Final Report Summary - APPROCEM (The Approximation Problem in Computational ElectroMagnetics)
The EU funded research project ApProCEM has been focused on various aspects of Grothendieck's approximation problem. This is a challenging question in the areas of pure and applied mathematics, playing a particularly relevant role in functional and numerical analysis, with respect to the solution of PDEs and other operator equations, where contributions related to the problem are expected to provide significant insights for improving the performance of approximation schemes of current use in computational electromagnetics and other fields, such as reduced basis methods and finite differences. In this respect, our results can help sheding light on a number of questions, provide a deeper understanding of their nature, and lead to new promising trends of research.
Specifically, the work by the fellow, Dr Salvatore Tringali, has mostly focused on the intimate interplay between approximation theory and additive theory, by the development of a very general framework which, somewhat surprisingly, brings under the umbrella of a unifying vision fundamental aspects of mathematical theories so far developed independently from each other, namely the theory of normed spaces and the theory of measure spaces. This has been achieved by the introduction of several breakthrough ideas, most notably including plots (sort of non-unital, non-associative categories with a relaxed composition law, where it is still possible to define key concepts such as isomorphisms, equivalences, funtoriality, limits, and colimits, etc., and which represent, among the other things, a comprehensive generalization of Mitchell's semicategories, Ehresmann's multiplicative graphs and Gabriel's quivers), semantic domains over plots (a broad attempt to re-invent the notion itself of "structure" by extending the scope of categorical logic far behind its current limits), premorphisms and prealgebraic theories (a generalization of morphisms and algebraic theories where equality is replaced by an arbitrary preorder and models are interpreted internally to a semantic domain), and normed structures (a remarkable generalization of normed spaces, normed groups, normed algebras, and analogous structures where the norm is viewed as a premorphism from the models of a prealgebraic theory into a fixed "target structure" such as the non-negative real numbers, endowed with their usual order and operations, or the tropical semiring).
Due to the depth and the broadness of many of these ideas, much has still to be done, and the research on these topics is only in an embryonic form at present, but the perspectives are exciting, for a systematic development of the theory would make it possible to apply methods and results otherwise restricted to much "richer settings" to significantly larger classes of problems.
A concrete example in this sense is a generalization of the classical Riesz's lemma for the closed unit ball of a (real or complex) normed space to the non-abelian setting of seminormed divisible groups (under a few additional restrictions); see [4]. In particular, the result is used to prove an extension of the well-known fact that infinite-dimensional normed spaces do not have an invariant Haar measure.
Another example is the extension of the classical Cauchy-Davenport theorem for groups of prime orders to the much more abstract context of cancellative semigroups [5, 6], which has led to the generalization and the strengthening of a number of analogous results from the additive theory of groups, such as Chowla's and Pillai's theorems on cyclic groups and Kemperman's inequality for torsion-free groups. The same work has also provided the solution to a conjecture related to Karolyi's theorem for finite groups and to an elementary proof of the result that avoids the Feit-Thompson theorem (contrarily to any other proof so far available).
On another hand, the work by the scientist in charge, Professor Yvon Maday, has mostly focused on the applied aspects of the approximation problem, with significant contributions to the existing literature on reduced basis methods, going far beyond the scope of the original project statement.
In [1], for instance, the authors have proposed, on the one hand, a reduced basis well adapted, out of a series of numerical simulations of a parameter dependent problem, to the approximation of the problem of interest, and on the other hand, an algorithm to approximate the solution for any value of the parameter, in an accurate and fast way. The approach is also consistent, insomuch as it gives back the solutions chosen as an entry of the routine.
A second paper along these lines, coming out from a collaboration with H. Herrero and F. Pla, has led to a new interpolation procedure within the frame of "sets of small complexity", where the small complexity of a set is measured through the Kolmogorov n-width. In particular, the authors have been able to provide a monitoring approach, based on numerical simulations and data assimilation, to extract pertinent information from a given series of data and recover a monitored signal.
Lastly, a joint work with B. Stamm and M. Bebendorf [3] has given a systematic presentation and pointed out some intimate connections between different approaches of model reduction to provide the proper minimal basis set representing a class of signals under some a priori hypothesis.
BIBLIOGRAPHY.
[1] M. Bebendorf, Y. Maday, and B. Stamm, "Comparison of some Reduced Representation Approximations", to appear.
[2] H. Herrero, Y. Maday, and F. Pla, "RB (Reduced basis) for RB (Rayleigh-Bénard)", Computer Methods in Applied Mechanics and Engineering, Vol. 261-262, pp. 132-141 (July 2013).
[3] O. Mula, Y. Maday and Gabriel Turinici, "A priori convergence of the Generalized Empirical Interpolation Method", to appear.
