Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations

TL;DR

This paper explores a nonlinear approximation method for high-dimensional PDEs, establishing convergence for a variational approach and highlighting challenges in a non-variational version, thus advancing understanding of solving complex PDEs.

Contribution

It links a nonlinear approximation approach to greedy algorithms and demonstrates convergence for a variational method, revealing difficulties in a non-variational approach.

Findings

Variational approach converges for the Poisson equation.

Non-variational approach faces theoretical and numerical challenges.

Several open issues motivate further research.

Abstract

We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the approach and the greedy algorithms of approximation theory studied e.g. in [R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical case of the Poisson equation, we show that a variational version of the approach, based on minimization of energies, converges. On the other hand, we show various theoretical and numerical difficulties arising with the non variational version of the approach, consisting of simply solving the first order optimality equations of the problem. Several unsolved issues are indicated in order to motivate further research.