Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

TL;DR

This paper introduces a new multilevel Picard approximation scheme for high-dimensional semilinear elliptic PDEs, demonstrating it overcomes the curse of dimensionality similar to previous methods for parabolic PDEs.

Contribution

The paper develops and analyzes a novel MLP scheme specifically for semilinear elliptic PDEs, extending the applicability of overcoming the curse of dimensionality.

Findings

The new MLP scheme effectively approximates solutions to high-dimensional elliptic PDEs.

The scheme's complexity grows polynomially with dimension, not exponentially.

It generalizes previous approaches used for parabolic PDEs.

Abstract

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.