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 )}) for any delta in (0,infty ) under suitable assumptions, where din {{mathbb {N}}} is the dimensionality of the problem and {varepsilon }in (0,infty ) is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.

Highlights

  • Introduction and main resultsHigh-dimensional partial differential equations (PDEs) arise naturally in many important areas including quantum mechanics, statistical physics, financial engineering, economics, etc

  • 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 has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results

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Summary

Introduction and main results

High-dimensional partial differential equations (PDEs) arise naturally in many important areas including quantum mechanics, statistical physics, financial engineering, economics, etc. This article is part of the topical collection “Deep learning and PDEs” edited by Arnulf Jentzen, Lin Lin, Siddhartha Mishra, and Lexing Ying

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Notation
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The approximation scheme
Special case: semilinear heat equations
Numerical simulations of high-dimensional semilinear PDEs
Convergence rate for the multilevel Picard iteration
Setting
Pseudocode
Sketch of the proof
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Preliminary results for the Gauß-Legendre quadrature rules
Preliminary results for the seminorms
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Error analysis for the multilevel Picard iteration
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Analysis of the computational complexity and overall rate of convergence
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An example PDE
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