Center for Modeling, Analysis and Computation:
EXCHANGE OF KNOW-HOW AND EXPERIENCE between members of the Archimedes Center for modeling, Analysis and Computation (ACMAC) and members of associated established Centers of Excellence of Applied Mathematics in Europe. The associated Established Centers are
• University of Bonn, Institute for Applied Mathematics, University of Bonn,
• Cambridge University, DAMTP, Cambridge University,
• Politecnico di Milano, MOX, Milan,
• University of Paris VI, Laboratoire J.L. Lions, Paris,
• University of Oxford, Computing Laboratory, Oxford.
A number of outgoing and incoming visits to and from Centres of Excellence and Collaborating Institutions were conducted, which were either supported collaborations or participation to programs carried out by the Centres of Excellence, or alternatively supported visits of members from the Centres of Excellence or Collaborating Institutions for research collaborations with ACMAC members or in order to take part in activities taking place at ACMAC.
Emphasis is also given to Collaborations with other important Centers in Europe, with which several of our faculty have ongoing collaborations. The Collaborating Institutions listed in the contract are: University of Erlangen; KTH (Royal Institute of Technology), Stockholm; University of Münster; University of Nice; University of Rome Tor Vergata; SISSA, Trieste; INRIA, Rocquencourt; CMAP, Ecole Polytechnique; University Carlos III, Madrid.
This is a continuing objective covering all reporting periods.
RECRUITMENT OF INCOMING EXPERIENCED RESEARCHERS. The Center is expected to hire Long-Term Research Associates. In addition targeted short-term appointments will be reserved for Senior Experts. These will include experts in specific tasks that we would like to perform as well as short-term appointments of Distinguished Scientists which are international leaders in fields covered by this proposal.
The following types of appointments were made: (i) Appointments of postdoctoral researchers; (ii) Visiting appointments of senior researchers; (iii) Visiting appointments of distinguished researchers. The senior researchers stay on site for a period of a month or longer, collaborate with ACMAC members, deliver a series of lectures, or alternatively help ACMAC to implement aspects of the program.
A total of nineteen postdoctoral positions appointment were made since the beginning of the program. The posts were advertised in job websites worldwide. The selected researchers have PhD from prominent Institutions from Europe but also other parts of the world. Twenty one researchers were hired and participated in the activities of ACMAC as senior visitors.
The following researchers were hired as postdocs researchers: P. D. Karagiorge (PhD from University of Bristol), N. Gupta (PhD from Indian Institute of Technology), C. Donadello (PhD from the International School of Advanced Study (SISSA), Y. Nagase (PhD from Hokkaido University), L. R. Bellet (PhD from University of Geneva), A. Frouvelle (PhD from Université Paul Sabatier), J. Olivier (PhD from Université de Savoie), I. Chremmos (PhD from National Technical University of Athens (NTUA)), D. Mitsoudis (PhD from University of Athens), Ch. Konaxis (PhD from University of Athens), Ch. Sourdis (PhD from University of Athens), J. Giesselmann (PhD from Bielefeld University), D. A. Smith (PhD form University of Reading), A. Kunoth (PhD from Berlin University), C. Lattanzio (PhD from University of Rome Tor Vergata), M. Palombaro (PhD from University of Rome “La Sapienza”), M. N. Mach (PhD from University of Pisa), A. Athanasoulis (PhD from Princeton University), I. Pantazis (PhD from University of Crete), E. Pulvirenti (PhD from Universita degli Studi Roma, Tre), N. Tymis (PhD from Loughborough University) and A. Rissanou (PhD from University of Crete).
In addition, the following researchers were hired as senior researchers: E. Georgoulis (Sussex), D. Shepelskiy (Verkin Inst, Ukraine), L. (Univ. L’Aquila), L. S. Keeling (U. Graz, Austria); P. Kaklis (NTUA, Athens), S. Venakides (Duke Univ.), E. Kyza (University of Crete); F. Faure (Institut de Physique Nucleaire); T. Pryer (University of Sussex);; A. Cangiani (University of Oxford); A. Vainchtein (Cornell University); K. Siebert (University of Freiburg); M. Jensen (University of Oxford); Sergey Dobrokhotov (Institute of Electronic Machinendesign), P. (S.I.S.S.A.) K. Ecker (Mathematisches Institut, Freie Universität Berlin), L. (Sapienza University of Rome), E. C. Alberti (University of the Basque Country (UPV/EHU))
Finally, the following researchers were hired as distinguished visitors: S. Venakides (Duke Univ.); Th. Zariphopoulou (Oxford); P. Souganidis (U. Chicago); Tai-Ping Liu (Stanford Univ and Academia Sinica, Taiwan); Jian-Guo Liu (Duke Univ.); Gen Nakamura (Hokkaido Univ.); Michael Sigal (University of Toronto). Timothy Healy (Cornell), Peter Bates (Michigan State University), Manoussos Grillakis (university of Maryland), Yaroslav Kurylev (University College London), Gery Bona (University of Illinois at Chicago), Ohannes Karakashian (University of Tenessee). A special effort was devoted in attracting to ACMAC prominent members of the Greek Scientific diaspora.
This is a continuing objective covering all reporting periods.
ORGANIZATION OF WORKSHOPS AND CONFERENCES. The organization of workshops and meetings will be an important tool for knowledge transfer among the involved research groups. These actions will enhance networking and collaborations and expose ACMAC members in an international scientific environment. These activities are organized around thematic programs, that involve short-term visits of scientists working in the area emphasized by the thematic program, as well as periods of concentrated activity of workshops and conferences.
A major part of the project is the support of incoming research visits and the organization of workshops under the framework of thematic programs.
The following thematic programs were completed:
1. The Thematic program on STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS, took place in the period May-June 2011. A workshop on “Stochastic PDE and its applications” was held in the facilities of the centre from 13 – 17 June 2011; website here.
2. The Thematic program on MATHEMATICAL MODELING OF MATERIALS ACROSS SCALES, took place in the period of Sep 2010 – Aug 2011. Two workshops were organized: The first entitled “Continuum and kinetic methods in the theory of shocks, fronts, dislocations and interfaces” was held on June 20-24, 2011, workshop's webpage. The date of June 24th was dedicated in honor of C.M. Dafermos (Alumnae Prof. at Brown University). The second workshop "Coarse–graining of many–body systems: analysis, computations and applications" was held on Jun 27 – Jul 1, 2011, workshop's webpage .
3. The Thematic program on COMPUTATIONAL PARTIAL DIFFERENTIAL EQUATIONS started on Sep 1, 2011 and and ended on Dec 31, 2011. A number of research visits were supported under this thematic program and two workshops were organized: The first workshop, entitled "Modern Techniques in the Numerical Solution of Partial Differential Equations", took place on Sep19 – Sep 23, 2011. The workshops's webpage may be found here. The second workshop, entitled "Discontinuous Galerkin Methods for Partial Differential Equations", was organized on Sep 26-28, 2011. The program can be found at the workshop's webpage.
4. The Thematic program on WAVE PROPAGATION IN COMPLEX ENVIRONMENTS took place in the period of April 2012 – June 2012. Three workshops were organized: The first workshop, “Wave propagation in complex media and applications”, was held on May 7-11, 2012, with workshop's webpage. The second workshop, "Semiclassical & multiscale aspects of wave propagation", was held on May 28 – Jun 1, 2012. The workshop's webpage may be found here. The third workshop, “Waves and imaging in complex media”, was held in the facilities of FORTH (Foundation for Research and Technology-Hellas) from June 7th to 8th (the introductory part) and from June 11th to 15th (the main part), 2012. The website of the workshop can be found here.
5. The Thematic program on SCIENTIFIC AND HIGH PERFORMANCE COMPUTING, started on Sep 1, 2012 and finished on Jan 31, 2013. A number of research visits were supported under this thematic program and two workshops were organized: The first workshop entitled "Software Frameworks for Challenging Computational Problems" took place from 14 – 18 January 2013. More information can be found at the workshops's webpage. The second workshop, entitled "Geometric Computing", took place in ACMAC during the period between 21st – 25th January, 2013, see the workshop's webpage.
6. A workshop entitled “Women in Applied Mathematics” took place in Heraklion on May 2-5, 2011, co-organized between ACMAC and the Institute of Applied and Comp. Math, FORTH. Apart from the excellent scientific program detailed in the workshop's webpage, a panel discussion took place on the special problems facing female scientists in their academic careers took place.
7. The workshop “Cell biology and physiology: PDE models”, was held in the facilities of ACMAC on October 4-6, 2012. The aim of this workshop was to bring together leading european groups and present the latest developments on mathematical modeling of biological cells and their various interactions. The program can be found at the workshop's webpage.
8. The thematic program on “NONLINEAR SCHROEDINGER EQUATIONS AND APPLICATIONS” was organized in the period March 1 – May 31, 2013. Three workshops were organized as part of this thematic program: The workshop on “Domain Microstructure and dynamics in magnetic elements” on April 8-11, 2013 whose program can be found at http://www.acmac.uoc.gr/%CE%BCMAGelm2013/program.php Second, the workshop on “Nonlinear Schroedinger equatons: Theory and Applications” that took place on May 20-24, 2013 and the program of which is found at http://www.acmac.uoc.gr/NLS2013 The last entitled “The 11th European Finite Element Fair” that took place on May 31- June 1, 2013, whose webpage can be found at http://www.acmac.uoc.gr/EFEF2013
9. The thematic program on “MATHEMATICAL METHODS FOR MULTISCALE SYSTEMS” was organized from June 1- July 28 encompassing as main even the two-week workshop “Kinetic description of multiscale phenomena” and one day workshop "Wave Phenomena: Numerical Methods and Analysis" on 29th July.
The first workshop fostered the interplay of propabilistic and kinetic techniques for problems on phase transitions. The workshop’s webpage can be found at the following link: http://www.acmac.uoc.gr/KDM2013 The second workshop on 29th July, whose program can be found at http://www.acmac.uoc.gr/WAVENUMA2013/program.php.
10. The workshop “Stochastic Methods in Finance and Physics” was organized in the period July 15- 19, 2013. The aim of this workshop was to bring together groups that work on stochastic particle systems on diverse problems arising in finance or in the physical sciences and to display the interconnections of the problems. The workshop’s program can be found at http://www.acmac.uoc.gr/SMFP2013/
11. The final event of the ACMAC project was the “International Conference on Applied Mathematics” that took place the week of September 16-20, 2013. The idea of this even was to serve as a major International Conference on the interplay of modelling, analysis and computation with a program that combines plenary talks on major developments in the general areas of the ACMAC project, and at the same time to have minisymosia organized in parallel that are on specific subjects of interest to the members of the Center. The program of this Conference can be found at http://www.acmac.uoc.gr/ICAM2013/
DISSEMINATION AND PROMOTIONAL ACTIVITIES The web page of the ACMAC Centre www.acmac.uoc.gr is operational from Jun 2010 and is improved and evolving continuously. There have been promotional pamphlets for the Centre and separately for each of the ongoing programs that are distributed in Greece and other European Universities and Research Centres.
A much emphasized task is the participation of ACMAC members in International Conferences and Workshops, as well as Research visits to Universities. Many such visits were supported in the course of running this project.
WP7. EVALUATION: An evaluation of the effort will be conducted at the end of the 42nd month. Four experts will conduct the evaluation of the research strategy and agenda of the ACMAC centre. These internationally recognized and independent experts will be chosen by the Commission services at the 22nd month of the project. They will receive a contract by the Applied Mathematics Department according to the terms outlined in WP7.
Project Results:
Research Results
The research activity at ACMAC was in the field of Applied Mathematics and specifically at the interface of modeling, analysis and computation. A large number of research papers were completed by ACMAC members. A preprint server was set up where the published results of ACMAC members and the postdoctoral researchers are posted. The web address of the preprint server is
http://preprints.acmac.uoc.gr/
Among the results we highlight works on the subjects:
modern multiscale methods for the simulation of coupling mechanisms between atomistic and continuum descriptions;
work on coherent interferometric imaging based on the back-propagation of local spacetime cross-correlations of array data, the discovery and subsequent experimental observations of autofocusing radially symmetric beams;
the development of a rigorous and computationally flexible framework for parallelization of Kinetic Monte Carlo methods and of spatial multilevel coarse-graining methods for Monte Carlo sampling and molecular simulation;
the development of a novel approach for patching inner and outer solutions applicable to problems with exchange of stability;
the study on the structure of the system of polyconvex elastodynamics, and the work on dynamic formation of cavities in isotropic elasticity;
finally, the works on stochastic particle models describing the onset of phase transitions.
A brief description of the most significant scientific results of ACMAC members now follows including a list of the three most significant publications per ACMAC member.
Markos Katsoulakis
Coarse-Graining:
Katsoulakis and collaborators introduced in 2003 hierarchical Coarse-Graining (CG) schemes for accelerating Kinetic Monte Carlo (KMC) simulations, showing that these new computational methods speed-up conventional KMC by several orders of magnitude. In ACMAC he subsequently developed multilevel CG methods for reliable simulation at mesoscopic length scales and where morphological features dictated by microscopics (patterns, interfaces, etc.) can be observed. Finally, in his ACMAC-sponsored research, the conjectured role of multi-body interactions in CG was rigorously analyzed, and efficient numerical implementations of such terms were developed for the first time. These works have appeared in the top disciplinary computational journals, as well as in the top applied mathematics journals:
Uncertainty Quantification (UQ) for Non-equilibrium Complex Systems:
In ACMAC he started developing path-wise information theory-based and goal-oriented sensitivity analysis and parameter identification methods for complex high-dimensional dynamics, as well as path-wise information-theoretic tools for parameterized CG of non-equilibrium extended systems. The combination of these novel methodologies provide the first methods in the literature which are capable to handle UQ questions for stochastic complex systems with some or all of the following features: (a) multi-scale models with a very large number of parameters, (b) spatially distributed systems such as Kinetic Monte Carlo or Langevin Dynamics, (c) non-equilibrium processes typically associated with coupled physico-chemical mechanisms, driven boundary conditions, etc. These very recent works have just appeared in top disciplinary computational journals, as well as in top applied mathematic journals.
Parallelization Methods for Kinetic Monte Carlo.
Here, Katsoulakis, students and collaborators addressed mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo (KMC) simulations. They developed a new mathematical formulation of KMC parallelization, based on hierarchical operator splitting, as means of systematically decomposing the computational and communication loads between multiple processors in a parallel architecture. This spatial decomposition of the KMC algorithm into a hierarchy of Markov operators was also shown to be highly appropriate for parallelizing on clusters of Graphical Processor Units (GPU). Furthermore, this mathematical framework allowed to rigorously assess and improve the numerical and statistical consistency of existing parallel KMC software such as SPPARKS, developed recently in Sandia National Laboratories. These recent works have appeared in the top journals in high-performance computing and numerical analysis.
List with the most significant publications
1) E. Kalligiannaki, M. Katsoulakis, P. Plechac and D. G. Vlachos. Multilevel coarse graining and nano-pattern discovery in many particle stochastic systems. J. Comp. Phys., 231, 2599-2620, (2012).
2) Giorgos Arampatzis, M. Katsoulakis, Petr Plechac, Michela Taufer, Lifan Xu. Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms J. Comp. Phys., 231, 7795-7814, (2012)
3) G. Arampatzis, M. Katsoulakis and P. Plechac. Parallelization, processor communication and error analysis in lattice kinetic Monte Carlo, SIAM J. Num. Analysis, in press, (2014).
Chrysoula Tsogka
Together with Liliana Borcea and George Papanicolaou we have been working on developing statistically stable methods for imaging in heterogeneous media with numerous small heterogeneities that scatter the waves and impede the imaging process. Our most recent results on the subject are poresented in the SIAM Review paper [1].
Together with Josselin Garnier, George Papanicolaou and Adrien Semin we have been working on imaging using cross-correlations of passive noise recordings. Here we exploit the fact that the Green's function (or the travel time) between two passive sensors can be estimated from cross-correlations of signals generated by ambient incoherent noise, which is due to sources that are randomly distributed in space and statistically stationary in time. In [2], we carried out a detailed analysis of the signal to noise ratio of the image and analyzed how it depends on the properties of the noise sources and the array of receivers.
Together with D. Mitsoudis and S. Papadimitropoulos we have been working on the problem of selective imaging extended scatterers in waveguides using the scattered field obtained on an active array (cf. [3]). Our most significant result concerns the characterization of the signal (and noise) subspace of the array response matrix which allows us to analyze the properties of the images obtained when projections on different parts of this subspace are used.
List with the most significant publications
1) L. Borcea, F. González del Cueto, G. Papanicolaou and C. Tsogka, Filtering Deterministic Layer Effects in Imaging, SIAM Review 2012,54 (4), pp. 757-798., 2012.
2) J. Garnier, G. Papanicolaou, A. Semin and C. Tsogka, Signal-to-Noise Ratio Estimation in Passive Correlation-Based Imaging, SIAM Journal on Imaging Science, Vol. 6, No. 2, pp. 1092--1110, 2013.
