Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities
Abstract
In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.
Keywords
Cite
@article{arxiv.2310.12545,
title = {Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities},
author = {Ariel Neufeld and Sizhou Wu},
journal= {arXiv preprint arXiv:2310.12545},
year = {2025}
}