Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

TL;DR

This paper demonstrates that for certain semilinear heat equations with Lipschitz nonlinearities, a multilevel Picard approximation method can efficiently approximate solutions without suffering from the curse of dimensionality, unlike previous general cases.

Contribution

It proves polynomial growth in computational effort for semilinear PDEs with Lipschitz nonlinearities, partially resolving an open problem about the curse of dimensionality.

Findings

Multilevel Picard approximations achieve polynomial complexity in dimension and accuracy

The method applies to semilinear heat equations with Lipschitz nonlinearities

The approach extends the class of PDEs solvable efficiently in high dimensions

Abstract

For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows polynomially both in the dimension and in the reciprocal of the required accuracy.