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Scalable Fixed-Point Framework for High-Dimensional Hamilton-Jacobi Equations

Yesom Park, Stanley Osher

TL;DR

High-dimensional Hamilton-Jacobi equations pose computational challenges for grid-based solvers. The paper develops a mesh-free fixed-point framework based on the Hopf-Lax formula to compute viscosity solutions and the associated optimal controls directly at any point. It establishes contraction-based convergence and explicit error bounds, and introduces a multi-initialization strategy to handle non-unique fixed points due to kink formation. Numerical experiments up to 100 dimensions demonstrate high accuracy, near dimension-independent performance, and improved reliability compared with grid-based and neural approaches, highlighting practical impact for high-dimensional optimal control and related PDEs.

Abstract

We propose a novel, mesh-free, and gradient-free fixed-point approach for computing viscosity solutions of high-dimensional Hamilton-Jacobi (HJ) equations. By leveraging the Hopf-Lax formula, our approach iteratively solves the associated variational problem via a Picard iteration, enabling efficient evaluation of both the solution and its corresponding control without relying on grids, characteristics, or differentiation. We demonstrate the practical efficacy and scalability of the approach through numerical experiments in up to 100 dimensions, including control problems and non-smooth solutions. Our results show that the proposed scheme achieves high accuracy, is highly efficient, and exhibits computational times that are largely independent of dimensionality, highlighting its suitability for high-dimensional problems.

Scalable Fixed-Point Framework for High-Dimensional Hamilton-Jacobi Equations

TL;DR

High-dimensional Hamilton-Jacobi equations pose computational challenges for grid-based solvers. The paper develops a mesh-free fixed-point framework based on the Hopf-Lax formula to compute viscosity solutions and the associated optimal controls directly at any point. It establishes contraction-based convergence and explicit error bounds, and introduces a multi-initialization strategy to handle non-unique fixed points due to kink formation. Numerical experiments up to 100 dimensions demonstrate high accuracy, near dimension-independent performance, and improved reliability compared with grid-based and neural approaches, highlighting practical impact for high-dimensional optimal control and related PDEs.

Abstract

We propose a novel, mesh-free, and gradient-free fixed-point approach for computing viscosity solutions of high-dimensional Hamilton-Jacobi (HJ) equations. By leveraging the Hopf-Lax formula, our approach iteratively solves the associated variational problem via a Picard iteration, enabling efficient evaluation of both the solution and its corresponding control without relying on grids, characteristics, or differentiation. We demonstrate the practical efficacy and scalability of the approach through numerical experiments in up to 100 dimensions, including control problems and non-smooth solutions. Our results show that the proposed scheme achieves high accuracy, is highly efficient, and exhibits computational times that are largely independent of dimensionality, highlighting its suitability for high-dimensional problems.
Paper Structure (19 sections, 4 theorems, 55 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 19 sections, 4 theorems, 55 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Method
  3. Fixed-Point Representation via the Hopf--Lax Formula
  4. Control-theoretic interpretation.
  5. Advantages.
  6. Convergence Analysis
  7. Fixed Point Solver with Multiple Initializations
  8. Experimental Results
  9. High-dimensional problems
  10. Burgers' equation
  11. LQR Control Problem
  12. Cubic Hamiltonian Problem
  13. Non-smooth solutions
  14. Absolute-value Quadratic Form
  15. Logarithmic Form
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $H: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable functions such that: Define the fixed-point operator Then, for any $x \in \mathbb{R}^n$ and any time step $t>0$ satisfying the iteration converges to the unique global minimizer of the Hopf--Lax formula eq:hopf_lax.

Figures (2)

  • Figure 1: Solution of Burgers' equation with non-smooth initial condition $g(x) = x|x|$, computed via fixed-point iteration. Sharp gradients (kinks) develop as time evolves.
  • Figure 2: Solution of Burgers' equation with non-smooth initial condition $g(x) = x^2\log(1+|x|)$, computed via fixed-point iteration. Sharp gradients (kinks) develop as time evolves.

Theorems & Definitions (8)

  • Theorem 2.1: Fixed-point convergence
  • Proof 1
  • Theorem 2.2: Error Analysis
  • Proof 2
  • Theorem 2.3: Iteration Complexity
  • Proof 3
  • Theorem 2.4: Probabilistic Completeness of Multi-Initialization
  • Proof 4