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Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities

Ariel Neufeld, Sizhou Wu

TL;DR

This work develops a multilevel Picard (MLP) scheme for general high-dimensional semilinear parabolic PDEs with gradient-dependent nonlinearities and non-constant coefficients. By formulating a stochastic fixed-point equation via Feynman-Kac and Bismut-Elworthy-Li representations, the authors prove existence and uniqueness of a viscosity solution $u^d$ with gradient $ abla_x u^d$, and show that a full-history recursive MLP algorithm converges with explicit rates. The framework yields polynomial-in-dimension and polynomial-in-$1/ ext{ε}$ complexity, effectively overcoming the curse of dimensionality, and is demonstrated with a numerical example up to $d=300$. The results bridge stochastic fixed-point theory and nonlinear PDEs, delivering rigorous convergence guarantees for high-dimensional gradient-nonlinear PDE solvers applicable to physics, finance, and beyond.

Abstract

In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to $300$ dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.

Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities

TL;DR

This work develops a multilevel Picard (MLP) scheme for general high-dimensional semilinear parabolic PDEs with gradient-dependent nonlinearities and non-constant coefficients. By formulating a stochastic fixed-point equation via Feynman-Kac and Bismut-Elworthy-Li representations, the authors prove existence and uniqueness of a viscosity solution with gradient , and show that a full-history recursive MLP algorithm converges with explicit rates. The framework yields polynomial-in-dimension and polynomial-in- complexity, effectively overcoming the curse of dimensionality, and is demonstrated with a numerical example up to . The results bridge stochastic fixed-point theory and nonlinear PDEs, delivering rigorous convergence guarantees for high-dimensional gradient-nonlinear PDE solvers applicable to physics, finance, and beyond.

Abstract

In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.
Paper Structure (26 sections, 35 theorems, 482 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 26 sections, 35 theorems, 482 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting and assumptions
  3. Multilevel Picard approximation scheme: the main results
  4. Euler approximations
  5. Multilevel Picard (MLP) approximation scheme
  6. Viscosity solutions of PDEs
  7. The main results
  8. Pseudocode
  9. A Numerical Example
  10. Stochastic fixed-point equations
  11. Semilinear Parabolic Partial Differential Equations
  12. Multilevel Picard approximations
  13. Setting
  14. Error Analysis of Multilevel Picard Approximations
  15. Proof of the main results
  16. ...and 11 more sections

Key Result

Theorem 3.3

Let Assumptions assumption Lip and growth, assumption ellip, and assumption gradient hold. Then the following holds.

Figures (1)

  • Figure 1: Standard deviations of MLP solutions of PDE \ref{['simple_PDE']} with different simulation levels

Theorems & Definitions (81)

  • Remark 2.2
  • Remark 2.5
  • Remark 2.7
  • Remark 2.8
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Lemma 4.1
  • ...and 71 more