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Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities

Ariel Neufeld, Tuan Anh Nguyen, Sizhou Wu

TL;DR

The paper proves that multilevel Picard (MLP) approximations overcome the curse of dimensionality for general semilinear PDEs with gradient-dependent nonlinearities by leveraging a stochastic fixed-point equation linked to the Feynman-Kac representation and the Bismut–Elworthy–Li formula. It establishes existence, uniqueness, and regularity of SFPE solutions, proves spatial Lipschitz and temporal continuity, and introduces an MLP scheme with convergence guarantees to the SFPE fixed point. The analysis combines Grönwall-type bounds, Euler–Maruyama discretizations, and a perturbation lemma to derive a complexity result: the computational cost grows polynomially in the dimension $d$ and the inverse accuracy $1/\varepsilon$, with explicit bounds $\mathfrak{C}^d_{N_{d,\varepsilon},|N_{d,\varepsilon}|^{1/3},|N_{d,\varepsilon}|^{N_{d,\varepsilon}/3}} \le C_\delta \varepsilon^{-(4+\delta)} \eta d^\eta$. This demonstrates a practical path to high-dimensional PDE solution with provable performance guarantees.

Abstract

Neufeld and Wu (arXiv:2310.12545) developed a multilevel Picard (MLP) algorithm which can approximately solve general semilinear parabolic PDEs with gradient-dependent nonlinearities, allowing also for coefficient functions of the corresponding PDE to be non-constant. By introducing a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula and identifying the first and second component of the unique fixed-point of the SFPE with the unique viscosity solution of the PDE and its gradient, they proved convergence of their algorithm. However, it remained an open question whether the proposed MLP schema in arXiv:2310.12545 does not suffer from the curse of dimensionality. In this paper, we prove that the MLP algorithm in arXiv:2310.12545 indeed can overcome the curse of dimensionality, i.e. that its computational complexity only grows polynomially in the dimension $d\in \mathbb{N}$ and the reciprocal of the accuracy $\varepsilon$, under some suitable assumptions on the nonlinear part of the corresponding PDE.

Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities

TL;DR

The paper proves that multilevel Picard (MLP) approximations overcome the curse of dimensionality for general semilinear PDEs with gradient-dependent nonlinearities by leveraging a stochastic fixed-point equation linked to the Feynman-Kac representation and the Bismut–Elworthy–Li formula. It establishes existence, uniqueness, and regularity of SFPE solutions, proves spatial Lipschitz and temporal continuity, and introduces an MLP scheme with convergence guarantees to the SFPE fixed point. The analysis combines Grönwall-type bounds, Euler–Maruyama discretizations, and a perturbation lemma to derive a complexity result: the computational cost grows polynomially in the dimension and the inverse accuracy , with explicit bounds . This demonstrates a practical path to high-dimensional PDE solution with provable performance guarantees.

Abstract

Neufeld and Wu (arXiv:2310.12545) developed a multilevel Picard (MLP) algorithm which can approximately solve general semilinear parabolic PDEs with gradient-dependent nonlinearities, allowing also for coefficient functions of the corresponding PDE to be non-constant. By introducing a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula and identifying the first and second component of the unique fixed-point of the SFPE with the unique viscosity solution of the PDE and its gradient, they proved convergence of their algorithm. However, it remained an open question whether the proposed MLP schema in arXiv:2310.12545 does not suffer from the curse of dimensionality. In this paper, we prove that the MLP algorithm in arXiv:2310.12545 indeed can overcome the curse of dimensionality, i.e. that its computational complexity only grows polynomially in the dimension and the reciprocal of the accuracy , under some suitable assumptions on the nonlinear part of the corresponding PDE.
Paper Structure (12 sections, 13 theorems, 291 equations)

This paper contains 12 sections, 13 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Main result
  4. Stochastic fixed point equations
  5. A Grönwall-type inequality
  6. Existence and uniqueness of solutions to SFPEs
  7. Spatial Lipschitz continuity of solutions to SFPEs
  8. Temporal regularity of the fixed point
  9. Perturbation lemma
  10. MLP approximations
  11. Euler-Maruyama approximations revisited
  12. Complexity analysis

Key Result

Theorem 1.1

Let $\Theta=\cup_{n\in {\mathbbm{N}}}{\mathbbm{Z}}^n$, $T\in (0,\infty)$, $\mathbf{k}\in [0,\infty)$, $c\in [1,\infty)$. Let $\lVert \cdot\rVert\colon \cup_{k,\ell\in{\mathbbm{N}}}{\mathbbm{R}}^{k\times \ell}\to[0,\infty)$ satisfy for all $k,\ell\in{\mathbbm{N}}$, $s=(s_{ij})_{i\in[1,k]\cap{\mathbbm For every $K\in {\mathbbm{N}}$ let $\lfloor \cdot\rfloor_K\colon {\mathbbm{R}}\to{\mathbbm{R}}$ sat

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof : Proof of \ref{['a15']}
  • Lemma 2.2
  • proof : Proof of \ref{['b02']}
  • Lemma 2.3: Grönwall-type inequality
  • proof : Proof of \ref{['a11c']}
  • Corollary 2.4
  • ...and 19 more