Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities

Abstract

Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome the so-called curse of dimensionality in the sense that the number of computational operations of the approximation algorithm needed to achieve an approximation precision of size $\varepsilon$ > 0 grows at most polynomially in both the PDE dimension $d \in \mathbb{N}$ and the reciprocal of the prescribed approximation accuracy $\varepsilon$. In particular, to the best of our knowledge there exists no approximation algorithm in the scientific literature which has been proven to overcome the curse of dimensionality in the case of a class of nonlinear PDEs with general time horizons and gradient-dependent nonlinearities. It is the key contribution of this article to overcome this difficulty. More specifically, it is the key contribution of this article (i) to propose a new full-history recursive multilevel Picard approximation algorithm for high-dimensional nonlinear heat equations with general time horizons and gradient-dependent nonlinearities and (ii) to rigorously prove that this full-history recursive multilevel Picard approximation algorithm does indeed overcome the curse of dimensionality in the case of such nonlinear heat equations with gradient-dependent nonlinearities.