Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities

In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main...

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Bibliographic Details
Main Authors Neufeld, Ariel, Wu, Sizhou
Format Journal Article
LanguageEnglish
Published 19.10.2023
Subjects
Computer Science - Numerical Analysis
Mathematics - Analysis of PDEs
Mathematics - Numerical Analysis
Mathematics - Probability
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DOI10.48550/arxiv.2310.12545

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Summary:In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to$300$dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.
DOI:10.48550/arxiv.2310.12545