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

Abstract

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.

Keywords: curse of dimensionality; high-dimensional PDEs; high-dimensional semilinear backward stochastic differential equations; information based complexity; multilevel Monte Carlo method; multilevel Picard approximations; tractability of multivariate problems.