3) C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Selective imaging of extended reflectors in two-dimensional waveguides, SIAM Journal on Imaging Science, Vol. 6, No. 4, pp. 2714--2739, 2013.
Athanasios Tzavaras
Structure and properties of polyconvex elasticity.
In previous work we have constructed a variational approximation scheme for the equations of polyconvex elasticity and showed that it produces dissipative measure-valued solutions. In [1] we produce a weak-strong uniqueness theorem for dissipative measure-valued solutions. The variational approximation scheme offers a time-discretization for polyconvex elasticity that exploits its Hamiltonian structure and it provides an interesting framework for the numerical computation of entropic weak solutions for the equations of polyconvex elasticity.
Cavitation for the equations of radial elasticity.
The system of radial isotropic elasticity is probably the simplest example where the effect of polyconvexity can be compared with the well-developed theory of one-dimensional hyperbolic conservation laws. To be interpreted as elastic motions, solutions have to satisfy certain constraints associated to the non-interpenetration of matter. The system of radial isotropic elasticity admits a class of self-similar solutions, identified by Pericak-Spector and Spector, that correspond to opening of a cavity in the material at some critical time. These cavitating solutions decrease the total mechanical energy and are an important example of non-uniqueness for hyperbolic systems due to point singularities. Their presence poses unsettling questions as to the adequacy of the usual entropy solution as a sufficiently general framework for multi-dimensional hyperbolic theory. If opening a cavity reduces the energy, then this induces an autocatalytic mechanism for failure and raises questions for the model. From a perspective of microscopic dynamics, it would appear that the emergence of cavities requires an energetic cost that is not accounted for in continuum theories. In [3] we raise the question whether the notion of weak solutions can be modified to account for the cavity creation costs. This is achieved by the notion of singular induced from continuum solutions (slic-solutions) developed in [3], which accounts for the energetic cost of creating the cavity and resolves the aforementioned paradox.
Relative entropy and diffusive relaxation.
The relative entropy method provides a powerful approach to compare an entropic approximating theory (or a weak entropy solution) with a conservative (smooth) solution of a hyperbolic system. A problem of interest is to compare two entropy weak solutions and this has not been accomplished in any level of sufficient generality. A related (but simpler) issue is to compare an approximating theory of a diffusion equation with the limiting diffusion problem. Indeed, this
situation appears in the context of diffusive limits. In [2] we successfully carry out such a calculation for the relaxation problem from multi-d Euler equations with frictional damping to the porous media equation.
List with the most significant publications
1) S. Demoulini and D.M.A. Stuart, and A.E. Tzavaras. Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Rational Mech. Anal. 205 (2012), 927-961.
2) C. Lattanzio and A.E. Tzavaras, Relative entropy in diffusive relaxation. SIAM J. Math. Anal. 45 (2013), 1563-1584.
3) J. Giesselmann and A.E. Tzavaras, Singular limiting induced from continuum solutions and the problem of dynamic cavitation. Arch. Rational Mech. Anal. 212 (2014) 241-281.
Phoebus Rosakis
Surface energies in discrete crystals:
substantial progress was made in identifying connections between discrete and continuous models for crystals. In [1] the energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two and three dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. Thus important connections are established with interesting number theoretic results. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal for arbitrary choice of the interatomic pair potential.
Discrete models of dynamics of slow dislocations: the result of [2] answers a long standing open question in the discrete dynamical modelling of dislocation motion. We construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel–Kontorova model, representing dislocation motion under stress. So far it was not known whether there exist admissible travelling wave solutions below a theshold velocity.The new solutions fill this gap for the first time. We find solutions for any value of the velocity. These results have important implications on the understanding of the stability of low velocity dislocation motion via discrete models.
Ghost-force free quasicontinuum methods in three dimensions:
in paper [3] we construct energy based numerical methods free of ghost forces in three-dimensional lattices arising in crystalline materials for the first time. The analysis hinges on establishing a connection of the coupled system to conforming finite elements. Key ingredients are: (i) a new rep- resentation of discrete derivatives related to long range interactions of atoms as volume integrals of gradients of piecewise linear functions over bond volumes, and (ii) the construction of an underlying globally continuous function representing the coupled modeling method. Our work solves an important open problem arising in the context of the quasicontinuum method for crystalline materials.
List with the most significant publications
4) P. Rosakis, Continuum surface energy from a lattice model, arXiv preprint arXiv:1201.0712 (2012). To appear in Networks and Inhomogeneous Media.
5) P. Rosakis and A Vainchtein, New Solutions for Slow Moving Kinks in a Forced Frenkel-Kontorova Chain, Journal of Nonlinear Science 23(6), 1089-1110 (2013).
6) C. Makridakis, D. Mitsoudis and P. Rosakis, On atomistic-to-continuum couplings without ghost forces in three dimensions, Applied Mathematics Research Express, abt005 (2013). doi:10.1093/amrx/abt0,0 arXiv preprint arXiv:1211.7158 (2013).
Charalambos Makridakis
In [1] a novel finite element consistency analysis of Cauchy-Born approximations to atomistic models of crystalline materials in two and three space dimensions is introduced. The analytical tools introduced and an intermediate model can be useful in the design and the analysis of consistent coupled methods in more than one space dimensions. Indeed, our approach is the first comprehensible consistency analysis in two and three dimensions regarding the accuracy of the CB model. Further, in [2], using the results of [1] and new ideas, we provide a systematic approach leading to ghost-force-free couplings in two and three dimensions. The results of [2] contribute to long standing algorithmic design issues such as the construction of consistent couplings of arbitrary high-order of accuracy and to the design of the first consistent coupled energies in three dimensions. (iii) In [3] we propose a new class of finite element methods for the Navier-Stokes-Korteweg system which are by design consistent with the energy dissipation structure of the problem. The methods introduced are of arbitrary high order of accuracy and provide physically relevant approximations free of numerical artifacts. It seems that these are the first methods in the literature enjoying these properties.
List with the most significant publications
1) Makridakis, Charalambos ; Süli, Endre . Finite element analysis of Cauchy-Born approximations to atomistic models. Arch. Ration. Mech. Anal. 207 (2013), no. 3, 813--843.
2) Makridakis, Charalambos ; Mitsoudis, Dimitrios ; Rosakis, Phoebus . On atomistic-to-continuum couplings without ghost forces in three dimensions. Appl. Math. Res. Express. AMRX 2014, no. 1, 87--113.
3) J. Giesselmann, C. Makridakis and T. Pryer, Energy consistent discontinuous
Galerkin methods for the Navier-Stokes-Korteweg system, Math. Comp (2014)
DOI: http://dx.doi.org/10.1090/S0025-5718-2014-02792-0
Nikolaos Efremidis
Abruptly autofocusing waves.
We have introduced the concept of abruptly autofocusing waves, which are waves whose peak intensity remains almost constant during propagation, while close to a particular focal point, they suddenly autofocus and, as a result, their peak intensity increases by orders of magnitude. We observed the predicted phenomena in collaboration with two different experimental groups and utilized these beams for particle manipulation and creation of ablation spots. The first proposed family of autofocusing waves consists of radially symmetric Airy beams. We were able to generalize these families of waves and propose methods to efficiently generate them. The generalized families of waves proposed can follow arbitrary convex trajectories such as power low before they collapse at the focusing spot.
Bessel-like beam that follow predefined trajectories.
We have also proposed a method for generating Bessel-like optical beams with arbitrary trajectories in free space. The method involves phase-modulating an optical wavefront so that conical bundles of rays are formed whose apexes write a continuous focal curve with pre-specified shape. These ray cones have circular bases on the input plane; thus their interference results in a Bessel-like transverse field profile that propagates along the specified trajectory with a remarkably invariant main lobe. Such beams can be useful as hybrids between non-accelerating and accelerating optical waves that share diffraction-resisting and self-healing properties. We have extended this method in the case of nonparaxial beams. Still, there is a new interesting feature that sets the nonparaxial case clearly apart: The symmetric Bessel-like profile forms on a plane that is normal to the focal trajectory. This further suggests that (in contrast to the paraxial case) nonparaxial diffraction-free (or diffraction-resisting) accelerating waves should be sought as propagation-invariant waves with a dynamically changing plane of observation that stays normal to the trajectory of the beam. Experimentally, we have observed the predicted phenomena in collaboration with Zhigang Chen’s group in San Francisco State University.
Curved beams in periodic index configurations.
We have studied curved trajectory dynamics and design in discrete array settings. We found that beams with power law phases produce curved caustics associated with the fold and cusp type catastrophes. A parabolic phase produces a focus that suffers from spherical aberrations. More important, we find that by designing the initial phase or wavefront of the beam we can construct trajectories with pure power law caustics as well as aberration-free focusing of discrete waves. Going beyond the discrete, tight-binding model, which we examined recently, we show how the exact band structure and the associated diffraction relations of a periodic waveguide lattice can be exploited to phase-engineer caustics with predetermined convex trajectories or to achieve optimum aberration-free focal spots.
List with the most significant publications
1) Ioannis Chremmos, Nikolaos K. Efremidis, and Demetrios N. Christodoulides, Pre-engineered abruptly autofocusing beams , Optics Letters, vol. 36, p. 1890 (2011).
2) Ioannis D. Chremmos, Zhigang Chen, Demetrios N. Christodoulides, and Nikolaos K. Efremidis, Bessel-like optical beams with arbitrary trajectories, Optics Letters, vol. 37, p. 5003 (2012).
3) Nikolaos K. Efremidis and Ioannis Chremmos, Caustic design in periodic lattices , Optics Letters, vol. 37, p. 1277 (2012).
Dimitrios Tsagkarogiannis
The first concerns the study of effective thermodynamic potentials for systems of interacting
particles in the continuum with Lennard-Jones type interactions. The main question is to
construct effective theories for the study of phenomena such as condensation and phase
transitions. A main technical tool is the cluster expansion method around some appropriately chosen regime. In particular we were interested in density expansions for mixtures of countably many different types of particles. This is known in the literature as multi-species virial expansion. We followed a combinatorial path and in order to prove convergence we used the Lagrange–Good inversion formula, which has other applications such as counting colored trees or studying probability generating functions in multi-type branching processes. We proved that the virial expansion converges absolutely in a domain of small densities. In addition, we established that the virial coefficients can be expressed in terms of two-connected graphs (see ref. 1).
The second is the continuation of previous work on the study of non equilibrium systems driven by current reservoirs. We are interested in microscopic systems of interacting particles in [-N,N] as natural models for current reservoirs and Fick's law. In previous work we have established the hydrodynamic limit and studied the stationary measure. In the recent works we investigated the exponential rate of convergence to the stationary measure, which we prove to be of the order N^2 (see ref. 3). As a technical tool we had to study a random walk with death in [-N,N] moving in a time dependent environment. The environment is a system of particles which describes a current flux from N to -N. Its evolution is influenced by the presence of the random walk and in turns it affects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in N) for the survival probability up to time t (see ref. 2).
List with the most significant publications
1) Sabine Jansen, Stephen J. Tate, Dimitrios Tsagkarogiannis, Daniel Ueltschi, Multispecies Virial Exapnsions, Comm. Math. Phys. 2014,(DOI)10.1007/s00220-014-2026-9
2) Anna DeMasi, Errico Presutti, Dimitrios Tsagkarogiannis, Maria Eulalia Vares, Extinction time for a random walk in a random environment, Bernoulli, 2014 to appear.
3) Anna DeMasi, Errico Presutti, Dimitrios Tsagkarogiannis, Maria Eulalia Vares, Exponential rate of convergence in current reservoirs, Bernoulli, 2014 to appear.
Vagelis Harmandaris
Computational modeling of polymers:
Through a systematic hierarchical simulation methodology we were able to study the properties of complex macromolecular systems at the molecular level. The simulation results were quantitatively compared against theoretical predictions and experimental data. In a further stage the dynamics of polymer blends were also studied and compared to the self-concentration model that requires a composition-dependent length scale.
Modeling of biomolecular systems:
We have performed detailed atomistic molecular dynamics simulations of peptides (Diphenylalanine, FF) in using an explicit solvent model. The self-assembling propensity of FF in water is obvious while in methanol a very weak self-assembling propensity is observed. We studied and compared structural properties of FF in the two different solvents and a comparison with a system of dialanine (AA) in the corresponding solvents was also performed.
Hierarchical multi-scale modeling of nanocomposites:
We have performed multi-scale simulations of graphene based polymer systems. Detailed atomistic simulations allowed us to explore the effect of graphene on the mobility of polymers by studying three well known and widely used polymers, polyethylene (PE), polystyrene (PS) and poly(methyl-methacrylate) (PMMA). Qualitative and quantitative differences in the dynamical properties of the polymer chains in particular at the polymer–graphene interface are detected.
List with the most significant publications
1) V. Harmandaris, G. Floudas, K. Kremer, “Dynamic heterogeneity in fully miscible blends of polystyrene with oligostyrene”, Phys. Rev. Let. 2013, 110, 165701.
2) A. Rissanou, E. Georgilis, M. Kasotaskis, A. Mitraki, V.A. Harmandaris, “Effect of solvent on the self-assembly of dialanine and diphenylalanine peptides”, J. Phys. Chem. B 2013, 117, 3962-3975.
3) A. Rissanou, V. Harmandaris, “Dynamics of various polymer/graphene interfacial systems through atomistic molecular dynamics simulations”, Soft Matter, 2014, 10, 2876-2888.
Spyros Kamvissis
A significant part of my ACMAC funded research has been on the question of integrability of initial/boundary value problems for so-called completely integrable equations like cubic NLS in 1 space dimension.
There is a so-called unified transform theory, constructed by Fokas and collaborators, which generalizes the inverse scattering theory for Cauchy problems for such equations. A main "gap" for this theory to be completely satisfactory is the study of the Dirichlet-to-Neumann map for the boundary (not the initial) data. It is crucial to be able to find nontrivial classes of functions such that for Dirichlet data in such a class, the Neumann data also belongs in the same class.
Together with A. Fokas himself and J. Lennels (at Cambridge), D. Shepelsky (in Ukraine and in Crete), S. Venakides (during his visit to Crete), ACMAC postdoc David Smith and University of Crete postdoc D. Antonopoulou we have spent a lot of time working on this problem.
There is a preprint with D. Shepelsky and a paper in progress with D. Antonopoulou which address the problem, see list below.
Other ACMAC funded work concerns general asymptotic problems on soliton equations. I would like to particularly point out a paper with D.Smith on asymptotics for a discretization (in both time and space) of KdV.
List with the most significant publications
1) D.Antonopoulou S.Kamvissis Schwartz initial data for the NLS and Fokas' transform, in progress.
2) S.Kamvissis D.Shepelsky Initial boundary value problem for the focusing NLS equation with periodic boundary data, preprint.
3) S.Kamvissis D.Smith Long time behavior of the IpKdV equation, in progress.
Georgia Karali
Non-radial solutions for a singular perturbation problem.
Contrary to previous approaches to such singular perturbation problems, that mainly relied on the construction of upper and lower solutions or weak convergence arguments, we use a perturbation argument and find a genuine solution close to a carefully constructed approximation by means of the contraction mapping principle, provided the singular parameter is chosen sufficiently small. We develop a novel approach for rigorously patching together inner and outer approximations, which is flexible enough to deal also with other related problems. Besides improving previously known estimates regarding stable solutions, where the method of upper and lower solutions applies, the perturbation approach allowed us to construct highly unstable solutions (radial and bifurcating non-radial), in the sense that their Morse index (in the general class) diverges as the small singular parameter tends to zero. Our new methodology enabled us to succeed in proving recently the existence of highly unstable solutions in 2 dimensions without assuming any symmetry assumptions. Also, we have answered an open interesting problem posed recently in Aftalion, Jerrard, and Royo-Letelier concerning ground states of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit, where the corresponding limit algebraic equation admits a pitchfork bifurcation at a submanifold of the domain.
One-dimensional stochastic Cahn-Hilliard equation.
We describe the joint motion of multiple kinks or interfaces for the one-dimensional Cahn-Hilliard equation on a bounded interval perturbed by small additive noise. The approach is based on an approximate slow manifold, where the motion of interfaces is described by the projection onto the manifold. This was used deterministically by Bates and Xun (1994/95) to verify metastable behavior of solutions with multiple kinks. We derive a stochastic differential equation for the motion of the interfaces. The approximation is (with high probability) valid until an interface breaks down, or until times that are large compared to any negative power of the small interaction length. The differences with the deterministic case are many and the stochastic case requires by far more tedious analysis and ideas. This is the first work in the literature in this direction.
List with the most significant publications
1) C. Sourdis and G. Karali, The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit. to appear Archive for Rational Mechanics and Analysis.
2) D. Antonopoulou, D. Bloemker and G. Karali, Front motion in the one-dimensional stochastic Cahn-Hilliard equation. SIAM J. Math. Anal. 44, 3242-3280, (2012).
3) C. Sourdis and G. Karali, Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities}, Ann. Inst. H. Poincare Anal. Non Lineaire, 29, 131-170, (2012).
