Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

TL;DR

This paper presents a novel multilevel Picard iteration algorithm combining Feynman-Kac and Bismut-Elworthy-Li formulas for efficiently solving high-dimensional semilinear parabolic PDEs with proven complexity bounds.

Contribution

It introduces a new numerical method with analytical tools for high-dimensional PDEs, achieving bounded computational complexity under certain conditions.

Findings

Algorithm performs well on physics and finance PDEs.

Complexity is bounded by O(d * ε^{-(4+δ)}) for accuracy ε.

New analytical tools support the algorithm's analysis.

Abstract

We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman-Kac and the Bismut-Elworthy-Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with very satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman-Kac formula, a new class of semi-norms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearity that the computational complexity of the proposed algorithm is bounded by $O(d\,\varepsilon^{-(4+\delta)})$ for any $\delta \in (0,\infty)$ under suitable assumptions, where $d\in \mathbb{N}$ is the dimensionality of the problem and $\varepsilon\in(0,\infty)$ is the prescribed accuracy.