Periodic Reporting for period 1 - RandomMultiScales (Computational Random Multiscale Problems)
Okres sprawozdawczy: 2020-08-01 do 2022-01-31
In this project, we aim to construct and analyze computational methods for multiscale problems under randomness and disorder. Our research is driven by astonishing physical phenomena such as cloaking with metamaterials or localization of quantum waves in disordered media. However, the underlying mathematical models of random partial differential equations are characterized by a complex interplay of effects on many non-separable or even a continuum of characteristic scales. Attempting to solve them in a direct numerical simulation easily exceeds current computational capabilities by several orders of magnitude. Therefore, we seek a new generation of computational multiscale methods that account for randomness and disorder in a hierarchical and adaptive manner. For the design and numerical analysis of such methods, we intertwine multiscale modelling and simulation, uncertainty quantification, and computational physics. Our research plan envisions the following four objectives:
(A) Numerical stochastic homogenisation with arbitrary random coefficients
(B) Homogenisation-based Multilevel Monte Carlo solvers
(C) Multiscale numerical scattering in high-contrast random media
(D) Multiscale simulation of quantum phase transitions
Parts (A)-(B) focus on prototypical model problems, illuminating the essential challenges of random multiscale problems. To ensure that the corresponding developments are of relevance in applications, modules (C)-(D) are devoted to their generalization to advanced linear and non-linear problems in the context of wave phenomena in disordered media. The project aims to provide reliable computational methods for wave simulation in extreme parameter regimes as they occur in metamaterials or phase transitions of ultracold bosonic gases.
(A) Numerical stochastic homogenisation with arbitrary random coefficients
(B) Homogenisation-based Multilevel Monte Carlo solvers
(C) Multiscale numerical scattering in high-contrast random media
(D) Multiscale simulation of quantum phase transitions
Parts (A)-(B) focus on prototypical model problems, illuminating the essential challenges of random multiscale problems. To ensure that the corresponding developments are of relevance in applications, modules (C)-(D) are devoted to their generalization to advanced linear and non-linear problems in the context of wave phenomena in disordered media. The project aims to provide reliable computational methods for wave simulation in extreme parameter regimes as they occur in metamaterials or phase transitions of ultracold bosonic gases.
At this stage of the project, we have obtained numerous results for all the above objectives (A)-(D).
First, in “A priori error analysis of a numerical stochastic homogenization method” we bridged the recent breakthrough of numerical homogenization with arbitrary rough coefficients and the celebrated quantitative theory of stochastic homogenisation and derived rigorous a priori error estimates.
The survey article “Numerical homogenization beyond scale separation” reviews modern numerical homogenization methods that can accurately handle problems with a continuum of scales. It serves as a template for more general multi-physics problems.
A different approach, presented in “Operator compression with deep neural networks” uses deep neural networks trained to compress fine scale information contained in the continuous operator to a finite-dimensional sparse object. This then replicates the effective behavior of the solution on a macroscopic scale of interest, even in the presence of unresolved oscillations of the underlying coefficient.
Taking a step back, in “Super-localization of elliptic multiscale problems” we have succeeded in finding a novel localization strategy that outperforms the state-of-the-art exponential decay of the Localized Orthogonal Decomposition (LOD). The new problem-adapted basis decays super-exponentially, promising to pave the way to more sophisticated methods that account for randomness and disorder.
Moreover, in “Multi-resolution Localized Orthogonal Decomposition for Helmholtz problems” we introduced a new multi-resolution LOD that merges the concepts of LOD and operator-adapted wavelets (gamblets). This method marks the first step in the direction of a homogenization-based multilevel Monte Carlo scheme, as it shows the potential of a LOD in a multilevel context.
As we intend to provide physically relevant numerical schemes, we applied the above methods to multiscale scattering problems represented by the Helmholtz equation in “Super-localized Orthogonal Decomposition for high-frequency Helmholtz problems” as well as “Rational-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots”. The results for deterministic and stochastic materials are promising and already cover a wide range of the frequency spectrum. Still, the combination with high contrast as well as the simulation of complete frequency bands is challenging.
