Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities

TL;DR

This paper introduces a multilevel Picard algorithm for solving complex semilinear parabolic PDEs with gradient-dependent nonlinearities, providing convergence analysis and demonstrating practical effectiveness in high-dimensional cases.

Contribution

The paper develops a novel multilevel Picard method for general semilinear parabolic PDEs with non-constant coefficients and gradient dependencies, including comprehensive convergence and complexity analysis.

Findings

Algorithm converges for high-dimensional PDEs up to 300 dimensions.

Provides a full theoretical analysis of convergence and complexity.

Demonstrates practical applicability with numerical examples.

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 $300$ dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.