Volume 55, Number 2, March-April 2021
|507 - 531
|16 March 2021
Nonlinear methods for model reduction
Department of Mathematics, Texas A&M University, College Station, TX 77840, USA
2 Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris, France
* Corresponding author: firstname.lastname@example.org
Accepted: 3 August 2020
Typical model reduction methods for parametric partial differential equations construct a linear space Vn 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 Vn by a nonlinear space Σn. 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.
Mathematics Subject Classification: 41A10 / 41A58 / 41A63 / 65N15
Key words: Parametric PDEs / reduced modeling / piecewise polynomials
© EDP Sciences, SMAI 2021
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