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

Abstract

For a long time it is 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 polynomially both in the dimension and in the reciprocal of the required accuracy.

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• Multilevel Picard iterations for solving smooth semilinear parabolic heat equations
• Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method
• Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
• Numerical methods for backward stochastic differential equations: A survey
• Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
• A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver
• Multilevel Picard approximations for McKean-Vlasov stochastic differential equations
• Towards fast weak adversarial training to solve high dimensional parabolic partial differential equations using XNODE-WAN
• Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations