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Stability of a regularized Newton method with two potentials

Boushra Abbas

Abstract

In a Hilbert space setting, we study the stability properties of the regularized continuous Newton method with two potentials, which aims at solving inclusions governed by structured monotone operators. The Levenberg-Marquardt regularization term acts in an open loop way. As a byproduct of our study, we can take the regularization coefficient of bounded variation. These stability results are directly related to the study of numerical algorithms that combine forward-backward and Newton's methods.

Stability of a regularized Newton method with two potentials

Abstract

In a Hilbert space setting, we study the stability properties of the regularized continuous Newton method with two potentials, which aims at solving inclusions governed by structured monotone operators. The Levenberg-Marquardt regularization term acts in an open loop way. As a byproduct of our study, we can take the regularization coefficient of bounded variation. These stability results are directly related to the study of numerical algorithms that combine forward-backward and Newton's methods.

Paper Structure

This paper contains 5 sections, 9 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. A priori estimates
  3. Stability results
  4. Bounded Variation Regularization Coefficient $\lambda\left(\cdot\right)$
  5. Perspective

Key Result

Proposition 2.1

Let $\left(x,\upsilon\right)$ be the strong solution of system (basica)-(basicc) on $\left[0,T\right]$, $T>0$. Then

Theorems & Definitions (18)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 8 more