Menelaos I. Karavelas
The exact maximum complexity of the Minkowski sum of convex polytopes of general dimension. Given r d-dimensional polytopes P_1, P_2,... P_r in the Euclidean d-dimensional space, we are interested in evaluating the maximum number of k-faces or their Minkowski sum P_1+P_2+...+P_r. The solution to this problem was previously know only in 2 and 3 dimensions (for 2- and 3-dimensional polytopes, respectively). Results in higher dimensions were only partial and sporadic. During the course of the project we managed to fully resolve the problem when r=2 or r=3 (i.e. for two and three summands), and for general dimension. More precisely, given r convex d-dimensional polytopes in d-dimensions, where r=2 or r=3, we have evaluated closed-form upper bound expressions on the number of k-faces of their Minkowski sum as a function of the number of vertices of the two or three polytopes. Our upper bounds are shown to be tight.
At a more technical level, our approach resembles, or rather generalizes the approach of McMullen for proving the upper bound theorem for polytopes (this is a famous result in Combinatorial Geometry, stating that any d-polytope with n vertices, cannot have more k-faces than the cyclic d-dimensional polytope with the same number of vertices). To prove our upper bound we have considered the Minkowski sum of the polytopes as a section of their Cayley polytope in dimension d+r-1 (recall that r=2 or 3). In the Cayley polytope we have identified a set of faces F that is in 1-1 correspondence with the faces of the Minkowski sum, and for this set F, we have considered its f-vector f(F) and h-vector h(F). We analyzed its h-vector, showed that it satisfies equations that generalized the well-known Dehn-Sommerville equations of simplicial polytopes, and have managed to prove a recurrence relation for the elements of h(F). From this recurrence relation we proved upper bounds on the individual elements of the h-vector, which in conjunction with the Dehn-Sommerville-like equations for h(F) have enabled us to compute upper bounds on the elements of f(F). Given the correspondence between the faces of Minkowski sum and the faces of F, the upper bounds for f(F) immediately give upper bounds for the faces of the Minkowski sum of the polytopes. These bounds have been shown to be tight by explicit constructions.
Besides two and three summands, we have been extending our methodology to an arbitrary number of summands. In particular, we have generalized the Dehn-Sommerville-like equations for r=2 or 3 to any generic r, and we have proved the recurrence relation on the elements of h(F) for any number of summands. We are currently working on finalizing our results, and proving tight asymptotic expressions for the number of k-faces for the Minkowski sum of r d-dimensional polytopes, for any r greater or equal to 4.
The robust and efficient computation of the 2D Apollonius diagram. The planar Apollonius diagram (also known as the additively weighted Voronoi diagram) is a generalization of the standard Voronoi diagram for points when the diagram generators are disks. A point in the space is assigned to the generator that is closest to it, where distance is measured from the boundary of the disk, ans is positive (resp., negative) if the point lies outside (resp., inside) the disk. As in the case of usual point Voronoi diagrams, degenerate configurations can occur. One such degenerate configuration occurs when four mutually disjoint disks are externally tangent to the same circle. Degenerate configurations pose various problems in geometric algorithms, and it is often the case that we perturb symbolically the input data so as to remove these degeneracies. Such symbolic perturbation schemes were lacking for the case of Apollonius diagrams; the reason for that is that the problem itself is quite complicated, while the predicates necessary for the evaluation of the diagram are quite complicated (for example, with respect to the case of points). During the course of the project, we devised a perturbation framework for dealing with these degeneracies. In short, the framework amounts to changing symbolically the radii/weights of the input generators, where disks with higher radius/weight are considered as perturbed more. Under these considerations we have been able to resolve all degeneracies of the 2D Apollonius diagram, as well as the degeneracies of all predicates appearing in the standard randomized incremental construction of the diagram. Out approach is close to being fully implemented within the framework of the Computational Geometry Algorithms Library (CGAL: www.cgal.org) for the 2D case, while we know how to handle the most basic predicate for the 3D case.
Besides handling degeneracies, we have been considering approaches for constructing the diagram in a non-incremental manner. More precisely, we are considering an offline algorithm that computes the Voronoi diagram of the centers first, and then increase the weights of the sites to compute the Apollonius diagram. We have already shown that this approach yield a randomized algorithm with O(n log(n)) expected running time, as opposed to O(n log^2(n)) expected running time required by the standard randomized incremental approach. Preliminary benchmarks have shown that, at least in the case of disjoint generators, the new algorithm can be much faster in practice than the randomized incremental one.
List with the most significant publications
1) Menelaos I. Karavelas and Eleni Tzanaki. The maximum number of faces of the Minkowski sum of two convex polytopes. In Proceedings of 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'12), pages 11-28, Kyoto, Japan, January 17-19, 2012. http://doi.acm.org/10.1145/2095116.2095118
2) Menelaos I. Karavelas, Christos Konaxis, and Eleni Tzanaki. The maximum number of faces of the Minkowski sum of three convex polytopes. In Proceedings of the 29th Annual ACM Symposium on Computational Geometry (SCG'13), pages 187-196, Rio de Janeiro, Brazil, June 17-20, 2013. http://dx.doi.org/10.1145/2462356.2462368
3) Olivier Devillers, Menelaos I. Karavelas, and Monique Teillaud. Qualitative Symbolic Perturbation: a new geometry-based perturbation framework. Research Report RR-8153, INRIA, 2012. http://hal.inria.fr/hal-00758631
Theodoros Katsaounis
Linear Schrödinger equation.
Development, analysis and implementation of a space-time adaptive algorithm for the solution of the linear Schrödinger equation. The adaptive algorithm is based on rigorous a posteriori error estimation. We use finite element spaces for the spatial discretisation and a modified Crank-Nicolson scheme as a time-stepping mechanism. The a posteriori error estimators/indicators are of optimal order in space and time. The adaptive algorithm is tested thoroughly and proven to be an efficient and very robust solver for the linear Schrödinger eq. Adaptivity proves to be beneficial, reduces the computational cost substantially while keeping the total error under a desired tolerance.
Shear Band formation.
We study numerically an instability mechanism for the formation of shear bands at high strain-rate deformations of metals. We use adaptive finite element methods of any order in the spatial discretisation, and implicit Runge-Kutta methods with variable step in time. The numerical schemes are of implicit- explicit type and provide adequate resolution of shear bands up to full development. We find that already from the initial stages, shear band formation is associated with collapse of stress diffusion across the band and that process intensifies as the band fully forms. For fully developed bands, heat conduction plays an important role in the subsequent evolution by causing a delay or even stopping the development of the band. Further we provide an explanation of the onset of localization and the formation of shear bands in high strain-rate plasticity of metals. We employ the Arrhenius constitutive model and show Hadamard instability for the linearized problem. For the nonlinear model, using an asymptotic procedure motivated by the theory of relaxation and the Chapman-Enskog expansion, we derive an effective equation for the evolution of the strain rate, which is backward parabolic with a small stabilizing fourth order correction. We construct self-similar solutions that describe the self-organization into a localized solution starting from well prepared data.
Dispersive water wave propagation.
We introduce a general finite volume / finite difference computational framework for simulating water wave propagation. The main application in question is the accurate prediction of tsunami propagation and landing. The main idea is to rewrite the system, describing two way propagation, in a conservation law form. The application of the finite volume method is adapted to discretise the advective and dispersive fluxes. Various numerical fluxes and reconstruction techniques are tested. The time stepping is done using Runge-Kutta methods which preserve the TVD property of finite volume scheme. The method is validated through a series of numerical experiments and compared against experimental data when possible. Extensive numerical studies were performed in the case of tsumani wave landing and computation of the runup. The dispersive models were compared with the classical Shallow Water eqs.
Image processing.
A model problem is used to represent a typical image processing problem of reconstructing an unknown in the face of incomplete data. A consistent discretization for a vanishing regularization solution is defined so that, in the absence of noise, limits first with respect to regularization and then with respect to grid refinement agree with a continuum solution defined in terms of a saddle point formulation. It is proved and demonstrated computationally for artificial examples and for magnetic resonance images that a mixed finite element discretization is consistent in the sense defined here. On the other hand, it is demonstrated computationally that a standard finite element discretization is not consistent, and the reason for the inconsistency is suggested on the basis of the theory presented for the mixed method.
List with the most significant publications
1) Th. Katsaounis & I. Kyza, A posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrödinger equation, to appear in Numerische Mathematik, 2014.
2) Th. Baxevanis, Th. Katsaounis, & A. Tzavaras, Adaptive finite element computations of shear band formation, Mathematical Models in Applied Sciences, vol. 20, no.3 pp.423-448 2010.
3) D. Dutykh, Th. Katsaounis & D. Mitsotakis, Finite volume schemes for dispersive wave propagation and runup, Journal of Computational Physics, vol. 230, no. 8, pp. 3035-3061, 2011.
Stavros Komineas
In the field of ultracold atoms the coupling of atoms to effective magnetic fields has been achieved in the last years. Effective fields are created by using coherent optical dressing of atomic levels which result in non-dynamical gauge fields. These techniques have the potential to generate non-Abelian gauge fields. We have considered a gas of bosonic atoms coupled to a gauge field with U(2) symmetry, and with constant effective magnetic field. We studied the effects of weak interactions by Gross-Pitaevskii mean-field ntheory on the vortex lattice phase induced by a uniform effective magnetic field. We showed that, with increasing non-Abelian gauge field, the nature of the ground state changes dramatically, with structural changes of the vortex lattice. These changes can be understood as produced by effective interactions with non-zero range.
The injection of a dc spin-polarised current through a ferromagnetic element can induce magnetisation oscillations and thus turn the nanoelement into a spin-transfer nano-oscillator. We analysed magnetization dynamics using the Landau-Lifshitz equation including a Slonczewski spin-torque term, in two types of magnets.
(a) In isotropic magnets, we established that a vortex-antivortex dipole is set in steady-state rotational motion due to the interaction between the vortices, while an external in-plane magnetic field can tune the frequency of rotation. The rotational motion was linked to the nonzero skyrmion number of the dipole. Three types of vortex-antivortex pairs were obtained as we vary the external field and spin-torque strength. Their character and stability properties were elucidated by an asymptotic analysis.
(b) In magnetic elements with perpendicular anisotropy the relevant excitations are magnetic bubbles or non-topological droplet solitons. Their dynamics under spin-polarised current produces magnetization oscillations.
We analysed the dynamics of dipolar and quadrapolar excitations. Exciton-polaritons (quasiparticles of exciton and photons) which are produced in semiconductors have been condensed in the laboratory in the sense of Bose-Einstein. These are non-equilibrium states in condensed matter and they are interesting because of rich phenomena taking place while the condensate is achieved at relatively high temperature. The mean-field description of the polariton system is given by a system of two Gross-Pitaevskii type equations which include a dispersive term for photons and a nonlinear term modelling exciton interactions. We study soliton solutions of the system of Gross-Pitaevskii equations and have developed a rigorous method to find a series of solitons. We have found solitons in the sense of the usual nonlinear Schroedinger equation but also solitons with surprising discontinuous profiles. Discontinuities are explained within the specific system of Gross-Pitaevskii equations.
List with the most significant publications
1) S. Komineas, “Frequency generation by a magnetic vortex-antivortex dipole in spin-polarized current”, Europhys. Lett. 98 (2012) 57002
2) Stavros Komineas, Nigel R. Cooper, “Vortex lattices for ultracold bosonic atoms in a non-Abelian gauge potential”, Phys. Rev. A 85, 053623 (2012)
3) G. Finocchio, V. Puliafito, S. Komineas, L. Torres, O. Ozatay, T. Hauet, B. Azzerboni, “Nanoscale spintronic oscillators based on the excitation of confined soliton modes”, J. Appl. Phys. 114, 163908 (2013)
Georgios Kossioris
In collaboration with G.E. Zouraris (Univ. of Crete) we considered an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate stochastic parabolic problem discretizing the noise using linear splines. Then we construct fully-discrete approximations to the solution of the approximate problem using, for the discretization in space, a Galerkin finite element method based on H 2 −piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates: for the error between the solution to the problem and the solution to the approximate problem, and for the numerical approximation error of the solution to the approximate problem.
In collaboration with O. Lakkis (Univ. of Sussex) and M. Romito (Univ. of Florence) we are in a final stage of completing the manuscript entitled “Strong convergence for numerical approximation of the stochastic Allen-Cahn in one dimension”. The initial equation, referred in the sequel as stochastic A-C equation, is approximated by replacing the space-time white noise by the proper piecewise space-time random function. A finite element approximation scheme of the approximating stochastic A-C equation is then analyzed. We first obtained space-time L p and L∞ stability estimates for the approximating linear stochastic heat equation leading, by a careful estimate analysis, to L p and L∞ error estimates. In order to study the nonlinear problem we study the corresponding nonlinear randomly shifted approximating stochastic A-C equation where we have a slightly better regularity. Using the corresponding estimates for the linear problem we obtained L p and L∞ error estimates for the randomly shifted approximating equation. In order to perform the finite element analysis we obtained L p and L∞ regularity error estimates for the approximating linear heat equation and the approximating nonlinear randomly shifted equation. Based on the above regularity estimates we obtained the L∞ error estimates for the finite element approximation scheme of the approximating heat equation. The derivation of the corresponding L∞ error estimate for the finite element scheme for the randomly shifted approximating stochastic A-C equation is under derivation.
In collaboration with M. Plexousakis (Univ. of Crete) and A. Yannacopoulos (Athens Univ. of Economics and Business) we study the stochastic Allen-Cahn equation in one space dimension via a Wiener chaos expansion. It is a well-known fact that as the scaling parameter ε tends to zero, the solution converges to a travelling wave solution centered at a point that follows a Brownian motion. Given the difficulty to handle the Wiener chaos of the full nonlinear equation and in order to get an insight of the dominant terms of the Wiener chaos expansion of the full solution for small ε, we first analyze the Wiener chaos expansion of the travelling wave solution which is the zero-order approximation term for sufficiently small ε.
In collaboration with M. Loulakis (National technical Univ. of Athens) and M. Plexousakis (Univ. of Crete). We study random dynamical systems with an emphasis on stochastic bifurcations for problems related to environmental economic modeling. Both analytical as well as numerical techniques were considered.More specifically, we studied the Stochastic Shallow Lake Problem. We proved uniqueness and regularity of the value function in the framework of viscosity solutions and we applied stochastic Pontryagin Maximum Principle to study the optimal dynamics of the problem. We considered appropriate numerical methods to reconstruct the value function and the corresponding optimal dynamics.
We analyze the surface wind variability over selected areas of the Greek territory by comparing a 3-km-spatial resolution simulation performed with the Weather Research and Forecasting (WRF) model with actual surface measurements. Daily 36hrs runs at 12 UTC were driven by FNL (1º× 1º) data for three time periods during winter, spring and summer, respectively, are conducted. Various verification statistics, such as bias, RMSE and DACC for wind speed and direction were used to gauge the mesoscale model performance. The following paper “Evaluation of WRF performance for the analysis of surface wind speeds over various Greek regions”
Numerical solution of fourth-order SPDES: In the following works it has been achieved the convergence analysis of fully discrete finite element approximations of the solution to some linearized versions of the stochastic Cahn-Hillird equation driven by an additive space-time white noise or the space derivative of an additive space-time white noise. Optimal error estimates have been obtained.
List with the most significant publications
1) Kossioris, Georgios T.and Zouraris, Georgios E. (2010) ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2). pp. 289-322.
2) Kossioris, Georgios T.and Zouraris, Georgios E. (2013) Finite element approximations for a linear fourth-order parabolic SPDE in two and three dimensions with additive space-time white noise. Applied Numerical Mathematics, 67 (Special Issue: NUMAN 2010). pp. 243-261.1
3) Kossioris, Georgios and Zouraris, Georgios E. (2013) Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise. Discrete and Continuous Dynamical Systems - Series B 18 (7). pp. 1845-1872.
Michail Loulakis
Zero-range processes with jump rates that decrease with the number of particles per site can exhibit a condensation transition, where a positive fraction of all particles condenses on a single site when the total density exceeds a critical value. In [1] we considered rates which decay as a power law or a stretched exponential to a non-zero limiting value, and studied the onset of condensation at the critical density. We described exactly how a condensate emerges as we go from subcritical to supercritical densities, we established a law of large numbers for the excess mass fraction in the maximum, as well as distributional limits for the fluctuations of the maximum and the fluctuations in the bulk.
Many of the results rely on a general theorem for Large Deviations of Sums of Subexponential Random Variables-the analogue of Gibbs's conditioning principle-that was
proved in [2]. Precisely, we proved that conditionally on a large deviation of the sum of N subexponential random variables, the a posteriori distribution of the variables asymptotically looks like a product of N-1 copies of the prior distribution and a randomly located lump that realizes the large deviation.