[4] S. Tringali, "Seminormed divisible groups and a generalization of Riesz's lemma", under review.
[5] S. Tringali, "A Cauchy-Davenport type theorem for semigroups", to appear on Uniform Distribution Theory.
[6] S. Tringali, "A Cauchy-Davenport type theorem for semigroups, II", under review.
Specifically, the work by the fellow, Dr Salvatore Tringali, has mostly focused on the intimate interplay between approximation theory and additive theory, by the development of a very general framework which, somewhat surprisingly, brings under the umbrella of a unifying vision fundamental aspects of mathematical theories so far developed independently from each other, namely the theory of normed spaces and the theory of measure spaces. This has been achieved by the introduction of several breakthrough ideas, most notably including plots (sort of non-unital, non-associative categories with a relaxed composition law, where it is still possible to define key concepts such as isomorphisms, equivalences, funtoriality, limits, and colimits, etc., and which represent, among the other things, a comprehensive generalization of Mitchell's semicategories, Ehresmann's multiplicative graphs and Gabriel's quivers), semantic domains over plots (a broad attempt to re-invent the notion itself of "structure" by extending the scope of categorical logic far behind its current limits), premorphisms and prealgebraic theories (a generalization of morphisms and algebraic theories where equality is replaced by an arbitrary preorder and models are interpreted internally to a semantic domain), and normed structures (a remarkable generalization of normed spaces, normed groups, normed algebras, and analogous structures where the norm is viewed as a premorphism from the models of a prealgebraic theory into a fixed "target structure" such as the non-negative real numbers, endowed with their usual order and operations, or the tropical semiring).
Due to the depth and the broadness of many of these ideas, much has still to be done, and the research on these topics is only in an embryonic form at present, but the perspectives are exciting, for a systematic development of the theory would make it possible to apply methods and results otherwise restricted to much "richer settings" to significantly larger classes of problems.
A concrete example in this sense is a generalization of the classical Riesz's lemma for the closed unit ball of a (real or complex) normed space to the non-abelian setting of seminormed divisible groups (under a few additional restrictions); see [4]. In particular, the result is used to prove an extension of the well-known fact that infinite-dimensional normed spaces do not have an invariant Haar measure.
Another example is the extension of the classical Cauchy-Davenport theorem for groups of prime orders to the much more abstract context of cancellative semigroups [5, 6], which has led to the generalization and the strengthening of a number of analogous results from the additive theory of groups, such as Chowla's and Pillai's theorems on cyclic groups and Kemperman's inequality for torsion-free groups. The same work has also provided the solution to a conjecture related to Karolyi's theorem for finite groups and to an elementary proof of the result that avoids the Feit-Thompson theorem (contrarily to any other proof so far available).
On another hand, the work by the scientist in charge, Professor Yvon Maday, has mostly focused on the applied aspects of the approximation problem, with significant contributions to the existing literature on reduced basis methods, going far beyond the scope of the original project statement.
In [1], for instance, the authors have proposed, on the one hand, a reduced basis well adapted, out of a series of numerical simulations of a parameter dependent problem, to the approximation of the problem of interest, and on the other hand, an algorithm to approximate the solution for any value of the parameter, in an accurate and fast way. The approach is also consistent, insomuch as it gives back the solutions chosen as an entry of the routine.
A second paper along these lines, coming out from a collaboration with H. Herrero and F. Pla, has led to a new interpolation procedure within the frame of "sets of small complexity", where the small complexity of a set is measured through the Kolmogorov n-width. In particular, the authors have been able to provide a monitoring approach, based on numerical simulations and data assimilation, to extract pertinent information from a given series of data and recover a monitored signal.
Lastly, a joint work with B. Stamm and M. Bebendorf [3] has given a systematic presentation and pointed out some intimate connections between different approaches of model reduction to provide the proper minimal basis set representing a class of signals under some a priori hypothesis.
BIBLIOGRAPHY.
[1] M. Bebendorf, Y. Maday, and B. Stamm, "Comparison of some Reduced Representation Approximations", to appear.
[2] H. Herrero, Y. Maday, and F. Pla, "RB (Reduced basis) for RB (Rayleigh-Bénard)", Computer Methods in Applied Mechanics and Engineering, Vol. 261-262, pp. 132-141 (July 2013).
[3] O. Mula, Y. Maday and Gabriel Turinici, "A priori convergence of the Generalized Empirical Interpolation Method", to appear.
[4] S. Tringali, "Seminormed divisible groups and a generalization of Riesz's lemma", under review.
[5] S. Tringali, "A Cauchy-Davenport type theorem for semigroups", to appear on Uniform Distribution Theory.
[6] S. Tringali, "A Cauchy-Davenport type theorem for semigroups, II", under review.