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

by Ch. Beck and L. Gonon and A. Jentzen

(Report number 2020-16)

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.

Keywords:

@Techreport{BGJ20_889,
  author = {Ch. Beck and L. Gonon and A. Jentzen},
  title = {Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-16.pdf },
  year = {2020}
}

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