Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
-
Christian Beck
,
Fabian Hornung
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.
Acknowledgment
The second author gratefully acknowledges a Research Travel Grant by the Karlsruhe House of Young Scientists (KHYS) supporting his stay at ETH Zurich. The third author gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via research grant HU 1889/6-1. The fourth author gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Muenster: Dynamics–Geometry–Structure.
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© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
- Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems
- P1-nonconforming divergence-free finite element method on square meshes for Stokes equations
Artikel in diesem Heft
- Frontmatter
- Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
- Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems
- P1-nonconforming divergence-free finite element method on square meshes for Stokes equations
