Steepest descent and differential equations
type: Item
instance of: scholarly article
title: Steepest descent and differential equations
zbMATH Open document ID: 0576.65053
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: Let H, K, S be real Hilbert spaces and \(F=H\to K\), B:H\(\to S\) operators with local Lipschitz derivatives. For the solution of (1) \(F(u)=0\), \(B(u)=0\) the steepest descent process (2) \(z'(t)=-P(x)(\nabla \phi)(z(t))\), \(t\geq 0\), \(z(0)=x^ 0\), is considered where \(\phi (x)=\| F(x)\|^ 2\) and P(x) is the orthogonal projection of H onto the nullspace of B'(x). In generalization of known results to this constrained case, it is shown that when (3) \(\| P(x)(\nabla \phi)(x)\| \geq c\| F(x)\|\), \(x\in \Omega \subset H\), \(c>0\), and (4) rge(z)\(\in \Omega\) then \(u=\lim_{\epsilon \to \infty}z(t)\) exists and \(F(u)=0\), \(B(u)=B(x^ 0)\). The conditions (3) and (4) are discussed in some detail and examples involving several boundary value problems are given.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 3921848
zbMATH Keywords: constrained operator equations
zbMATH Keywords: convergence
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: steepest descent
author: J. W. Neuberger
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03720187
OpenAlex ID: W2112708152
DOI: 10.2969/JMSJ/03720187
description: scientific article; zbMATH DE number 3921848