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

Christian Beck (), Lukas Gonon () and Arnulf Jentzen ()
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Christian Beck: ETH Zurich
Lukas Gonon: Imperial College London
Arnulf Jentzen: The Chinese University of Hong Kong (CUHK-Shenzhen)

Partial Differential Equations and Applications, 2024, vol. 5, issue 6, 1-47

Abstract: 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: Numerical analysis; Monte Carlo methods; Full-history recursive multilevel Picard approximations; Semilinear elliptic partial differential equations; 65Cxx; 65C05; 65Nxx; 65N75 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s42985-024-00272-4

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