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Continuous modified Newton's-type method for nonlinear operator equations

A. G. Ramm, A. B. Smirnova, A. Favini

TL;DR

The paper studies nonlinear operator equations $F(x)=0$ in a Hilbert space using continuous dynamical systems methods (DSM) and continuous Newton-type schemes. It proves exponential convergence for well-posed cases via autonomous flows such as $\dot x(t)=\Phi(x(t))$, including schemes that avoid inverting $F'(x)$ by updating an auxiliary operator $B(t)$ or using a fixed inverse. For ill-posed problems it introduces a continuously regularized Newton scheme with a time-dependent parameter $\varepsilon(t)$, providing convergence and stability results and a stopping rule to handle data noise. Overall, it unifies well-posed and ill-posed nonlinear problem solving within the DSM framework, offering rigorous convergence guarantees and practical relevance for inverse problems.

Abstract

A nonlinear operator equation $F(x)=0$, $F:H\to H,$ in a Hilbert space is considered. Continuous Newton's-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point $x_0$ and becoming asymptotically close to a solution of $F(x)=0$ as $t\to +\infty$ are discussed. Well-posed and ill-posed problems are investigated.

Continuous modified Newton's-type method for nonlinear operator equations

TL;DR

The paper studies nonlinear operator equations in a Hilbert space using continuous dynamical systems methods (DSM) and continuous Newton-type schemes. It proves exponential convergence for well-posed cases via autonomous flows such as , including schemes that avoid inverting by updating an auxiliary operator or using a fixed inverse. For ill-posed problems it introduces a continuously regularized Newton scheme with a time-dependent parameter , providing convergence and stability results and a stopping rule to handle data noise. Overall, it unifies well-posed and ill-posed nonlinear problem solving within the DSM framework, offering rigorous convergence guarantees and practical relevance for inverse problems.

Abstract

A nonlinear operator equation , in a Hilbert space is considered. Continuous Newton's-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point and becoming asymptotically close to a solution of as are discussed. Well-posed and ill-posed problems are investigated.

Paper Structure

This paper contains 3 sections, 6 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Continuous modified Newton's schemes for well-posed problems
  3. Ill-posed case. Continuously regularized modified Newton's scheme

Key Result

Lemma 2.1

Let $H$ be a real Hilbert space, $F,\Phi: H\to H$. Suppose that there exist some positive numbers $c_1$ and $c_2$ such that $F$ and $\Phi$ are Fréchet differentiable in $U(r,x_0):=\{x\in H,\,\,||x-x_0||\le r\}$, $r:=\frac{c_2||F(x_0)||}{c_1}$ and $\,\forall h\in U(r,x_0)$ the following conditions ho and Then: 1. there exists a global solution $x=x(t)$ to problem (ch1) in the ball $U(r,x_0)$; 2. t

Theorems & Definitions (10)

  • Lemma 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Corollary 3.4