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arXiv · 1607.03295

Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

Weinan E·Martin Hutzenthaler·Arnulf Jentzen·Thomas Kruse
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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 $O(d\,\varepsilon^{-(4+δ)})$ for any $δ\in (0,\infty)$ under suitable assumptions, where $d\in \mathbb{N}$ is the dimensionality of the problem and $\varepsilon\in(0,\infty)$ is the prescribed accuracy.

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Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse. 2019-02-22. Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. https://doi.org/10.1007/s42985-021-00089-5

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