Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
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Date
2020-12-16
Publication Type
Journal Article
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yes
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Abstract
One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction–diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.
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published
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Book title
Journal / series
Volume
28 (4)
Pages / Article No.
197 - 222
Publisher
De Gruyter
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Edition / version
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Software
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Date collected
Date created
Subject
Parabolic partial differential equations; Multilevel Picard approximations; Feynman-Kac representation; Curse of dimensionality; Numerical analysis; Applied stochastic analysis; 60H30; 65C05; 65M75
Organisational unit
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
03874 - Hungerbühler, Norbert / Hungerbühler, Norbert
Notes
It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.
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