List with the most significant publications
1) Inés Armendáriz, Stefan Grosskinsky, Michail Loulakis: Zero Range Condensation at Criticality, Stoch. Proc. Appl.,123 (2013) no. 9, pp 3466-3496. DOI: 10.1016/j.spa.2013.04.021
2) Inés Armendáriz, Michail Loulakis: Conditional Distribution of Heavy Tailed Random Variables on Large Deviations of their Sum, Stoch. Proc. Appl. 121 (2011) no. 5, pp 1138-1147. DOI:10.1016/j.spa.2011.01.011
Georgios Makrakis
Wave-packet evolution in phase space:
We studied the singular semiclassical initial value problem for the phase space Schrodinger equation which is used in physical chemistry for computing certain mean values in realistic applications. In particular, we approximated the semiclassical quantum evolution in phase space by analyzing initial states as superpositions of Gaussian wave packets and applying individually semiclassical anisotropic Gaussian wave packet dynamics, which is based on the the nearby orbit approximation. We accordingly constructed a semiclassical approximation of the phase space propagator, the so-called semiclassical wave packet propagator. We analyzed the relation of this propagator ith the initial value repsresentations, which have been introduced long time ago as a computational tool in physical chemistry, and which are currently the subject of inensive study in semiclassical microlocal analysis.. By this semiclassical propagator we constructed asymptotic solutions of the phase space Schrodinger equation, and we tested the validity of the approximation using certain analytical examples.
Asymptotic theory of wave beams:
We constructed a new representation of Maslov’s canonical operator in a neighborhood of caustics using a special class of coordinate systems (eikonal coordinates) on Lagrangian manifolds and we explained why this is a much more convenient implementation, useful in specific physical problems, for example, those related to the asymptotic behavior of solutions of the scattering problem and of the Green’s function, and to linear hyperbolic systems with variable coefficients (e.g. the wave equation) with localized initial data, etc. We also clarified that our formulas are special cases of the general formulas in the theory of Fourier integral operators. Our main result is a constructive formula and an algorithm for constructing these formulas, which can in particular be used in combination with software like Mathematica or MatLab in numerical computations.
We also studied how to localize certain exact solutions of the Schrodinger equation (with non-constant coefficients), which can represented as the product of the Airy function (Berry-Balazs solutions) by the Bessel function, known as the Airy-Bessel beams in the paraxial approximation in optics. To this end, we representeded the localized solutions in the form of Maslov's canonical operator acting on compactly supported functions on special Lagrangian manifolds. Then, by exploiting some results by Hormander, we use the formula for the commutation of a pseudodierential operator with Maslov's canonical operator to “move" the compactly supported amplitudes outside the canonical operator, and thus obtain formulae preserving the structure based on the Airy and Bessel functions. This made possible to discuss the dispersive structure of the beams.
List with the most significant publications
1) 1. P. D. Karageorge and G.N. Makrakis, Asymptotic solutions of the phase space Schrodinger equation: Anisotropic Gaussian approximation (submitted) (http://arxiv.org/abs/1402.6854)
2) 2. S. Yu. Dobrokhotov, G. N. Makrakis, V. E. Nazaikinskii and T. Ya. Tudorovskii, New formulas of Maslov’s canonical operator in a neighborhood of focal points and caustics in 2D semiclassical asymptotics, Theoretical and Mathematical Physics, Vol. 177, No. 3, pp. 1579-1605, 2013.
3) 3. S. Yu. Dobrokhotov, G. N. Makrakis and V. E. Nazaikinski, Fourier integrals and a new representation of Maslov’s canonical operator near caustics, Proceedings of the International Conference "Spectral Theory and Differential Equations (STDE-2012)" in honor of Vladimir A. Marchenko’s 90th birthday, AMS, 2014 (to appear).
4) 4. S.Yu Dobrokhotov, G.N. Makrakis and V.E.Nazaiksinskii Maslov’s Canonical Operator, Hörmander’s Formula, and Localization of Berry–Balazs’ Solution in the Theory of Wave Beams, Conference "Mathematical Physics. Vladimirov-90" dedicated to the 90th anniversary of academician V. S. Vladimirov, November 13, 2013 (http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=7963)
Michael Plexousakis
The paper entitled Crank-Nicolson finite element discretizations for a two dimensional linear Schrodinger equation posed in a noncylindrical domain, with D. C. Antonopoulou, G. D. Karali, and G. E. Zouraris, studies space-time finite element methods for the linear Schrodinger equation, a main topic in my research on numerical methods for partial differential equations. Motivated by the paraxial narrowangle approximation of the Helmholtz equation in domains of variable topography, we considered an initial- and boundary-value problem for a general Schrodinger type equation posed on a two-dimensional noncylindrical domain with mixed boundary conditions. The problem was transformed into an equivalent one posed on a rectangular domain and we approximated its solution by a CrankNicolson nite element method. For the proposed numerical method, we derived an optimal order error estimate in the L2 norm and proved a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments were presented which verified the optimal order of convergence of the method.
The paper entitled Helmholtz equation with artificial boundary conditions in a two-dimensional waveguide, with D. A. Mitsoudis and Ch. Makridakis is the second in a series of papers devoted on numerical methods for the Helmholtz equation. In this work we developed and analysed a model for wave propagation based on the Helmholtz equation in the context of a realistic environment widely used in applications, especially underwater acoustics. We considered a two-dimensional waveguide in Cartesian coordinates consisting of a homogeneous water column confined between a horizontal pressure-release sea surface and an acoustically soft sea floor. The original infinite domain was truncated with two artificial boundaries and we formulated a model in the resulting bounded domain by introducing suitable nonlocal conditions on these artificial boundaries. The proposed model simulated efficiently the effect of the source and the backscattered field from the rest of the waveguide and proved appropriate for finite element computations.This work dealt with the well-posedness of the model. We showed stability estimates in appropriate Sobolev norms which, in turn, implied existence and uniqueness of the solution.
Our investigation on mesoscopic models of pattern{forming systems, with N. M. Abukhdeir, D. G. Vlachos, and M. Katsoulakis resulted in a publication in the Journal of Computational Physics entitled Long-time integration methods for mesoscopic models of pattern{forming systems. In this work spectral methods for simulation of a mesoscopic diffusion model of surface pattern formation were evaluated with the utter goal of achieving long simulation times. Backwards-differencing time-integration, coupled with an underlying Newton Krylov nonlinear solver (SUNDIALSCVODE), was found to substantially accelerate simulations, without the typical requirement of preconditioning. Quasi-equilibrium simulations of patterned phases predicted by the model were shown to agree well with linear stability analysis. Simulation results of the effect of repulsive particle-particle interactions on pattern relaxation time and short/long-range order were discussed and analyzed extensively.
List with the most significant publications
1) Crank-Nicolson finite element discretisations for a two-dimensional linear Schrodinger equation posed in a noncylindrical domain , with D. C. Antonopoulou, G. D. Karali, and G. E. Zouraris. To appear in Math. Comp.
2) Helmholtz equation with artificial boundary conditions in a two-dimensional waveguide, with D. A. Mitsoudis and Ch. Makridakis. SIAM J. Math. Anal. 44 (2012), no. 6, 4320- 4344.
3) Long-time integration methods for mesoscopic models of pattern{forming systems, with N. M. Abukhdeir, D. G. Vlachos, and M. Katsoulakis. J. Comp. Phys. 230 (2011), no. 14, 5704-5715.
Emmanouil Skarsoulis
Sensitivity of underwater acoustic observables.
The work on sensitivity kernels over the last 4 years focused on the study of finite-frequency observables such as peak arrival times, amplitudes and phase arrival times and their sensitivity behavior using wave-theoretic modeling approaches. A significant result deals with the long-range behavior of vertical sensitivity kernels of peak arrival times. While at short propagation ranges the wave-theoretic kernels at low frequencies exhibit significant deviations from the corresponding ray-theoretic ones (high-frequency asymptotic approximations), they converge to the latter with increasing range even at low frequencies. This result is particularly important since ray theory has been commonly used for the analysis of long-range ocean acoustic tomography experiments. An asymptotic approach for finite frequencies and long ranges has been developed, which confirms the above-mentioned behavior and provides a method for estimation of the ranges beyond which the ray-theoretic sensitivity kernels are sufficient, depending on the ocean environment (sound-speed distribution). Further results refer to the study of phase arrivals. Phase arrivals offer increased time resolution compared to peak arrivals, controlled by the central frequency, rather than the source bandwidth. It has been shown that phase arrival times exhibit a stable and predictable perturbation behavior, based on sensitivity kernels, and can be used as efficient observables in short- and medium-range propagation experiments.
List with the most significant publications
1) E. Skarsoulis, B. Cornuelle, M. Dzieciuch, Perturbation relations and sensitivity kernels for arrival amplitude and phase, Proc. 11th European Conf. on Underwater Acoustics, Edinburgh, 2012.
2) E. Skarsoulis, B. Cornuelle, M. Dzieciuch, Sensitivity behavior of phase and peak arrival times, Proc. 1st International Conference on Underwater Acoustics, pp. 1221-1228, Corfu, 2013.
Georgios Zouraris
Numerical solution of fourth-order SPDES: we manage to prove the convergence analysis of fully discrete finite element approximations of the solution to some linearized versions of the stochastic Cahn-Hillird equation drived by an additive space-time white noise or the space derivative of an additive space-time white noise.
Numerical approximation of Dirac type equations: we presented an existence and uniqueness theory for the solution to a nonlinear Dirac system along with the error analysis of an new implicit/explicit finite difference method for its approximation. We show the performance of the proposed method by applying it to standard model problems.
Numerical approximation of underwater acoustics problems: we propose numerical methods for the approximation of the solution of pde models known in the underwater acoustic area as ‘parabolic approximations’ when the sea environment has variable bottom. For the proposed methods we prove optimal order error estimates and show their performance by exposing results from numerical experiments.
Numerical solution of financial models: we approximate the solution of an HJM term structure financial model by Monte-Carlo Euler methods providing a strong approximation error eanalysis along with a-posteriori estimates for the week approximation error. Results from numerical experiments also included.
List with the most significant publications
1) Kossioris, Georgios T.and Zouraris, Georgios E. (2010) Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise. ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2). pp. 289-322.
2) Bournaveas, Nikolaos and Zouraris, Georgios E. (2012)Theory and numerical approximations for a nonlinear 1+1 Dirac system. ESAIM: Mathematical Modelling and Numerical Analysis , 46 (4). pp. 841-874.
3) Antonopoulou, D. C. and Karali, Georgia D and Plexousakis, M. and Zouraris, Georgios E. (2011) Crank-Nicolson finite element discretizations for a 2D linear Schrödinger-type equation posed in a noncylindrical domain. Mathematics of Computations (accepted on Oct 10, 2013).
Potential Impact:
Integration of the DApplM-UoC in the European Research Area.
A main benefit of the ACMAC project has been the significant increase in visibility for the group comprising the Department of Applied Mathematics of the University of Crete (DApplM-UoC), and the integration in the
European Research Area.
This was significantly enhanced by the partnership and interaction between the Center ACMAC and leading Centers of Applied Mathematics in Europe. A key impact of this partnership is the acquisition of the necessary know-how in building, promoting and maintaining a Center of Excellence. Even though the current financial crisis in Greece makes the financing of such a project through Greek sources very difficult (if not impossible), it is expected that the benefits of this partnership and collaboration will reach deep into the future in several ways. We expect that the academic bond of DApplM-UoC and the Center with these Centers of Excellence will lead to joint activities, creation of research networks, and higher mobility especially for young researchers, through future common proposals and actions.
Another important impact of the project has been the overall activity around the thematic programs, conferences and workshops. The resulting interaction between group members and incoming experienced researchers, senior experts and distinguished scientists brought valuable know-how to the group members and initiated interactions with leading researchers and scientists of Europe. This gave the opportunity to young members of our group to get acquainted with top scientists and be exposed to cutting edge research and innovation, and to gain valuable professional experiences in a diversified scientific environment.
Enhanced visibility is an absolute necessity for the better integration of DApplM-UoC in the European Research Area and in the international research community, and all actions contributed significantly towards this goal. Notably, the Conference activities in Crete and the Thematic Programs had far more impact on the visibility of our department than the usual participation of our faculty in workshops and big Conferences outside Greece. Also, it enhanced significantly the carreer prospects of the postdoctoral fellows hired in ACMAC and most of them continue successfully in their academic carreers having benefited significantly from the exposure to international contacts that they acquired during their tenure at ACMAC.
Upgrading the Research Capacity and Capability of DApplM-UoC.
The ACMAC project also had an essential impact to the improvement of personnel capability through the interaction between group members and young and senior scientists visiting ACMAC and through the secondments with the partner Centers of Excellence.
The project also gave the opportunity to involve to activities in Greece young and senior Greek scientists with careers abroad, striving towards a reversal of a brain-drain process that characterized Greek science in the past decades. Many of them were involved in multiple Centre activities and initiated or further developed contacts with Greek based scientists. The original goal of attracting them back to Greece was unfortunately not fulfilled due to the extraneous circumstances of the finances of the Greek state. Nevertheless, the contacts will remain and might present an opportunity for future hires when the circumstances change.
Third, this project permitted us to upgrade and reinforce our scientific equipment. A goof part of our research needs require heavy computing and therefore the acquired “cluster facility” is essential to our group in order to continue the development of state-of-the-art numerical methods for realistic application problems.
Finally, the development of the program and many of our activities acted as a bridge between team members and applied scientists from other disciplines like computer science and engineering thus initiating fruitful interactions and collaborations. Their participation in Thematic Programs as co-organizers or visitors catalyzed their interactions with ACMAC members. Also, some activities served as seed to initiate collaborations of scientific sub-communities and researchers are planning their continuation through other venues.
Improved Research Capacity and Regional Economic and Social Development.
The project impacted Regional Research Institutions in several ways.
It improved the research capabilities of the University of Crete. Members of the Applied Mathematics Department initiated a Masters program in Computational and Applied Mathematics. Also the Ph.D. program in Applied Mathematics became much more attractive and now has a number of Ph.D. students doing thesis work with members of the group.
Second, the existence of a major research facility in the Region, the Foundation of Research and Technology (FORTH) with its strong applied sciences/engineering and Research & Development teams, provided unique opportunities for mutual growth and collaboration with the members of the ACMAC center. Indeed, existing ties got stronger and there exists a closer integration of the two groups. Areas of collaboration emerged from the activity at ACMAC that includes polymers, biomechanics, nanotechnologies, seismology, and medical imaging.
Improved potential to participate in FP7 projects.
In the course of the ACMAC project, our faculty applied and was successful in many projects financed through the structural funds, and currently support their research through these projects. ACMAC has been instrumental in helping to develop the contacts and expertise to be successful in these projects. This blend of basic and applied research is considered to be of crucial importance for the ability of our group to attract funding in the future and continue research and development in this area.
Broader Impact.
The proposed activities and training gave young researchers at ACMAC new opportunities to pursue careers both in academia or industry, integrating in the EU knowledge-based economy.
By its very nature research in applied mathematics has indirect impact to technological innovation, i.e. techniques are introduced which then are taken up by developers who tailor them to the needs of the society. However, understanding the needs of applications is important in directing our research in the right directions and this is something we pay a lot of attention. The ACMAC center focused its activity around applied research problems ranging from polymers and nanomaterials, to seismology and medical imaging. These are areas of great interest as they constitute European as well as global priorities due to their potential economic and societal impact.
The ACMAC project was also of benefit to the international scientific community. The organization of Thematic Programs at ACMAC was designed in order to take into account related activity in other Institutes in EU and USA. Our faculty has traditionally been involved in many of these external activities as participants or organizers, and hence they were familiar with other upcoming yearly programs and organized our own Thematic Programs so as to avoid overlaps and conflicts, and enhance synergies where possible, emphasizing the ACMAC trademark in the synergy of Mathematical Modeling, Analysis and Computation.
Overall, the Archimedes Center for Modeling, Analysis, and Computation had a transformative impact on the evolution of Applied Mathematics in Crete and more generally in Greece, with ACMAC becoming an internationally recognized center in modern computational and applied mathematics. We hope this will have future implications in enhancing the regional development and attractiveness of European Science in the whole region of South-Eastern Europe.
List of Websites:
Web page of ACMAC. The web page of the project is:
www.acmac.uoc.gr
An executive summary
ACMAC
The Project and its objectives
The objective of the ACMAC project was to strengthen the research profile of the Applied Mathematics Department and to enable its members to enhance their contacts with major research centers in Europe. To this end certain measures were taken that reinforced the Department’s research capability in areas where it had recognized expertise. These measures were organized around a Center, named Archimedes Center for Modeling, Analysis, and Computation (ACMAC), aiming to promote academic excellence in Applied Mathematics and fostering the interface of modeling, analysis and computation.
The main objectives of the ACMAC project were:
• Exchange of know-how and expertise between ACMAC and established Centers of Excellence and other Collaborating Institutions with which ACMAC members had ongoing projects. The Centers of Excellence were:
• University of Bonn, Institute for Applied Mathematics, University of Bonn,
• Cambridge University, DAMTP, Cambridge University,
• Politecnico di Milano, MOX, Milan,
• University of Paris VI, Laboratoire J.L. Lions, Paris,
• University of Oxford, Computing Laboratory, Oxford.