Finally, we move closer to the simulation of multiscale quantum phase transitions. The simplest, yet no less challenging, model equation for the localization behavior of Bose-Einstein condensates in disorder potentials is the time-dependent Gross-Pitaevskii equation (GPE), whose steady states are characterized by the Gross-Pitaevskii eigenvalue problem (GPEVP). To this end, in “The J-method for the Gross-Pitaevskii eigenvalue problem” we analyzed the application of a slightly modified J-method to the GPEVP and derived a global convergence result. An additional quantified convergence analysis is performed in “Energy-adaptive Riemannian optimization on the Stiefel manifold” where we provided a novel Riemannian gradient descent method that allows finding critical points of energy minimization problems characterized by nonlinear eigenvector problems such as the GPE. Further mathematical and numerical insights we gained in the case of the GPEVP allow the prediction of physically relevant regimes where localization of ground states may be observed experimentally, see “Localization and delocalization of ground states of Bose-Einstein condensates under disorder”.
First, in “A priori error analysis of a numerical stochastic homogenization method” we bridged the recent breakthrough of numerical homogenization with arbitrary rough coefficients and the celebrated quantitative theory of stochastic homogenisation and derived rigorous a priori error estimates.
The survey article “Numerical homogenization beyond scale separation” reviews modern numerical homogenization methods that can accurately handle problems with a continuum of scales. It serves as a template for more general multi-physics problems.
A different approach, presented in “Operator compression with deep neural networks” uses deep neural networks trained to compress fine scale information contained in the continuous operator to a finite-dimensional sparse object. This then replicates the effective behavior of the solution on a macroscopic scale of interest, even in the presence of unresolved oscillations of the underlying coefficient.
Taking a step back, in “Super-localization of elliptic multiscale problems” we have succeeded in finding a novel localization strategy that outperforms the state-of-the-art exponential decay of the Localized Orthogonal Decomposition (LOD). The new problem-adapted basis decays super-exponentially, promising to pave the way to more sophisticated methods that account for randomness and disorder.
Moreover, in “Multi-resolution Localized Orthogonal Decomposition for Helmholtz problems” we introduced a new multi-resolution LOD that merges the concepts of LOD and operator-adapted wavelets (gamblets). This method marks the first step in the direction of a homogenization-based multilevel Monte Carlo scheme, as it shows the potential of a LOD in a multilevel context.
As we intend to provide physically relevant numerical schemes, we applied the above methods to multiscale scattering problems represented by the Helmholtz equation in “Super-localized Orthogonal Decomposition for high-frequency Helmholtz problems” as well as “Rational-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots”. The results for deterministic and stochastic materials are promising and already cover a wide range of the frequency spectrum. Still, the combination with high contrast as well as the simulation of complete frequency bands is challenging.
Finally, we move closer to the simulation of multiscale quantum phase transitions. The simplest, yet no less challenging, model equation for the localization behavior of Bose-Einstein condensates in disorder potentials is the time-dependent Gross-Pitaevskii equation (GPE), whose steady states are characterized by the Gross-Pitaevskii eigenvalue problem (GPEVP). To this end, in “The J-method for the Gross-Pitaevskii eigenvalue problem” we analyzed the application of a slightly modified J-method to the GPEVP and derived a global convergence result. An additional quantified convergence analysis is performed in “Energy-adaptive Riemannian optimization on the Stiefel manifold” where we provided a novel Riemannian gradient descent method that allows finding critical points of energy minimization problems characterized by nonlinear eigenvector problems such as the GPE. Further mathematical and numerical insights we gained in the case of the GPEVP allow the prediction of physically relevant regimes where localization of ground states may be observed experimentally, see “Localization and delocalization of ground states of Bose-Einstein condensates under disorder”.
All the results obtained so far go beyond the state-of-the-art by introducing new methods and their numerical analysis or by identifying news surprising links between existing techniques. In addition to our achievements to date, the new super-localization method for numerical homogenization marks a highlight as it
sheds light on the long-standing question in numerical analysis, whether truly local numerical homogenization is possible without logarithmic overhead.
From here, we are confident that by the end of the project we will be able to overcome the current limitations of fast methods for UQ of PDEs with random coefficients at short correlation lengths and take an important step toward understanding how to deal with randomness and high contrast in indefinite scattering problems at high frequencies.
Moreover, we believe that we will provide a methodology for multiscale simulation of Bose-Einstein condensates that goes beyond current capabilities and even allows the exploration of new physical phenomena.
sheds light on the long-standing question in numerical analysis, whether truly local numerical homogenization is possible without logarithmic overhead.
From here, we are confident that by the end of the project we will be able to overcome the current limitations of fast methods for UQ of PDEs with random coefficients at short correlation lengths and take an important step toward understanding how to deal with randomness and high contrast in indefinite scattering problems at high frequencies.
Moreover, we believe that we will provide a methodology for multiscale simulation of Bose-Einstein condensates that goes beyond current capabilities and even allows the exploration of new physical phenomena.