Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

E. Weinan, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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(dε-(4+δ)) for any δ∈ (0 , ∞) under suitable assumptions, where d∈ N is the dimensionality of the problem and 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.

Original languageEnglish (US)
Article number80
JournalPartial Differential Equations and Applications
Volume2
Issue number6
DOIs
StatePublished - Dec 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Keywords

  • Curse of dimensionality
  • High-dimensional PDEs
  • High-dimensional semilinear BSDEs
  • Multilevel Monte Carlo method
  • Multilevel Picard iteration

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