• Recruitment of incoming experienced researchers
• Organization of thematic programs taking place in the “Archimedes Center”, Crete, which included research visits and the organization of workshops in targeted themes. This was a principal activity envisaged to bring in Crete and involved in our project several internationally leading Applied Mathematicians.
• Dissemination of research results through participation of ACMAC members in conferences, workshops and research visits.
• Upgrading the existing computational infrastructure and enhancing the computational capacity of the group.
• Evaluation of the project at the end of the reporting periods.
ACMAC had a sizable impact to the evolution of Applied Mathematics in Crete as an internationally recognized centre in modern computational and applied mathematics.
Project Context and Objectives:
The main objectives of the ACMAC project, listed in Annex I of the contract, is to strengthen the research profile of the Applied Mathematics Department and to enable its members to enhance their contacts with major research centers in Europe.
This is achieved by a series of actions, detailed below, and organized in a research center called “Archimedes Center for Modeling, Analysis and Computation:
EXCHANGE OF KNOW-HOW AND EXPERIENCE between members of the Archimedes Center for modeling, Analysis and Computation (ACMAC) and members of associated established Centers of Excellence of Applied Mathematics in Europe. The associated Established Centers are
• University of Bonn, Institute for Applied Mathematics, University of Bonn,
• Cambridge University, DAMTP, Cambridge University,
• Politecnico di Milano, MOX, Milan,
• University of Paris VI, Laboratoire J.L. Lions, Paris,
• University of Oxford, Computing Laboratory, Oxford.
A number of outgoing and incoming visits to and from Centres of Excellence and Collaborating Institutions were conducted, which were either supported collaborations or participation to programs carried out by the Centres of Excellence, or alternatively supported visits of members from the Centres of Excellence or Collaborating Institutions for research collaborations with ACMAC members or in order to take part in activities taking place at ACMAC.
Emphasis is also given to Collaborations with other important Centers in Europe, with which several of our faculty have ongoing collaborations. The Collaborating Institutions listed in the contract are: University of Erlangen; KTH (Royal Institute of Technology), Stockholm; University of Münster; University of Nice; University of Rome Tor Vergata; SISSA, Trieste; INRIA, Rocquencourt; CMAP, Ecole Polytechnique; University Carlos III, Madrid.
This is a continuing objective covering all reporting periods.
RECRUITMENT OF INCOMING EXPERIENCED RESEARCHERS. The Center is expected to hire Long-Term Research Associates. In addition targeted short-term appointments will be reserved for Senior Experts. These will include experts in specific tasks that we would like to perform as well as short-term appointments of Distinguished Scientists which are international leaders in fields covered by this proposal.
The following types of appointments were made: (i) Appointments of postdoctoral researchers; (ii) Visiting appointments of senior researchers; (iii) Visiting appointments of distinguished researchers. The senior researchers stay on site for a period of a month or longer, collaborate with ACMAC members, deliver a series of lectures, or alternatively help ACMAC to implement aspects of the program.
A total of nineteen postdoctoral positions appointment were made since the beginning of the program. The posts were advertised in job websites worldwide. The selected researchers have PhD from prominent Institutions from Europe but also other parts of the world. Twenty one researchers were hired and participated in the activities of ACMAC as senior visitors.
The following researchers were hired as postdocs researchers: P. D. Karagiorge (PhD from University of Bristol), N. Gupta (PhD from Indian Institute of Technology), C. Donadello (PhD from the International School of Advanced Study (SISSA), Y. Nagase (PhD from Hokkaido University), L. R. Bellet (PhD from University of Geneva), A. Frouvelle (PhD from Université Paul Sabatier), J. Olivier (PhD from Université de Savoie), I. Chremmos (PhD from National Technical University of Athens (NTUA)), D. Mitsoudis (PhD from University of Athens), Ch. Konaxis (PhD from University of Athens), Ch. Sourdis (PhD from University of Athens), J. Giesselmann (PhD from Bielefeld University), D. A. Smith (PhD form University of Reading), A. Kunoth (PhD from Berlin University), C. Lattanzio (PhD from University of Rome Tor Vergata), M. Palombaro (PhD from University of Rome “La Sapienza”), M. N. Mach (PhD from University of Pisa), A. Athanasoulis (PhD from Princeton University), I. Pantazis (PhD from University of Crete), E. Pulvirenti (PhD from Universita degli Studi Roma, Tre), N. Tymis (PhD from Loughborough University) and A. Rissanou (PhD from University of Crete).
In addition, the following researchers were hired as senior researchers: E. Georgoulis (Sussex), D. Shepelskiy (Verkin Inst, Ukraine), L. (Univ. L’Aquila), L. S. Keeling (U. Graz, Austria); P. Kaklis (NTUA, Athens), S. Venakides (Duke Univ.), E. Kyza (University of Crete); F. Faure (Institut de Physique Nucleaire); T. Pryer (University of Sussex);; A. Cangiani (University of Oxford); A. Vainchtein (Cornell University); K. Siebert (University of Freiburg); M. Jensen (University of Oxford); Sergey Dobrokhotov (Institute of Electronic Machinendesign), P. (S.I.S.S.A.) K. Ecker (Mathematisches Institut, Freie Universität Berlin), L. (Sapienza University of Rome), E. C. Alberti (University of the Basque Country (UPV/EHU))
Finally, the following researchers were hired as distinguished visitors: S. Venakides (Duke Univ.); Th. Zariphopoulou (Oxford); P. Souganidis (U. Chicago); Tai-Ping Liu (Stanford Univ and Academia Sinica, Taiwan); Jian-Guo Liu (Duke Univ.); Gen Nakamura (Hokkaido Univ.); Michael Sigal (University of Toronto). Timothy Healy (Cornell), Peter Bates (Michigan State University), Manoussos Grillakis (university of Maryland), Yaroslav Kurylev (University College London), Gery Bona (University of Illinois at Chicago), Ohannes Karakashian (University of Tenessee). A special effort was devoted in attracting to ACMAC prominent members of the Greek Scientific diaspora.
This is a continuing objective covering all reporting periods.
ORGANIZATION OF WORKSHOPS AND CONFERENCES. The organization of workshops and meetings will be an important tool for knowledge transfer among the involved research groups. These actions will enhance networking and collaborations and expose ACMAC members in an international scientific environment. These activities are organized around thematic programs, that involve short-term visits of scientists working in the area emphasized by the thematic program, as well as periods of concentrated activity of workshops and conferences.
A major part of the project is the support of incoming research visits and the organization of workshops under the framework of thematic programs.
The following thematic programs were completed:
1. The Thematic program on STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS, took place in the period May-June 2011. A workshop on “Stochastic PDE and its applications” was held in the facilities of the centre from 13 – 17 June 2011; website here.
2. The Thematic program on MATHEMATICAL MODELING OF MATERIALS ACROSS SCALES, took place in the period of Sep 2010 – Aug 2011. Two workshops were organized: The first entitled “Continuum and kinetic methods in the theory of shocks, fronts, dislocations and interfaces” was held on June 20-24, 2011, workshop's webpage. The date of June 24th was dedicated in honor of C.M. Dafermos (Alumnae Prof. at Brown University). The second workshop "Coarse–graining of many–body systems: analysis, computations and applications" was held on Jun 27 – Jul 1, 2011, workshop's webpage .
3. The Thematic program on COMPUTATIONAL PARTIAL DIFFERENTIAL EQUATIONS started on Sep 1, 2011 and and ended on Dec 31, 2011. A number of research visits were supported under this thematic program and two workshops were organized: The first workshop, entitled "Modern Techniques in the Numerical Solution of Partial Differential Equations", took place on Sep19 – Sep 23, 2011. The workshops's webpage may be found here. The second workshop, entitled "Discontinuous Galerkin Methods for Partial Differential Equations", was organized on Sep 26-28, 2011. The program can be found at the workshop's webpage.
4. The Thematic program on WAVE PROPAGATION IN COMPLEX ENVIRONMENTS took place in the period of April 2012 – June 2012. Three workshops were organized: The first workshop, “Wave propagation in complex media and applications”, was held on May 7-11, 2012, with workshop's webpage. The second workshop, "Semiclassical & multiscale aspects of wave propagation", was held on May 28 – Jun 1, 2012. The workshop's webpage may be found here. The third workshop, “Waves and imaging in complex media”, was held in the facilities of FORTH (Foundation for Research and Technology-Hellas) from June 7th to 8th (the introductory part) and from June 11th to 15th (the main part), 2012. The website of the workshop can be found here.
5. The Thematic program on SCIENTIFIC AND HIGH PERFORMANCE COMPUTING, started on Sep 1, 2012 and finished on Jan 31, 2013. A number of research visits were supported under this thematic program and two workshops were organized: The first workshop entitled "Software Frameworks for Challenging Computational Problems" took place from 14 – 18 January 2013. More information can be found at the workshops's webpage. The second workshop, entitled "Geometric Computing", took place in ACMAC during the period between 21st – 25th January, 2013, see the workshop's webpage.
6. A workshop entitled “Women in Applied Mathematics” took place in Heraklion on May 2-5, 2011, co-organized between ACMAC and the Institute of Applied and Comp. Math, FORTH. Apart from the excellent scientific program detailed in the workshop's webpage, a panel discussion took place on the special problems facing female scientists in their academic careers took place.
7. The workshop “Cell biology and physiology: PDE models”, was held in the facilities of ACMAC on October 4-6, 2012. The aim of this workshop was to bring together leading european groups and present the latest developments on mathematical modeling of biological cells and their various interactions. The program can be found at the workshop's webpage.
8. The thematic program on “NONLINEAR SCHROEDINGER EQUATIONS AND APPLICATIONS” was organized in the period March 1 – May 31, 2013. Three workshops were organized as part of this thematic program: The workshop on “Domain Microstructure and dynamics in magnetic elements” on April 8-11, 2013 whose program can be found at http://www.acmac.uoc.gr/%CE%BCMAGelm2013/program.php Second, the workshop on “Nonlinear Schroedinger equatons: Theory and Applications” that took place on May 20-24, 2013 and the program of which is found at http://www.acmac.uoc.gr/NLS2013 The last entitled “The 11th European Finite Element Fair” that took place on May 31- June 1, 2013, whose webpage can be found at http://www.acmac.uoc.gr/EFEF2013
9. The thematic program on “MATHEMATICAL METHODS FOR MULTISCALE SYSTEMS” was organized from June 1- July 28 encompassing as main even the two-week workshop “Kinetic description of multiscale phenomena” and one day workshop "Wave Phenomena: Numerical Methods and Analysis" on 29th July.
The first workshop fostered the interplay of propabilistic and kinetic techniques for problems on phase transitions. The workshop’s webpage can be found at the following link: http://www.acmac.uoc.gr/KDM2013 The second workshop on 29th July, whose program can be found at http://www.acmac.uoc.gr/WAVENUMA2013/program.php.
10. The workshop “Stochastic Methods in Finance and Physics” was organized in the period July 15- 19, 2013. The aim of this workshop was to bring together groups that work on stochastic particle systems on diverse problems arising in finance or in the physical sciences and to display the interconnections of the problems. The workshop’s program can be found at http://www.acmac.uoc.gr/SMFP2013/
11. The final event of the ACMAC project was the “International Conference on Applied Mathematics” that took place the week of September 16-20, 2013. The idea of this even was to serve as a major International Conference on the interplay of modelling, analysis and computation with a program that combines plenary talks on major developments in the general areas of the ACMAC project, and at the same time to have minisymosia organized in parallel that are on specific subjects of interest to the members of the Center. The program of this Conference can be found at http://www.acmac.uoc.gr/ICAM2013/
DISSEMINATION AND PROMOTIONAL ACTIVITIES The web page of the ACMAC Centre www.acmac.uoc.gr is operational from Jun 2010 and is improved and evolving continuously. There have been promotional pamphlets for the Centre and separately for each of the ongoing programs that are distributed in Greece and other European Universities and Research Centres.
A much emphasized task is the participation of ACMAC members in International Conferences and Workshops, as well as Research visits to Universities. Many such visits were supported in the course of running this project.
WP7. EVALUATION: An evaluation of the effort will be conducted at the end of the 42nd month. Four experts will conduct the evaluation of the research strategy and agenda of the ACMAC centre. These internationally recognized and independent experts will be chosen by the Commission services at the 22nd month of the project. They will receive a contract by the Applied Mathematics Department according to the terms outlined in WP7.
Project Results:
Research Results
The research activity at ACMAC was in the field of Applied Mathematics and specifically at the interface of modeling, analysis and computation. A large number of research papers were completed by ACMAC members. A preprint server was set up where the published results of ACMAC members and the postdoctoral researchers are posted. The web address of the preprint server is
http://preprints.acmac.uoc.gr/
Among the results we highlight works on the subjects:
modern multiscale methods for the simulation of coupling mechanisms between atomistic and continuum descriptions;
work on coherent interferometric imaging based on the back-propagation of local spacetime cross-correlations of array data, the discovery and subsequent experimental observations of autofocusing radially symmetric beams;
the development of a rigorous and computationally flexible framework for parallelization of Kinetic Monte Carlo methods and of spatial multilevel coarse-graining methods for Monte Carlo sampling and molecular simulation;
the development of a novel approach for patching inner and outer solutions applicable to problems with exchange of stability;
the study on the structure of the system of polyconvex elastodynamics, and the work on dynamic formation of cavities in isotropic elasticity;
finally, the works on stochastic particle models describing the onset of phase transitions.
A brief description of the most significant scientific results of ACMAC members now follows including a list of the three most significant publications per ACMAC member.
Markos Katsoulakis
Coarse-Graining:
Katsoulakis and collaborators introduced in 2003 hierarchical Coarse-Graining (CG) schemes for accelerating Kinetic Monte Carlo (KMC) simulations, showing that these new computational methods speed-up conventional KMC by several orders of magnitude. In ACMAC he subsequently developed multilevel CG methods for reliable simulation at mesoscopic length scales and where morphological features dictated by microscopics (patterns, interfaces, etc.) can be observed. Finally, in his ACMAC-sponsored research, the conjectured role of multi-body interactions in CG was rigorously analyzed, and efficient numerical implementations of such terms were developed for the first time. These works have appeared in the top disciplinary computational journals, as well as in the top applied mathematics journals:
Uncertainty Quantification (UQ) for Non-equilibrium Complex Systems:
In ACMAC he started developing path-wise information theory-based and goal-oriented sensitivity analysis and parameter identification methods for complex high-dimensional dynamics, as well as path-wise information-theoretic tools for parameterized CG of non-equilibrium extended systems. The combination of these novel methodologies provide the first methods in the literature which are capable to handle UQ questions for stochastic complex systems with some or all of the following features: (a) multi-scale models with a very large number of parameters, (b) spatially distributed systems such as Kinetic Monte Carlo or Langevin Dynamics, (c) non-equilibrium processes typically associated with coupled physico-chemical mechanisms, driven boundary conditions, etc. These very recent works have just appeared in top disciplinary computational journals, as well as in top applied mathematic journals.
Parallelization Methods for Kinetic Monte Carlo.
Here, Katsoulakis, students and collaborators addressed mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo (KMC) simulations. They developed a new mathematical formulation of KMC parallelization, based on hierarchical operator splitting, as means of systematically decomposing the computational and communication loads between multiple processors in a parallel architecture. This spatial decomposition of the KMC algorithm into a hierarchy of Markov operators was also shown to be highly appropriate for parallelizing on clusters of Graphical Processor Units (GPU). Furthermore, this mathematical framework allowed to rigorously assess and improve the numerical and statistical consistency of existing parallel KMC software such as SPPARKS, developed recently in Sandia National Laboratories. These recent works have appeared in the top journals in high-performance computing and numerical analysis.
List with the most significant publications
1) E. Kalligiannaki, M. Katsoulakis, P. Plechac and D. G. Vlachos. Multilevel coarse graining and nano-pattern discovery in many particle stochastic systems. J. Comp. Phys., 231, 2599-2620, (2012).
2) Giorgos Arampatzis, M. Katsoulakis, Petr Plechac, Michela Taufer, Lifan Xu. Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms J. Comp. Phys., 231, 7795-7814, (2012)
3) G. Arampatzis, M. Katsoulakis and P. Plechac. Parallelization, processor communication and error analysis in lattice kinetic Monte Carlo, SIAM J. Num. Analysis, in press, (2014).
Chrysoula Tsogka
Together with Liliana Borcea and George Papanicolaou we have been working on developing statistically stable methods for imaging in heterogeneous media with numerous small heterogeneities that scatter the waves and impede the imaging process. Our most recent results on the subject are poresented in the SIAM Review paper [1].
Together with Josselin Garnier, George Papanicolaou and Adrien Semin we have been working on imaging using cross-correlations of passive noise recordings. Here we exploit the fact that the Green's function (or the travel time) between two passive sensors can be estimated from cross-correlations of signals generated by ambient incoherent noise, which is due to sources that are randomly distributed in space and statistically stationary in time. In [2], we carried out a detailed analysis of the signal to noise ratio of the image and analyzed how it depends on the properties of the noise sources and the array of receivers.
