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Article Dans Une Revue ESAIM: Mathematical Modelling and Numerical Analysis Année : 2021

Nonlinear methods for model reduction

Andrea Bonito
  • Fonction : Auteur
Ronald Devore
  • Fonction : Auteur
Diane Guignard
  • Fonction : Auteur
Peter Jantsch
  • Fonction : Auteur
Guergana Petrova
  • Fonction : Auteur

Résumé

Typical model reduction methods for parametric partial differential equations construct a linear space V n which approximates well the solution manifold M consisting of all solutions u ( y ) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n -term approximation, and certain tree-based methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space V n by a nonlinear space Σ n . Little is known in terms of their performance guarantees, and most existing numerical experiments use a parameter dimension of at most two. In this work, we make a step towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation, we give a first comparison of their performance with the performance of standard linear approximation for any compact set. We then study these methods for solution manifolds of parametrized elliptic PDEs. We study a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and give performance guarantees with respect to accuracy of approximation versus m and N .
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Dates et versions

hal-03884662 , version 1 (05-12-2022)

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Andrea Bonito, Albert Cohen, Ronald Devore, Diane Guignard, Peter Jantsch, et al.. Nonlinear methods for model reduction. ESAIM: Mathematical Modelling and Numerical Analysis, 2021, 55 (2), pp.507-531. ⟨10.1051/m2an/2020057⟩. ⟨hal-03884662⟩
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