Numerical Analysis and Computation of Nonlinear Partial Differential Equations from Applied Mathematics

1993 ◽  
Author(s):  
Donald A. French
Keyword(s):  
Numerical Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Applied Mathematics ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential

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

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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