Together with D. Mitsoudis and S. Papadimitropoulos we have been working on the problem of selective imaging extended scatterers in waveguides using the scattered field obtained on an active array (cf. [3]). Our most significant result concerns the characterization of the signal (and noise) subspace of the array response matrix which allows us to analyze the properties of the images obtained when projections on different parts of this subspace are used.
List with the most significant publications
1) L. Borcea, F. González del Cueto, G. Papanicolaou and C. Tsogka, Filtering Deterministic Layer Effects in Imaging, SIAM Review 2012,54 (4), pp. 757-798., 2012.
2) J. Garnier, G. Papanicolaou, A. Semin and C. Tsogka, Signal-to-Noise Ratio Estimation in Passive Correlation-Based Imaging, SIAM Journal on Imaging Science, Vol. 6, No. 2, pp. 1092--1110, 2013.
3) C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Selective imaging of extended reflectors in two-dimensional waveguides, SIAM Journal on Imaging Science, Vol. 6, No. 4, pp. 2714--2739, 2013.
Athanasios Tzavaras
Structure and properties of polyconvex elasticity.
In previous work we have constructed a variational approximation scheme for the equations of polyconvex elasticity and showed that it produces dissipative measure-valued solutions. In [1] we produce a weak-strong uniqueness theorem for dissipative measure-valued solutions. The variational approximation scheme offers a time-discretization for polyconvex elasticity that exploits its Hamiltonian structure and it provides an interesting framework for the numerical computation of entropic weak solutions for the equations of polyconvex elasticity.
Cavitation for the equations of radial elasticity.
The system of radial isotropic elasticity is probably the simplest example where the effect of polyconvexity can be compared with the well-developed theory of one-dimensional hyperbolic conservation laws. To be interpreted as elastic motions, solutions have to satisfy certain constraints associated to the non-interpenetration of matter. The system of radial isotropic elasticity admits a class of self-similar solutions, identified by Pericak-Spector and Spector, that correspond to opening of a cavity in the material at some critical time. These cavitating solutions decrease the total mechanical energy and are an important example of non-uniqueness for hyperbolic systems due to point singularities. Their presence poses unsettling questions as to the adequacy of the usual entropy solution as a sufficiently general framework for multi-dimensional hyperbolic theory. If opening a cavity reduces the energy, then this induces an autocatalytic mechanism for failure and raises questions for the model. From a perspective of microscopic dynamics, it would appear that the emergence of cavities requires an energetic cost that is not accounted for in continuum theories. In [3] we raise the question whether the notion of weak solutions can be modified to account for the cavity creation costs. This is achieved by the notion of singular induced from continuum solutions (slic-solutions) developed in [3], which accounts for the energetic cost of creating the cavity and resolves the aforementioned paradox.
Relative entropy and diffusive relaxation.
The relative entropy method provides a powerful approach to compare an entropic approximating theory (or a weak entropy solution) with a conservative (smooth) solution of a hyperbolic system. A problem of interest is to compare two entropy weak solutions and this has not been accomplished in any level of sufficient generality. A related (but simpler) issue is to compare an approximating theory of a diffusion equation with the limiting diffusion problem. Indeed, this
situation appears in the context of diffusive limits. In [2] we successfully carry out such a calculation for the relaxation problem from multi-d Euler equations with frictional damping to the porous media equation.
List with the most significant publications
1) S. Demoulini and D.M.A. Stuart, and A.E. Tzavaras. Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Rational Mech. Anal. 205 (2012), 927-961.
2) C. Lattanzio and A.E. Tzavaras, Relative entropy in diffusive relaxation. SIAM J. Math. Anal. 45 (2013), 1563-1584.
3) J. Giesselmann and A.E. Tzavaras, Singular limiting induced from continuum solutions and the problem of dynamic cavitation. Arch. Rational Mech. Anal. 212 (2014) 241-281.
Phoebus Rosakis
Surface energies in discrete crystals:
substantial progress was made in identifying connections between discrete and continuous models for crystals. In [1] the energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two and three dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. Thus important connections are established with interesting number theoretic results. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal for arbitrary choice of the interatomic pair potential.
Discrete models of dynamics of slow dislocations: the result of [2] answers a long standing open question in the discrete dynamical modelling of dislocation motion. We construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel–Kontorova model, representing dislocation motion under stress. So far it was not known whether there exist admissible travelling wave solutions below a theshold velocity.The new solutions fill this gap for the first time. We find solutions for any value of the velocity. These results have important implications on the understanding of the stability of low velocity dislocation motion via discrete models.
Ghost-force free quasicontinuum methods in three dimensions:
in paper [3] we construct energy based numerical methods free of ghost forces in three-dimensional lattices arising in crystalline materials for the first time. The analysis hinges on establishing a connection of the coupled system to conforming finite elements. Key ingredients are: (i) a new rep- resentation of discrete derivatives related to long range interactions of atoms as volume integrals of gradients of piecewise linear functions over bond volumes, and (ii) the construction of an underlying globally continuous function representing the coupled modeling method. Our work solves an important open problem arising in the context of the quasicontinuum method for crystalline materials.
List with the most significant publications
4) P. Rosakis, Continuum surface energy from a lattice model, arXiv preprint arXiv:1201.0712 (2012). To appear in Networks and Inhomogeneous Media.
5) P. Rosakis and A Vainchtein, New Solutions for Slow Moving Kinks in a Forced Frenkel-Kontorova Chain, Journal of Nonlinear Science 23(6), 1089-1110 (2013).
6) C. Makridakis, D. Mitsoudis and P. Rosakis, On atomistic-to-continuum couplings without ghost forces in three dimensions, Applied Mathematics Research Express, abt005 (2013). doi:10.1093/amrx/abt0,0 arXiv preprint arXiv:1211.7158 (2013).
Charalambos Makridakis
In [1] a novel finite element consistency analysis of Cauchy-Born approximations to atomistic models of crystalline materials in two and three space dimensions is introduced. The analytical tools introduced and an intermediate model can be useful in the design and the analysis of consistent coupled methods in more than one space dimensions. Indeed, our approach is the first comprehensible consistency analysis in two and three dimensions regarding the accuracy of the CB model. Further, in [2], using the results of [1] and new ideas, we provide a systematic approach leading to ghost-force-free couplings in two and three dimensions. The results of [2] contribute to long standing algorithmic design issues such as the construction of consistent couplings of arbitrary high-order of accuracy and to the design of the first consistent coupled energies in three dimensions. (iii) In [3] we propose a new class of finite element methods for the Navier-Stokes-Korteweg system which are by design consistent with the energy dissipation structure of the problem. The methods introduced are of arbitrary high order of accuracy and provide physically relevant approximations free of numerical artifacts. It seems that these are the first methods in the literature enjoying these properties.
List with the most significant publications
1) Makridakis, Charalambos ; Süli, Endre . Finite element analysis of Cauchy-Born approximations to atomistic models. Arch. Ration. Mech. Anal. 207 (2013), no. 3, 813--843.
2) Makridakis, Charalambos ; Mitsoudis, Dimitrios ; Rosakis, Phoebus . On atomistic-to-continuum couplings without ghost forces in three dimensions. Appl. Math. Res. Express. AMRX 2014, no. 1, 87--113.
3) J. Giesselmann, C. Makridakis and T. Pryer, Energy consistent discontinuous
Galerkin methods for the Navier-Stokes-Korteweg system, Math. Comp (2014)
DOI: http://dx.doi.org/10.1090/S0025-5718-2014-02792-0
Nikolaos Efremidis
Abruptly autofocusing waves.
We have introduced the concept of abruptly autofocusing waves, which are waves whose peak intensity remains almost constant during propagation, while close to a particular focal point, they suddenly autofocus and, as a result, their peak intensity increases by orders of magnitude. We observed the predicted phenomena in collaboration with two different experimental groups and utilized these beams for particle manipulation and creation of ablation spots. The first proposed family of autofocusing waves consists of radially symmetric Airy beams. We were able to generalize these families of waves and propose methods to efficiently generate them. The generalized families of waves proposed can follow arbitrary convex trajectories such as power low before they collapse at the focusing spot.
Bessel-like beam that follow predefined trajectories.
We have also proposed a method for generating Bessel-like optical beams with arbitrary trajectories in free space. The method involves phase-modulating an optical wavefront so that conical bundles of rays are formed whose apexes write a continuous focal curve with pre-specified shape. These ray cones have circular bases on the input plane; thus their interference results in a Bessel-like transverse field profile that propagates along the specified trajectory with a remarkably invariant main lobe. Such beams can be useful as hybrids between non-accelerating and accelerating optical waves that share diffraction-resisting and self-healing properties. We have extended this method in the case of nonparaxial beams. Still, there is a new interesting feature that sets the nonparaxial case clearly apart: The symmetric Bessel-like profile forms on a plane that is normal to the focal trajectory. This further suggests that (in contrast to the paraxial case) nonparaxial diffraction-free (or diffraction-resisting) accelerating waves should be sought as propagation-invariant waves with a dynamically changing plane of observation that stays normal to the trajectory of the beam. Experimentally, we have observed the predicted phenomena in collaboration with Zhigang Chen’s group in San Francisco State University.
Curved beams in periodic index configurations.
We have studied curved trajectory dynamics and design in discrete array settings. We found that beams with power law phases produce curved caustics associated with the fold and cusp type catastrophes. A parabolic phase produces a focus that suffers from spherical aberrations. More important, we find that by designing the initial phase or wavefront of the beam we can construct trajectories with pure power law caustics as well as aberration-free focusing of discrete waves. Going beyond the discrete, tight-binding model, which we examined recently, we show how the exact band structure and the associated diffraction relations of a periodic waveguide lattice can be exploited to phase-engineer caustics with predetermined convex trajectories or to achieve optimum aberration-free focal spots.
List with the most significant publications
1) Ioannis Chremmos, Nikolaos K. Efremidis, and Demetrios N. Christodoulides, Pre-engineered abruptly autofocusing beams , Optics Letters, vol. 36, p. 1890 (2011).
2) Ioannis D. Chremmos, Zhigang Chen, Demetrios N. Christodoulides, and Nikolaos K. Efremidis, Bessel-like optical beams with arbitrary trajectories, Optics Letters, vol. 37, p. 5003 (2012).
3) Nikolaos K. Efremidis and Ioannis Chremmos, Caustic design in periodic lattices , Optics Letters, vol. 37, p. 1277 (2012).
Dimitrios Tsagkarogiannis
The first concerns the study of effective thermodynamic potentials for systems of interacting
particles in the continuum with Lennard-Jones type interactions. The main question is to
construct effective theories for the study of phenomena such as condensation and phase
transitions. A main technical tool is the cluster expansion method around some appropriately chosen regime. In particular we were interested in density expansions for mixtures of countably many different types of particles. This is known in the literature as multi-species virial expansion. We followed a combinatorial path and in order to prove convergence we used the Lagrange–Good inversion formula, which has other applications such as counting colored trees or studying probability generating functions in multi-type branching processes. We proved that the virial expansion converges absolutely in a domain of small densities. In addition, we established that the virial coefficients can be expressed in terms of two-connected graphs (see ref. 1).
The second is the continuation of previous work on the study of non equilibrium systems driven by current reservoirs. We are interested in microscopic systems of interacting particles in [-N,N] as natural models for current reservoirs and Fick's law. In previous work we have established the hydrodynamic limit and studied the stationary measure. In the recent works we investigated the exponential rate of convergence to the stationary measure, which we prove to be of the order N^2 (see ref. 3). As a technical tool we had to study a random walk with death in [-N,N] moving in a time dependent environment. The environment is a system of particles which describes a current flux from N to -N. Its evolution is influenced by the presence of the random walk and in turns it affects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in N) for the survival probability up to time t (see ref. 2).
List with the most significant publications
1) Sabine Jansen, Stephen J. Tate, Dimitrios Tsagkarogiannis, Daniel Ueltschi, Multispecies Virial Exapnsions, Comm. Math. Phys. 2014,(DOI)10.1007/s00220-014-2026-9
2) Anna DeMasi, Errico Presutti, Dimitrios Tsagkarogiannis, Maria Eulalia Vares, Extinction time for a random walk in a random environment, Bernoulli, 2014 to appear.
3) Anna DeMasi, Errico Presutti, Dimitrios Tsagkarogiannis, Maria Eulalia Vares, Exponential rate of convergence in current reservoirs, Bernoulli, 2014 to appear.
Vagelis Harmandaris
Computational modeling of polymers:
Through a systematic hierarchical simulation methodology we were able to study the properties of complex macromolecular systems at the molecular level. The simulation results were quantitatively compared against theoretical predictions and experimental data. In a further stage the dynamics of polymer blends were also studied and compared to the self-concentration model that requires a composition-dependent length scale.
Modeling of biomolecular systems:
We have performed detailed atomistic molecular dynamics simulations of peptides (Diphenylalanine, FF) in using an explicit solvent model. The self-assembling propensity of FF in water is obvious while in methanol a very weak self-assembling propensity is observed. We studied and compared structural properties of FF in the two different solvents and a comparison with a system of dialanine (AA) in the corresponding solvents was also performed.
Hierarchical multi-scale modeling of nanocomposites:
We have performed multi-scale simulations of graphene based polymer systems. Detailed atomistic simulations allowed us to explore the effect of graphene on the mobility of polymers by studying three well known and widely used polymers, polyethylene (PE), polystyrene (PS) and poly(methyl-methacrylate) (PMMA). Qualitative and quantitative differences in the dynamical properties of the polymer chains in particular at the polymer–graphene interface are detected.
List with the most significant publications
1) V. Harmandaris, G. Floudas, K. Kremer, “Dynamic heterogeneity in fully miscible blends of polystyrene with oligostyrene”, Phys. Rev. Let. 2013, 110, 165701.
2) A. Rissanou, E. Georgilis, M. Kasotaskis, A. Mitraki, V.A. Harmandaris, “Effect of solvent on the self-assembly of dialanine and diphenylalanine peptides”, J. Phys. Chem. B 2013, 117, 3962-3975.
3) A. Rissanou, V. Harmandaris, “Dynamics of various polymer/graphene interfacial systems through atomistic molecular dynamics simulations”, Soft Matter, 2014, 10, 2876-2888.
Spyros Kamvissis
A significant part of my ACMAC funded research has been on the question of integrability of initial/boundary value problems for so-called completely integrable equations like cubic NLS in 1 space dimension.
There is a so-called unified transform theory, constructed by Fokas and collaborators, which generalizes the inverse scattering theory for Cauchy problems for such equations. A main "gap" for this theory to be completely satisfactory is the study of the Dirichlet-to-Neumann map for the boundary (not the initial) data. It is crucial to be able to find nontrivial classes of functions such that for Dirichlet data in such a class, the Neumann data also belongs in the same class.
Together with A. Fokas himself and J. Lennels (at Cambridge), D. Shepelsky (in Ukraine and in Crete), S. Venakides (during his visit to Crete), ACMAC postdoc David Smith and University of Crete postdoc D. Antonopoulou we have spent a lot of time working on this problem.
There is a preprint with D. Shepelsky and a paper in progress with D. Antonopoulou which address the problem, see list below.
Other ACMAC funded work concerns general asymptotic problems on soliton equations. I would like to particularly point out a paper with D.Smith on asymptotics for a discretization (in both time and space) of KdV.
List with the most significant publications
1) D.Antonopoulou S.Kamvissis Schwartz initial data for the NLS and Fokas' transform, in progress.
2) S.Kamvissis D.Shepelsky Initial boundary value problem for the focusing NLS equation with periodic boundary data, preprint.
3) S.Kamvissis D.Smith Long time behavior of the IpKdV equation, in progress.
Georgia Karali
Non-radial solutions for a singular perturbation problem.
Contrary to previous approaches to such singular perturbation problems, that mainly relied on the construction of upper and lower solutions or weak convergence arguments, we use a perturbation argument and find a genuine solution close to a carefully constructed approximation by means of the contraction mapping principle, provided the singular parameter is chosen sufficiently small. We develop a novel approach for rigorously patching together inner and outer approximations, which is flexible enough to deal also with other related problems. Besides improving previously known estimates regarding stable solutions, where the method of upper and lower solutions applies, the perturbation approach allowed us to construct highly unstable solutions (radial and bifurcating non-radial), in the sense that their Morse index (in the general class) diverges as the small singular parameter tends to zero. Our new methodology enabled us to succeed in proving recently the existence of highly unstable solutions in 2 dimensions without assuming any symmetry assumptions. Also, we have answered an open interesting problem posed recently in Aftalion, Jerrard, and Royo-Letelier concerning ground states of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit, where the corresponding limit algebraic equation admits a pitchfork bifurcation at a submanifold of the domain.
One-dimensional stochastic Cahn-Hilliard equation.
