Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy.

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Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

Journal of Numerical Mathematics ◽

10.1515/jnma-2019-0074 ◽

2020 ◽

Vol 28 (4) ◽

pp. 197-222

Author(s):

Christian Beck ◽

Fabian Hornung ◽

Martin Hutzenthaler ◽

Arnulf Jentzen ◽

Thomas Kruse

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Applied Mathematics ◽

Nonlinear Partial Differential Equations ◽

Reaction Diffusion ◽

Curse Of Dimensionality ◽

Computational Effort ◽

Approximation Schemes ◽

Partial Differential ◽

Diffusion Type

AbstractOne 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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Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210515000712 ◽

2016 ◽

Vol 146 (4) ◽

pp. 777-832 ◽

Cited By ~ 1

Author(s):

J. C. Meyer ◽

D. J. Needham

Keyword(s):

Partial Differential Equations ◽

Initial Data ◽

Differential Equations ◽

Continuous Dependence ◽

Parabolic Partial Differential Equations ◽

Well Posedness ◽

Partial Differential ◽

Hölder Continuous ◽

Lipschitz Continuous ◽

Semilinear Parabolic

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

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