Nonlinear methods for model reduction
Abstract
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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Mar 01, 2021
- Source ID
- 10.1051/m2an/2020057
Entities
People
- Albert Cohen
- Andrea Bonito
- Diane Guignard
- Guergana Petrova
- Peter Jantsch
- Ronald DeVore
Organizations
- Isaac Newton Institute
- National Science Foundation
- Office of Naval Research
- Swiss National Science Foundation