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A least-squares Galerkin approach to gradient recovery for Hamilton-Jacobi-Bellman equation with Cordes coefficients

Omar Lakkis, Amireh Mousavi

TL;DR

The convergence of the continuum semismooth Newton method for the fully non- linear Hamilton-Jacobi-Bellman equation is shown and this linearization for the equation yields a recursive sequence of linear elliptic boundary value prob- lems in nondivergence form.

Abstract

We propose a conforming finite element method to approximate the strong solution of the second order Hamilton-Jacobi-Bellman equation with Dirichlet boundary and coefficients satisfying Cordes condition. We show the convergence of the continuum semismooth Newton method for the fully nonlinear Hamilton-Jacobi-Bellman equation. Applying this linearization for the equation yields a recursive sequence of linear elliptic boundary value problems in nondivergence form. We deal numerically with such BVPs via the least-squares gradient recovery of Lakkis & Mousavi [2021, arxiv:1909.00491]. We provide an optimal-rate apriori and aposteriori error bounds for the approximation. The aposteriori error are used to drive an adaptive refinement procedure. We close with computer experiments on uniform and adaptive meshes to reconcile the theoretical findings.

A least-squares Galerkin approach to gradient recovery for Hamilton-Jacobi-Bellman equation with Cordes coefficients

TL;DR

The convergence of the continuum semismooth Newton method for the fully non- linear Hamilton-Jacobi-Bellman equation is shown and this linearization for the equation yields a recursive sequence of linear elliptic boundary value prob- lems in nondivergence form.

Abstract

We propose a conforming finite element method to approximate the strong solution of the second order Hamilton-Jacobi-Bellman equation with Dirichlet boundary and coefficients satisfying Cordes condition. We show the convergence of the continuum semismooth Newton method for the fully nonlinear Hamilton-Jacobi-Bellman equation. Applying this linearization for the equation yields a recursive sequence of linear elliptic boundary value problems in nondivergence form. We deal numerically with such BVPs via the least-squares gradient recovery of Lakkis & Mousavi [2021, arxiv:1909.00491]. We provide an optimal-rate apriori and aposteriori error bounds for the approximation. The aposteriori error are used to drive an adaptive refinement procedure. We close with computer experiments on uniform and adaptive meshes to reconcile the theoretical findings.
Paper Structure (24 sections, 19 theorems, 90 equations, 2 figures)

This paper contains 24 sections, 19 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Set-up and notation
  3. Assumptions on the data
  4. Homogeneous Dirichlet problem reformulation of (\ref{['eq:HJB0-inhomogeneous']})
  5. Semismooth Newton method
  6. Policy and set-valued maps
  7. Assumption on convergence of policy sequence
  8. Assumption (Uniform convergence of policy sequences)
  9. The Newton derivative of the HJB operator
  10. Howard's algorithm
  11. A least-squares Galerkin gradient recovery for linear nondivergence form elliptic PDEs
  12. Least-squares Galerkin for linear nondivergence form PDE
  13. An equivalent problem to the equation with nonzero boundary
  14. Galerkin finite element discretization
  15. Discretization
  16. ...and 9 more sections

Key Result

Theorem 2.3

Suppose that $\varOmega\xspace$ is a bounded convex domain in $\mathbb R\xspace^{d}$, $\mathcal{A}$ is a compact metric space under $d_{\mathcal{A}}$, and that ${\vec{ { A } },\vec{b},c,f}\in\operatorname C\xspace^{0}({\varOmega\xspace\times\mathcal{A}};X)$, for $X=\operatorname{Sym}{(

Figures (2)

  • Figure 1: Convergence rates for Test problem \ref{['test:non-homogeneous-boundary']}.
  • Figure 2: (A): Generated adaptive mesh and (B), (C), (D): convergence rate in the uniform and adaptive refinement for Test problem \ref{['test:adaptive-disk-domain']} with $\mathbb P\xspace^{2}$ elements.

Theorems & Definitions (20)

  • Theorem 2.3: existence and uniqueness of a strong solution SmeSul-2
  • Lemma 1: state-to-policy operator is well defined and continuous
  • Proposition 1: policy-to-coefficient lower semicontinuity
  • proof
  • Theorem 3.5: Newton differentiability of the HJB operator
  • Lemma 2: bounded invertibility of the Newton derivatives
  • Theorem 3.6: superlinear convergence of the semismooth Newton method
  • Lemma 3: Maxwell's inequality
  • Lemma 4: generalized Maxwell's inequality
  • Lemma 5: a Miranda-Talenti estimate
  • ...and 10 more