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Policy iteration for nonconvex viscous Hamilton--Jacobi equations

Xiaoqin Guo, Hung Vinh Tran, Yuming Paul Zhang

TL;DR

This work develops and analyzes a discrete space-time policy-iteration framework for nonconvex viscous Hamilton–Jacobi equations with potentially degenerate diffusion. It proves an exponential convergence rate of the PI iterations to the discretized solution and then derives quantitative convergence rates from the discrete problem to the true viscosity solution via regularization and convolution techniques. The rates depend on diffusion nondegeneracy: near $O(h^{\alpha/2})$ in the nondegenerate case and $O(h^{2\alpha/(9+7\alpha)})$ in the degenerate case, for any $\alpha\in(0,1)$. These results extend PI analysis to nonconvex, degenerate parabolic HJ equations and provide practical guidance on grid design and parameter choices for numerical schemes.

Abstract

We study the convergence rates of policy iteration (PI) for nonconvex viscous Hamilton--Jacobi equations using a discrete space-time scheme, where both space and time variables are discretized. We analyze the case with an uncontrolled diffusion term, which corresponds to a possibly degenerate viscous Hamilton--Jacobi equation. We first obtain an exponential convergent result of PI for the discrete space-time schemes. We then investigate the discretization error.

Policy iteration for nonconvex viscous Hamilton--Jacobi equations

TL;DR

This work develops and analyzes a discrete space-time policy-iteration framework for nonconvex viscous Hamilton–Jacobi equations with potentially degenerate diffusion. It proves an exponential convergence rate of the PI iterations to the discretized solution and then derives quantitative convergence rates from the discrete problem to the true viscosity solution via regularization and convolution techniques. The rates depend on diffusion nondegeneracy: near in the nondegenerate case and in the degenerate case, for any . These results extend PI analysis to nonconvex, degenerate parabolic HJ equations and provide practical guidance on grid design and parameter choices for numerical schemes.

Abstract

We study the convergence rates of policy iteration (PI) for nonconvex viscous Hamilton--Jacobi equations using a discrete space-time scheme, where both space and time variables are discretized. We analyze the case with an uncontrolled diffusion term, which corresponds to a possibly degenerate viscous Hamilton--Jacobi equation. We first obtain an exponential convergent result of PI for the discrete space-time schemes. We then investigate the discretization error.

Paper Structure

This paper contains 10 sections, 17 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Settings
  3. Discrete space-time schemes
  4. Main results
  5. Preliminaries
  6. Proof of Theorem \ref{['T.4.2']}
  7. Convergence of the discrete equations
  8. Regularity of continuous equations
  9. Convolution regularization
  10. The quantitative convergence result

Key Result

Theorem 1.1

Assume N2 and N4. For $h\in (0,1)$, let $\Lambda^h>\lambda^h\geq \nu_h$ be such that Then, for all $n\geq 1$, where $C_h:= e^{C_1T/\lambda^h}\left({12}\|g\|_\infty^2+T{\lambda^h}\right)$ and $C_1:=48\max\{\|c\|^2_\infty,2d\|f\|_\infty^2\}$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 27 more