Multilevel Picard approximations overcome the curse of dimensionality when approximating semilinear heat equations with gradient-dependent nonlinearities in $L^p$-sense

TL;DR

This paper demonstrates that multilevel Picard approximations can efficiently approximate solutions to high-dimensional semilinear heat equations with gradient-dependent nonlinearities in the $L^p$-sense, overcoming the curse of dimensionality.

Contribution

The authors establish that multilevel Picard methods achieve polynomial growth in computational effort relative to dimension and accuracy, even with gradient-dependent nonlinearities.

Findings

Polynomial growth of computational effort in dimension and accuracy

Effective approximation of solutions in $L^p$-sense for high-dimensional problems

Overcoming the curse of dimensionality in semilinear heat equations

Abstract

We prove that multilevel Picard approximations are capable of approximating solutions of semilinear heat equations in $L^{p}$-sense, ${p}\in [2,\infty)$, in the case of gradient-dependent, Lipschitz-continuous nonlinearities, in the sense that the computational effort of the multilevel Picard approximations grow at most polynomially in both the dimension $d$ and the reciprocal $1/\epsilon$ of the prescribed accuracy $\epsilon$.