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

2020/03/01 by Christian Beck, Beck, Christian, Lukas Gonon +3 · 2 citations
Engineering · Mathematics · Physics and Astronomy · #65Cxx #65Mxx #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2003.00596

openalex publication_date 2020/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

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