Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities

Abstract

Neufeld and Wu (arXiv:2310.12545) developed a multilevel Picard (MLP) algorithm which can approximately solve general semilinear parabolic PDEs with gradient-dependent nonlinearities, allowing also for coefficient functions of the corresponding PDE to be non-constant. By introducing a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula and identifying the first and second component of the unique fixed-point of the SFPE with the unique viscosity solution of the PDE and its gradient, they proved convergence of their algorithm. However, it remained an open question whether the proposed MLP schema in arXiv:2310.12545 does not suffer from the curse of dimensionality. In this paper, we prove that the MLP algorithm in arXiv:2310.12545 indeed can overcome the curse of dimensionality, i.e. that its computational complexity only grows polynomially in the dimension $d\in \mathbb{N}$ and the reciprocal of the accuracy $\varepsilon$, under some suitable assumptions on the nonlinear part of the corresponding PDE.