We describe the joint motion of multiple kinks or interfaces for the one-dimensional Cahn-Hilliard equation on a bounded interval perturbed by small additive noise. The approach is based on an approximate slow manifold, where the motion of interfaces is described by the projection onto the manifold. This was used deterministically by Bates and Xun (1994/95) to verify metastable behavior of solutions with multiple kinks. We derive a stochastic differential equation for the motion of the interfaces. The approximation is (with high probability) valid until an interface breaks down, or until times that are large compared to any negative power of the small interaction length. The differences with the deterministic case are many and the stochastic case requires by far more tedious analysis and ideas. This is the first work in the literature in this direction.
List with the most significant publications
1) C. Sourdis and G. Karali, The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit. to appear Archive for Rational Mechanics and Analysis.
2) D. Antonopoulou, D. Bloemker and G. Karali, Front motion in the one-dimensional stochastic Cahn-Hilliard equation. SIAM J. Math. Anal. 44, 3242-3280, (2012).
3) C. Sourdis and G. Karali, Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities}, Ann. Inst. H. Poincare Anal. Non Lineaire, 29, 131-170, (2012).
Menelaos I. Karavelas
The exact maximum complexity of the Minkowski sum of convex polytopes of general dimension. Given r d-dimensional polytopes P_1, P_2,... P_r in the Euclidean d-dimensional space, we are interested in evaluating the maximum number of k-faces or their Minkowski sum P_1+P_2+...+P_r. The solution to this problem was previously know only in 2 and 3 dimensions (for 2- and 3-dimensional polytopes, respectively). Results in higher dimensions were only partial and sporadic. During the course of the project we managed to fully resolve the problem when r=2 or r=3 (i.e. for two and three summands), and for general dimension. More precisely, given r convex d-dimensional polytopes in d-dimensions, where r=2 or r=3, we have evaluated closed-form upper bound expressions on the number of k-faces of their Minkowski sum as a function of the number of vertices of the two or three polytopes. Our upper bounds are shown to be tight.
At a more technical level, our approach resembles, or rather generalizes the approach of McMullen for proving the upper bound theorem for polytopes (this is a famous result in Combinatorial Geometry, stating that any d-polytope with n vertices, cannot have more k-faces than the cyclic d-dimensional polytope with the same number of vertices). To prove our upper bound we have considered the Minkowski sum of the polytopes as a section of their Cayley polytope in dimension d+r-1 (recall that r=2 or 3). In the Cayley polytope we have identified a set of faces F that is in 1-1 correspondence with the faces of the Minkowski sum, and for this set F, we have considered its f-vector f(F) and h-vector h(F). We analyzed its h-vector, showed that it satisfies equations that generalized the well-known Dehn-Sommerville equations of simplicial polytopes, and have managed to prove a recurrence relation for the elements of h(F). From this recurrence relation we proved upper bounds on the individual elements of the h-vector, which in conjunction with the Dehn-Sommerville-like equations for h(F) have enabled us to compute upper bounds on the elements of f(F). Given the correspondence between the faces of Minkowski sum and the faces of F, the upper bounds for f(F) immediately give upper bounds for the faces of the Minkowski sum of the polytopes. These bounds have been shown to be tight by explicit constructions.
Besides two and three summands, we have been extending our methodology to an arbitrary number of summands. In particular, we have generalized the Dehn-Sommerville-like equations for r=2 or 3 to any generic r, and we have proved the recurrence relation on the elements of h(F) for any number of summands. We are currently working on finalizing our results, and proving tight asymptotic expressions for the number of k-faces for the Minkowski sum of r d-dimensional polytopes, for any r greater or equal to 4.
The robust and efficient computation of the 2D Apollonius diagram. The planar Apollonius diagram (also known as the additively weighted Voronoi diagram) is a generalization of the standard Voronoi diagram for points when the diagram generators are disks. A point in the space is assigned to the generator that is closest to it, where distance is measured from the boundary of the disk, ans is positive (resp., negative) if the point lies outside (resp., inside) the disk. As in the case of usual point Voronoi diagrams, degenerate configurations can occur. One such degenerate configuration occurs when four mutually disjoint disks are externally tangent to the same circle. Degenerate configurations pose various problems in geometric algorithms, and it is often the case that we perturb symbolically the input data so as to remove these degeneracies. Such symbolic perturbation schemes were lacking for the case of Apollonius diagrams; the reason for that is that the problem itself is quite complicated, while the predicates necessary for the evaluation of the diagram are quite complicated (for example, with respect to the case of points). During the course of the project, we devised a perturbation framework for dealing with these degeneracies. In short, the framework amounts to changing symbolically the radii/weights of the input generators, where disks with higher radius/weight are considered as perturbed more. Under these considerations we have been able to resolve all degeneracies of the 2D Apollonius diagram, as well as the degeneracies of all predicates appearing in the standard randomized incremental construction of the diagram. Out approach is close to being fully implemented within the framework of the Computational Geometry Algorithms Library (CGAL: www.cgal.org) for the 2D case, while we know how to handle the most basic predicate for the 3D case.
Besides handling degeneracies, we have been considering approaches for constructing the diagram in a non-incremental manner. More precisely, we are considering an offline algorithm that computes the Voronoi diagram of the centers first, and then increase the weights of the sites to compute the Apollonius diagram. We have already shown that this approach yield a randomized algorithm with O(n log(n)) expected running time, as opposed to O(n log^2(n)) expected running time required by the standard randomized incremental approach. Preliminary benchmarks have shown that, at least in the case of disjoint generators, the new algorithm can be much faster in practice than the randomized incremental one.
List with the most significant publications
1) Menelaos I. Karavelas and Eleni Tzanaki. The maximum number of faces of the Minkowski sum of two convex polytopes. In Proceedings of 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'12), pages 11-28, Kyoto, Japan, January 17-19, 2012. http://doi.acm.org/10.1145/2095116.2095118
2) Menelaos I. Karavelas, Christos Konaxis, and Eleni Tzanaki. The maximum number of faces of the Minkowski sum of three convex polytopes. In Proceedings of the 29th Annual ACM Symposium on Computational Geometry (SCG'13), pages 187-196, Rio de Janeiro, Brazil, June 17-20, 2013. http://dx.doi.org/10.1145/2462356.2462368
3) Olivier Devillers, Menelaos I. Karavelas, and Monique Teillaud. Qualitative Symbolic Perturbation: a new geometry-based perturbation framework. Research Report RR-8153, INRIA, 2012. http://hal.inria.fr/hal-00758631
Theodoros Katsaounis
Linear Schrödinger equation.
Development, analysis and implementation of a space-time adaptive algorithm for the solution of the linear Schrödinger equation. The adaptive algorithm is based on rigorous a posteriori error estimation. We use finite element spaces for the spatial discretisation and a modified Crank-Nicolson scheme as a time-stepping mechanism. The a posteriori error estimators/indicators are of optimal order in space and time. The adaptive algorithm is tested thoroughly and proven to be an efficient and very robust solver for the linear Schrödinger eq. Adaptivity proves to be beneficial, reduces the computational cost substantially while keeping the total error under a desired tolerance.
Shear Band formation.
We study numerically an instability mechanism for the formation of shear bands at high strain-rate deformations of metals. We use adaptive finite element methods of any order in the spatial discretisation, and implicit Runge-Kutta methods with variable step in time. The numerical schemes are of implicit- explicit type and provide adequate resolution of shear bands up to full development. We find that already from the initial stages, shear band formation is associated with collapse of stress diffusion across the band and that process intensifies as the band fully forms. For fully developed bands, heat conduction plays an important role in the subsequent evolution by causing a delay or even stopping the development of the band. Further we provide an explanation of the onset of localization and the formation of shear bands in high strain-rate plasticity of metals. We employ the Arrhenius constitutive model and show Hadamard instability for the linearized problem. For the nonlinear model, using an asymptotic procedure motivated by the theory of relaxation and the Chapman-Enskog expansion, we derive an effective equation for the evolution of the strain rate, which is backward parabolic with a small stabilizing fourth order correction. We construct self-similar solutions that describe the self-organization into a localized solution starting from well prepared data.
Dispersive water wave propagation.
We introduce a general finite volume / finite difference computational framework for simulating water wave propagation. The main application in question is the accurate prediction of tsunami propagation and landing. The main idea is to rewrite the system, describing two way propagation, in a conservation law form. The application of the finite volume method is adapted to discretise the advective and dispersive fluxes. Various numerical fluxes and reconstruction techniques are tested. The time stepping is done using Runge-Kutta methods which preserve the TVD property of finite volume scheme. The method is validated through a series of numerical experiments and compared against experimental data when possible. Extensive numerical studies were performed in the case of tsumani wave landing and computation of the runup. The dispersive models were compared with the classical Shallow Water eqs.
Image processing.
A model problem is used to represent a typical image processing problem of reconstructing an unknown in the face of incomplete data. A consistent discretization for a vanishing regularization solution is defined so that, in the absence of noise, limits first with respect to regularization and then with respect to grid refinement agree with a continuum solution defined in terms of a saddle point formulation. It is proved and demonstrated computationally for artificial examples and for magnetic resonance images that a mixed finite element discretization is consistent in the sense defined here. On the other hand, it is demonstrated computationally that a standard finite element discretization is not consistent, and the reason for the inconsistency is suggested on the basis of the theory presented for the mixed method.
List with the most significant publications
1) Th. Katsaounis & I. Kyza, A posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrödinger equation, to appear in Numerische Mathematik, 2014.
2) Th. Baxevanis, Th. Katsaounis, & A. Tzavaras, Adaptive finite element computations of shear band formation, Mathematical Models in Applied Sciences, vol. 20, no.3 pp.423-448 2010.
3) D. Dutykh, Th. Katsaounis & D. Mitsotakis, Finite volume schemes for dispersive wave propagation and runup, Journal of Computational Physics, vol. 230, no. 8, pp. 3035-3061, 2011.
Stavros Komineas
In the field of ultracold atoms the coupling of atoms to effective magnetic fields has been achieved in the last years. Effective fields are created by using coherent optical dressing of atomic levels which result in non-dynamical gauge fields. These techniques have the potential to generate non-Abelian gauge fields. We have considered a gas of bosonic atoms coupled to a gauge field with U(2) symmetry, and with constant effective magnetic field. We studied the effects of weak interactions by Gross-Pitaevskii mean-field ntheory on the vortex lattice phase induced by a uniform effective magnetic field. We showed that, with increasing non-Abelian gauge field, the nature of the ground state changes dramatically, with structural changes of the vortex lattice. These changes can be understood as produced by effective interactions with non-zero range.
The injection of a dc spin-polarised current through a ferromagnetic element can induce magnetisation oscillations and thus turn the nanoelement into a spin-transfer nano-oscillator. We analysed magnetization dynamics using the Landau-Lifshitz equation including a Slonczewski spin-torque term, in two types of magnets.
(a) In isotropic magnets, we established that a vortex-antivortex dipole is set in steady-state rotational motion due to the interaction between the vortices, while an external in-plane magnetic field can tune the frequency of rotation. The rotational motion was linked to the nonzero skyrmion number of the dipole. Three types of vortex-antivortex pairs were obtained as we vary the external field and spin-torque strength. Their character and stability properties were elucidated by an asymptotic analysis.
(b) In magnetic elements with perpendicular anisotropy the relevant excitations are magnetic bubbles or non-topological droplet solitons. Their dynamics under spin-polarised current produces magnetization oscillations.
We analysed the dynamics of dipolar and quadrapolar excitations. Exciton-polaritons (quasiparticles of exciton and photons) which are produced in semiconductors have been condensed in the laboratory in the sense of Bose-Einstein. These are non-equilibrium states in condensed matter and they are interesting because of rich phenomena taking place while the condensate is achieved at relatively high temperature. The mean-field description of the polariton system is given by a system of two Gross-Pitaevskii type equations which include a dispersive term for photons and a nonlinear term modelling exciton interactions. We study soliton solutions of the system of Gross-Pitaevskii equations and have developed a rigorous method to find a series of solitons. We have found solitons in the sense of the usual nonlinear Schroedinger equation but also solitons with surprising discontinuous profiles. Discontinuities are explained within the specific system of Gross-Pitaevskii equations.
List with the most significant publications
1) S. Komineas, “Frequency generation by a magnetic vortex-antivortex dipole in spin-polarized current”, Europhys. Lett. 98 (2012) 57002
2) Stavros Komineas, Nigel R. Cooper, “Vortex lattices for ultracold bosonic atoms in a non-Abelian gauge potential”, Phys. Rev. A 85, 053623 (2012)
3) G. Finocchio, V. Puliafito, S. Komineas, L. Torres, O. Ozatay, T. Hauet, B. Azzerboni, “Nanoscale spintronic oscillators based on the excitation of confined soliton modes”, J. Appl. Phys. 114, 163908 (2013)
Georgios Kossioris
In collaboration with G.E. Zouraris (Univ. of Crete) we considered an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate stochastic parabolic problem discretizing the noise using linear splines. Then we construct fully-discrete approximations to the solution of the approximate problem using, for the discretization in space, a Galerkin finite element method based on H 2 −piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates: for the error between the solution to the problem and the solution to the approximate problem, and for the numerical approximation error of the solution to the approximate problem.
In collaboration with O. Lakkis (Univ. of Sussex) and M. Romito (Univ. of Florence) we are in a final stage of completing the manuscript entitled “Strong convergence for numerical approximation of the stochastic Allen-Cahn in one dimension”. The initial equation, referred in the sequel as stochastic A-C equation, is approximated by replacing the space-time white noise by the proper piecewise space-time random function. A finite element approximation scheme of the approximating stochastic A-C equation is then analyzed. We first obtained space-time L p and L∞ stability estimates for the approximating linear stochastic heat equation leading, by a careful estimate analysis, to L p and L∞ error estimates. In order to study the nonlinear problem we study the corresponding nonlinear randomly shifted approximating stochastic A-C equation where we have a slightly better regularity. Using the corresponding estimates for the linear problem we obtained L p and L∞ error estimates for the randomly shifted approximating equation. In order to perform the finite element analysis we obtained L p and L∞ regularity error estimates for the approximating linear heat equation and the approximating nonlinear randomly shifted equation. Based on the above regularity estimates we obtained the L∞ error estimates for the finite element approximation scheme of the approximating heat equation. The derivation of the corresponding L∞ error estimate for the finite element scheme for the randomly shifted approximating stochastic A-C equation is under derivation.
In collaboration with M. Plexousakis (Univ. of Crete) and A. Yannacopoulos (Athens Univ. of Economics and Business) we study the stochastic Allen-Cahn equation in one space dimension via a Wiener chaos expansion. It is a well-known fact that as the scaling parameter ε tends to zero, the solution converges to a travelling wave solution centered at a point that follows a Brownian motion. Given the difficulty to handle the Wiener chaos of the full nonlinear equation and in order to get an insight of the dominant terms of the Wiener chaos expansion of the full solution for small ε, we first analyze the Wiener chaos expansion of the travelling wave solution which is the zero-order approximation term for sufficiently small ε.
In collaboration with M. Loulakis (National technical Univ. of Athens) and M. Plexousakis (Univ. of Crete). We study random dynamical systems with an emphasis on stochastic bifurcations for problems related to environmental economic modeling. Both analytical as well as numerical techniques were considered.More specifically, we studied the Stochastic Shallow Lake Problem. We proved uniqueness and regularity of the value function in the framework of viscosity solutions and we applied stochastic Pontryagin Maximum Principle to study the optimal dynamics of the problem. We considered appropriate numerical methods to reconstruct the value function and the corresponding optimal dynamics.
We analyze the surface wind variability over selected areas of the Greek territory by comparing a 3-km-spatial resolution simulation performed with the Weather Research and Forecasting (WRF) model with actual surface measurements. Daily 36hrs runs at 12 UTC were driven by FNL (1º× 1º) data for three time periods during winter, spring and summer, respectively, are conducted. Various verification statistics, such as bias, RMSE and DACC for wind speed and direction were used to gauge the mesoscale model performance. The following paper “Evaluation of WRF performance for the analysis of surface wind speeds over various Greek regions”
Numerical solution of fourth-order SPDES: In the following works it has been achieved the convergence analysis of fully discrete finite element approximations of the solution to some linearized versions of the stochastic Cahn-Hillird equation driven by an additive space-time white noise or the space derivative of an additive space-time white noise. Optimal error estimates have been obtained.
List with the most significant publications
1) Kossioris, Georgios T.and Zouraris, Georgios E. (2010) ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2). pp. 289-322.
2) Kossioris, Georgios T.and Zouraris, Georgios E. (2013) Finite element approximations for a linear fourth-order parabolic SPDE in two and three dimensions with additive space-time white noise. Applied Numerical Mathematics, 67 (Special Issue: NUMAN 2010). pp. 243-261.1
3) Kossioris, Georgios and Zouraris, Georgios E. (2013) Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise. Discrete and Continuous Dynamical Systems - Series B 18 (7). pp. 1845-1872.
