We consider model reduction methods for parametric/random partial differential equations. The usual approach to model reduction is to construct a linear space $V_n$ of (hopefully low) dimension $n$ which accurately approximates the parameter-to-solution map, and then use it to build an efficient forward solver. However, the construction of a suitable linear space is not always feasible. It is well-known that numerical methods based on nonlinear approximation outperform linear methods in many contexts. In a so-called library approximation, the idea is to replace the linear space $V_n$ by a collection of linear/affine spaces $V^1,\ldots,V^N$ of dimension $m \le n$.

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, illustrate its performance through numerical experiments, and discuss possible extensions.

This is joint work with A. Bonito, A. Cohen, R. DeVore, P. Jantsch, and G. Petrova.