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Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone

Gerard Awanou

TL;DR

It is proved the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation.

Abstract

For finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Ampère equation, we give the first proof of uniqueness. The boundary condition is enforced through the use of the notion of asymptotic cone while the differential operator is discretized based on a discrete analogue of the subdifferential. We establish the convergence of a subsequence of a damped Newton's method for the nonlinear system resulting from the discretization, thereby proving the existence of a solution. Using related arguments we then prove that such a solution is necessarily unique. Convergence of the discretization as well as numerical experiments are given.

Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone

TL;DR

It is proved the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation.

Abstract

For finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Ampère equation, we give the first proof of uniqueness. The boundary condition is enforced through the use of the notion of asymptotic cone while the differential operator is discretized based on a discrete analogue of the subdifferential. We establish the convergence of a subsequence of a damped Newton's method for the nonlinear system resulting from the discretization, thereby proving the existence of a solution. Using related arguments we then prove that such a solution is necessarily unique. Convergence of the discretization as well as numerical experiments are given.

Paper Structure

This paper contains 19 sections, 19 theorems, 84 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Weak solutions of the Monge-Ampère equation
  3. Description of the numerical method
  4. Relation of the asymptotic cone approach with other work
  5. Notable features of our analysis of the damped Newton's method
  6. Organization of the paper
  7. Preliminaries
  8. The damped Newton's method
  9. Convergence of the damped Newton's method for the discretization
  10. The set $\mathbb{U}$ is open and Assumption \ref{['a1']} holds
  11. $C^1$ continuity of the mapping $G$
  12. Invertibility of the Jacobian of $G$ for $w\neq 0$
  13. Proof of Theorem \ref{['main0']}
  14. Uniqueness of solutions
  15. Limit equations
  16. ...and 4 more sections

Key Result

Lemma 2.1

Awanou-second-sym If $f>0$ on $\Omega$, then solutions of m2d4 for $V=V_{max}$ are unique up to an additive constant.

Figures (1)

  • Figure 1: Test2: Left: exact solution. Right: approximation for $h=3/(8(1/2+49))$.

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.2
  • Remark 4.1
  • Theorem 4.2
  • Theorem 4.3
  • ...and 26 more