Infinite-dimensional moment-SOS hierarchy for nonlinear partial differential equations

TL;DR

This paper introduces an infinite-dimensional moment-SOS hierarchy that reformulates nonlinear PDEs as linear optimization problems on measures, enabling convergence proofs and approximate solutions through finite-dimensional relaxations, demonstrated on a heat equation example.

Contribution

It develops a novel infinite-dimensional hierarchy using SOS techniques to solve nonlinear PDEs via linear optimization on measures, with proven convergence.

Findings

Hierarchy converges to true solutions of nonlinear PDEs.

Finite relaxations provide accurate approximate solutions.

Numerical experiments validate the approach on a heat equation with nonlinear perturbation.

Abstract

We formulate a class of nonlinear {evolution} partial differential equations (PDEs) as linear optimization problems on moments of positive measures supported on infinite-dimensional vector spaces. Using sums of squares (SOS) representations of polynomials in these spaces, we can prove convergence of a hierarchy of finite-dimensional semidefinite relaxations solving approximately these infinite-dimensional optimization problems. As an illustration, we report on numerical experiments for solving the heat equation subject to a nonlinear perturbation.