Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

Weinan E (), Martin Hutzenthaler (), Arnulf Jentzen () and Thomas Kruse ()
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Weinan E: Princeton University
Martin Hutzenthaler: University of Duisburg-Essen
Arnulf Jentzen: ETH Zurich
Thomas Kruse: University of Gießen

Partial Differential Equations and Applications, 2021, vol. 2, issue 6, 1-31

Abstract: 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 satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by $$O(d\,{\varepsilon }^{-(4+\delta )})$$ O ( d ε - ( 4 + δ ) ) for any $$\delta \in (0,\infty )$$ δ ∈ ( 0 , ∞ ) under suitable assumptions, where $$d\in {{\mathbb {N}}}$$ d ∈ N is the dimensionality of the problem and $${\varepsilon }\in (0,\infty )$$ ε ∈ ( 0 , ∞ ) is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.

Keywords: Curse of dimensionality; High-dimensional PDEs; High-dimensional semilinear BSDEs; Multilevel Picard iteration; Multilevel Monte Carlo method; 65M75 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (7)

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DOI: 10.1007/s42985-021-00089-5

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