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

Author:

Hutzenthaler Martin1,Jentzen Arnulf23ORCID,Kruse Thomas4,Anh Nguyen Tuan1,von Wurstemberger Philippe3

Affiliation:

1. University of Duisburg-Essen, Duisburg, Germany

2. University of Münster, Münster, Germany

3. Eidgenössische Technische Hochschule Zürich, Zurich, Switzerland

4. Justus Liebig Universität Giessen, Giessen, Germany

Abstract

For a long time it has been 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 at most polynomially both in the dimension and in the reciprocal of the required accuracy.

Funder

Deutsche Forschungsgemeinschaft

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

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