Typical model reduction methods for parametric partial differential equations construct a linear space V$$ which approximates well the solution manifold M consisting of all solutions u(y) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n-term approximation, and certain tree-based methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space V$$ by a nonlinear space Σ$$. 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.
Keywords: Parametric PDEs, reduced modeling, piecewise polynomials
@article{M2AN_2021__55_2_507_0,
author = {Bonito, Andrea and Cohen, Albert and Devore, Ronald and Guignard, Diane and Jantsch, Peter and Petrova, Guergana},
title = {Nonlinear methods for model reduction},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {507--531},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {2},
doi = {10.1051/m2an/2020057},
mrnumber = {4230422},
zbl = {1476.41004},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020057/}
}
TY - JOUR AU - Bonito, Andrea AU - Cohen, Albert AU - Devore, Ronald AU - Guignard, Diane AU - Jantsch, Peter AU - Petrova, Guergana TI - Nonlinear methods for model reduction JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 507 EP - 531 VL - 55 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020057/ DO - 10.1051/m2an/2020057 LA - en ID - M2AN_2021__55_2_507_0 ER -
%0 Journal Article %A Bonito, Andrea %A Cohen, Albert %A Devore, Ronald %A Guignard, Diane %A Jantsch, Peter %A Petrova, Guergana %T Nonlinear methods for model reduction %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 507-531 %V 55 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020057/ %R 10.1051/m2an/2020057 %G en %F M2AN_2021__55_2_507_0
Bonito, Andrea; Cohen, Albert; Devore, Ronald; Guignard, Diane; Jantsch, Peter; Petrova, Guergana. Nonlinear methods for model reduction. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 507-531. doi: 10.1051/m2an/2020057
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