Nonlinear reduced models for parametric PDEs

Diane Guignard
University of Ottawa

We consider model reduction methods for parametric partial differential equations. The usual approach to model reduction is to construct a low dimensional linear space which accurately approximates the parameter-to-solution map, and use it to build an efficient forward solver. However, the construction of a suitable linear space is not always feasible numerically. It is well-known that nonlinear methods may provide improved efficiency. In a so-called library approximation, the idea is to replace the linear space by a collection of linear (or affine) spaces of smaller dimension.
In this talk, we first review standard linear methods for model reduction. Then, we present a strategy which can be used to generate a nonlinear reduced model, namely a library based on piecewise (Taylor) polynomials. We provide an analysis of the method, in particular the derivation of an upper bound on the size of the library, and illustrate its performance through several numerical experiments.