Michail Loulakis
Zero-range processes with jump rates that decrease with the number of particles per site can exhibit a condensation transition, where a positive fraction of all particles condenses on a single site when the total density exceeds a critical value. In [1] we considered rates which decay as a power law or a stretched exponential to a non-zero limiting value, and studied the onset of condensation at the critical density. We described exactly how a condensate emerges as we go from subcritical to supercritical densities, we established a law of large numbers for the excess mass fraction in the maximum, as well as distributional limits for the fluctuations of the maximum and the fluctuations in the bulk.
Many of the results rely on a general theorem for Large Deviations of Sums of Subexponential Random Variables-the analogue of Gibbs's conditioning principle-that was
proved in [2]. Precisely, we proved that conditionally on a large deviation of the sum of N subexponential random variables, the a posteriori distribution of the variables asymptotically looks like a product of N-1 copies of the prior distribution and a randomly located lump that realizes the large deviation.
List with the most significant publications
1) Inés Armendáriz, Stefan Grosskinsky, Michail Loulakis: Zero Range Condensation at Criticality, Stoch. Proc. Appl.,123 (2013) no. 9, pp 3466-3496. DOI: 10.1016/j.spa.2013.04.021
2) Inés Armendáriz, Michail Loulakis: Conditional Distribution of Heavy Tailed Random Variables on Large Deviations of their Sum, Stoch. Proc. Appl. 121 (2011) no. 5, pp 1138-1147. DOI:10.1016/j.spa.2011.01.011
Georgios Makrakis
Wave-packet evolution in phase space:
We studied the singular semiclassical initial value problem for the phase space Schrodinger equation which is used in physical chemistry for computing certain mean values in realistic applications. In particular, we approximated the semiclassical quantum evolution in phase space by analyzing initial states as superpositions of Gaussian wave packets and applying individually semiclassical anisotropic Gaussian wave packet dynamics, which is based on the the nearby orbit approximation. We accordingly constructed a semiclassical approximation of the phase space propagator, the so-called semiclassical wave packet propagator. We analyzed the relation of this propagator ith the initial value repsresentations, which have been introduced long time ago as a computational tool in physical chemistry, and which are currently the subject of inensive study in semiclassical microlocal analysis.. By this semiclassical propagator we constructed asymptotic solutions of the phase space Schrodinger equation, and we tested the validity of the approximation using certain analytical examples.
Asymptotic theory of wave beams:
We constructed a new representation of Maslov’s canonical operator in a neighborhood of caustics using a special class of coordinate systems (eikonal coordinates) on Lagrangian manifolds and we explained why this is a much more convenient implementation, useful in specific physical problems, for example, those related to the asymptotic behavior of solutions of the scattering problem and of the Green’s function, and to linear hyperbolic systems with variable coefficients (e.g. the wave equation) with localized initial data, etc. We also clarified that our formulas are special cases of the general formulas in the theory of Fourier integral operators. Our main result is a constructive formula and an algorithm for constructing these formulas, which can in particular be used in combination with software like Mathematica or MatLab in numerical computations.
We also studied how to localize certain exact solutions of the Schrodinger equation (with non-constant coefficients), which can represented as the product of the Airy function (Berry-Balazs solutions) by the Bessel function, known as the Airy-Bessel beams in the paraxial approximation in optics. To this end, we representeded the localized solutions in the form of Maslov's canonical operator acting on compactly supported functions on special Lagrangian manifolds. Then, by exploiting some results by Hormander, we use the formula for the commutation of a pseudodierential operator with Maslov's canonical operator to “move" the compactly supported amplitudes outside the canonical operator, and thus obtain formulae preserving the structure based on the Airy and Bessel functions. This made possible to discuss the dispersive structure of the beams.
List with the most significant publications
1) 1. P. D. Karageorge and G.N. Makrakis, Asymptotic solutions of the phase space Schrodinger equation: Anisotropic Gaussian approximation (submitted) (http://arxiv.org/abs/1402.6854)
2) 2. S. Yu. Dobrokhotov, G. N. Makrakis, V. E. Nazaikinskii and T. Ya. Tudorovskii, New formulas of Maslov’s canonical operator in a neighborhood of focal points and caustics in 2D semiclassical asymptotics, Theoretical and Mathematical Physics, Vol. 177, No. 3, pp. 1579-1605, 2013.
3) 3. S. Yu. Dobrokhotov, G. N. Makrakis and V. E. Nazaikinski, Fourier integrals and a new representation of Maslov’s canonical operator near caustics, Proceedings of the International Conference "Spectral Theory and Differential Equations (STDE-2012)" in honor of Vladimir A. Marchenko’s 90th birthday, AMS, 2014 (to appear).
4) 4. S.Yu Dobrokhotov, G.N. Makrakis and V.E.Nazaiksinskii Maslov’s Canonical Operator, Hörmander’s Formula, and Localization of Berry–Balazs’ Solution in the Theory of Wave Beams, Conference "Mathematical Physics. Vladimirov-90" dedicated to the 90th anniversary of academician V. S. Vladimirov, November 13, 2013 (http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=7963)
Michael Plexousakis
The paper entitled Crank-Nicolson finite element discretizations for a two dimensional linear Schrodinger equation posed in a noncylindrical domain, with D. C. Antonopoulou, G. D. Karali, and G. E. Zouraris, studies space-time finite element methods for the linear Schrodinger equation, a main topic in my research on numerical methods for partial differential equations. Motivated by the paraxial narrowangle approximation of the Helmholtz equation in domains of variable topography, we considered an initial- and boundary-value problem for a general Schrodinger type equation posed on a two-dimensional noncylindrical domain with mixed boundary conditions. The problem was transformed into an equivalent one posed on a rectangular domain and we approximated its solution by a CrankNicolson nite element method. For the proposed numerical method, we derived an optimal order error estimate in the L2 norm and proved a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments were presented which verified the optimal order of convergence of the method.
The paper entitled Helmholtz equation with artificial boundary conditions in a two-dimensional waveguide, with D. A. Mitsoudis and Ch. Makridakis is the second in a series of papers devoted on numerical methods for the Helmholtz equation. In this work we developed and analysed a model for wave propagation based on the Helmholtz equation in the context of a realistic environment widely used in applications, especially underwater acoustics. We considered a two-dimensional waveguide in Cartesian coordinates consisting of a homogeneous water column confined between a horizontal pressure-release sea surface and an acoustically soft sea floor. The original infinite domain was truncated with two artificial boundaries and we formulated a model in the resulting bounded domain by introducing suitable nonlocal conditions on these artificial boundaries. The proposed model simulated efficiently the effect of the source and the backscattered field from the rest of the waveguide and proved appropriate for finite element computations.This work dealt with the well-posedness of the model. We showed stability estimates in appropriate Sobolev norms which, in turn, implied existence and uniqueness of the solution.
Our investigation on mesoscopic models of pattern{forming systems, with N. M. Abukhdeir, D. G. Vlachos, and M. Katsoulakis resulted in a publication in the Journal of Computational Physics entitled Long-time integration methods for mesoscopic models of pattern{forming systems. In this work spectral methods for simulation of a mesoscopic diffusion model of surface pattern formation were evaluated with the utter goal of achieving long simulation times. Backwards-differencing time-integration, coupled with an underlying Newton Krylov nonlinear solver (SUNDIALSCVODE), was found to substantially accelerate simulations, without the typical requirement of preconditioning. Quasi-equilibrium simulations of patterned phases predicted by the model were shown to agree well with linear stability analysis. Simulation results of the effect of repulsive particle-particle interactions on pattern relaxation time and short/long-range order were discussed and analyzed extensively.
List with the most significant publications
1) Crank-Nicolson finite element discretisations for a two-dimensional linear Schrodinger equation posed in a noncylindrical domain , with D. C. Antonopoulou, G. D. Karali, and G. E. Zouraris. To appear in Math. Comp.
2) Helmholtz equation with artificial boundary conditions in a two-dimensional waveguide, with D. A. Mitsoudis and Ch. Makridakis. SIAM J. Math. Anal. 44 (2012), no. 6, 4320- 4344.
3) Long-time integration methods for mesoscopic models of pattern{forming systems, with N. M. Abukhdeir, D. G. Vlachos, and M. Katsoulakis. J. Comp. Phys. 230 (2011), no. 14, 5704-5715.
Emmanouil Skarsoulis
Sensitivity of underwater acoustic observables.
The work on sensitivity kernels over the last 4 years focused on the study of finite-frequency observables such as peak arrival times, amplitudes and phase arrival times and their sensitivity behavior using wave-theoretic modeling approaches. A significant result deals with the long-range behavior of vertical sensitivity kernels of peak arrival times. While at short propagation ranges the wave-theoretic kernels at low frequencies exhibit significant deviations from the corresponding ray-theoretic ones (high-frequency asymptotic approximations), they converge to the latter with increasing range even at low frequencies. This result is particularly important since ray theory has been commonly used for the analysis of long-range ocean acoustic tomography experiments. An asymptotic approach for finite frequencies and long ranges has been developed, which confirms the above-mentioned behavior and provides a method for estimation of the ranges beyond which the ray-theoretic sensitivity kernels are sufficient, depending on the ocean environment (sound-speed distribution). Further results refer to the study of phase arrivals. Phase arrivals offer increased time resolution compared to peak arrivals, controlled by the central frequency, rather than the source bandwidth. It has been shown that phase arrival times exhibit a stable and predictable perturbation behavior, based on sensitivity kernels, and can be used as efficient observables in short- and medium-range propagation experiments.
List with the most significant publications
1) E. Skarsoulis, B. Cornuelle, M. Dzieciuch, Perturbation relations and sensitivity kernels for arrival amplitude and phase, Proc. 11th European Conf. on Underwater Acoustics, Edinburgh, 2012.
2) E. Skarsoulis, B. Cornuelle, M. Dzieciuch, Sensitivity behavior of phase and peak arrival times, Proc. 1st International Conference on Underwater Acoustics, pp. 1221-1228, Corfu, 2013.
Georgios Zouraris
Numerical solution of fourth-order SPDES: we manage to prove the convergence analysis of fully discrete finite element approximations of the solution to some linearized versions of the stochastic Cahn-Hillird equation drived by an additive space-time white noise or the space derivative of an additive space-time white noise.
Numerical approximation of Dirac type equations: we presented an existence and uniqueness theory for the solution to a nonlinear Dirac system along with the error analysis of an new implicit/explicit finite difference method for its approximation. We show the performance of the proposed method by applying it to standard model problems.
Numerical approximation of underwater acoustics problems: we propose numerical methods for the approximation of the solution of pde models known in the underwater acoustic area as ‘parabolic approximations’ when the sea environment has variable bottom. For the proposed methods we prove optimal order error estimates and show their performance by exposing results from numerical experiments.
Numerical solution of financial models: we approximate the solution of an HJM term structure financial model by Monte-Carlo Euler methods providing a strong approximation error eanalysis along with a-posteriori estimates for the week approximation error. Results from numerical experiments also included.
List with the most significant publications
1) Kossioris, Georgios T.and Zouraris, Georgios E. (2010) Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise. ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2). pp. 289-322.
2) Bournaveas, Nikolaos and Zouraris, Georgios E. (2012)Theory and numerical approximations for a nonlinear 1+1 Dirac system. ESAIM: Mathematical Modelling and Numerical Analysis , 46 (4). pp. 841-874.
3) Antonopoulou, D. C. and Karali, Georgia D and Plexousakis, M. and Zouraris, Georgios E. (2011) Crank-Nicolson finite element discretizations for a 2D linear Schrödinger-type equation posed in a noncylindrical domain. Mathematics of Computations (accepted on Oct 10, 2013).
Potential Impact:
Integration of the DApplM-UoC in the European Research Area.
A main benefit of the ACMAC project has been the significant increase in visibility for the group comprising the Department of Applied Mathematics of the University of Crete (DApplM-UoC), and the integration in the
European Research Area.
This was significantly enhanced by the partnership and interaction between the Center ACMAC and leading Centers of Applied Mathematics in Europe. A key impact of this partnership is the acquisition of the necessary know-how in building, promoting and maintaining a Center of Excellence. Even though the current financial crisis in Greece makes the financing of such a project through Greek sources very difficult (if not impossible), it is expected that the benefits of this partnership and collaboration will reach deep into the future in several ways. We expect that the academic bond of DApplM-UoC and the Center with these Centers of Excellence will lead to joint activities, creation of research networks, and higher mobility especially for young researchers, through future common proposals and actions.
Another important impact of the project has been the overall activity around the thematic programs, conferences and workshops. The resulting interaction between group members and incoming experienced researchers, senior experts and distinguished scientists brought valuable know-how to the group members and initiated interactions with leading researchers and scientists of Europe. This gave the opportunity to young members of our group to get acquainted with top scientists and be exposed to cutting edge research and innovation, and to gain valuable professional experiences in a diversified scientific environment.
Enhanced visibility is an absolute necessity for the better integration of DApplM-UoC in the European Research Area and in the international research community, and all actions contributed significantly towards this goal. Notably, the Conference activities in Crete and the Thematic Programs had far more impact on the visibility of our department than the usual participation of our faculty in workshops and big Conferences outside Greece. Also, it enhanced significantly the carreer prospects of the postdoctoral fellows hired in ACMAC and most of them continue successfully in their academic carreers having benefited significantly from the exposure to international contacts that they acquired during their tenure at ACMAC.
Upgrading the Research Capacity and Capability of DApplM-UoC.
The ACMAC project also had an essential impact to the improvement of personnel capability through the interaction between group members and young and senior scientists visiting ACMAC and through the secondments with the partner Centers of Excellence.
The project also gave the opportunity to involve to activities in Greece young and senior Greek scientists with careers abroad, striving towards a reversal of a brain-drain process that characterized Greek science in the past decades. Many of them were involved in multiple Centre activities and initiated or further developed contacts with Greek based scientists. The original goal of attracting them back to Greece was unfortunately not fulfilled due to the extraneous circumstances of the finances of the Greek state. Nevertheless, the contacts will remain and might present an opportunity for future hires when the circumstances change.
Third, this project permitted us to upgrade and reinforce our scientific equipment. A goof part of our research needs require heavy computing and therefore the acquired “cluster facility” is essential to our group in order to continue the development of state-of-the-art numerical methods for realistic application problems.
Finally, the development of the program and many of our activities acted as a bridge between team members and applied scientists from other disciplines like computer science and engineering thus initiating fruitful interactions and collaborations. Their participation in Thematic Programs as co-organizers or visitors catalyzed their interactions with ACMAC members. Also, some activities served as seed to initiate collaborations of scientific sub-communities and researchers are planning their continuation through other venues.
Improved Research Capacity and Regional Economic and Social Development.
The project impacted Regional Research Institutions in several ways.
It improved the research capabilities of the University of Crete. Members of the Applied Mathematics Department initiated a Masters program in Computational and Applied Mathematics. Also the Ph.D. program in Applied Mathematics became much more attractive and now has a number of Ph.D. students doing thesis work with members of the group.
Second, the existence of a major research facility in the Region, the Foundation of Research and Technology (FORTH) with its strong applied sciences/engineering and Research & Development teams, provided unique opportunities for mutual growth and collaboration with the members of the ACMAC center. Indeed, existing ties got stronger and there exists a closer integration of the two groups. Areas of collaboration emerged from the activity at ACMAC that includes polymers, biomechanics, nanotechnologies, seismology, and medical imaging.
Improved potential to participate in FP7 projects.
In the course of the ACMAC project, our faculty applied and was successful in many projects financed through the structural funds, and currently support their research through these projects. ACMAC has been instrumental in helping to develop the contacts and expertise to be successful in these projects. This blend of basic and applied research is considered to be of crucial importance for the ability of our group to attract funding in the future and continue research and development in this area.
Broader Impact.
The proposed activities and training gave young researchers at ACMAC new opportunities to pursue careers both in academia or industry, integrating in the EU knowledge-based economy.
By its very nature research in applied mathematics has indirect impact to technological innovation, i.e. techniques are introduced which then are taken up by developers who tailor them to the needs of the society. However, understanding the needs of applications is important in directing our research in the right directions and this is something we pay a lot of attention. The ACMAC center focused its activity around applied research problems ranging from polymers and nanomaterials, to seismology and medical imaging. These are areas of great interest as they constitute European as well as global priorities due to their potential economic and societal impact.
The ACMAC project was also of benefit to the international scientific community. The organization of Thematic Programs at ACMAC was designed in order to take into account related activity in other Institutes in EU and USA. Our faculty has traditionally been involved in many of these external activities as participants or organizers, and hence they were familiar with other upcoming yearly programs and organized our own Thematic Programs so as to avoid overlaps and conflicts, and enhance synergies where possible, emphasizing the ACMAC trademark in the synergy of Mathematical Modeling, Analysis and Computation.
Overall, the Archimedes Center for Modeling, Analysis, and Computation had a transformative impact on the evolution of Applied Mathematics in Crete and more generally in Greece, with ACMAC becoming an internationally recognized center in modern computational and applied mathematics. We hope this will have future implications in enhancing the regional development and attractiveness of European Science in the whole region of South-Eastern Europe.
List of Websites:
Web page of ACMAC. The web page of the project is:
www.acmac.uoc.gr