Convergence of the gradient method for ill-posed problems

Stefan Kindermann

doi:10.3934/ipi.2017033

Article Contents

2017, Volume 11, Issue 4: 703-720. Doi: 10.3934/ipi.2017033

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Convergence of the gradient method for ill-posed problems

Stefan Kindermann^,

Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstrasse 69,4040 Linz, Austria

Received: May 31, 2016

Revised: April 30, 2017

Published: October 2017

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Abstract

We study the convergence of the gradient descent method for solving ill-posed problems where the solution is characterized as a global minimum of a differentiable functional in a Hilbert space. The classical least-squares functional for nonlinear operator equations is a special instance of this framework, and the gradient method then reduces to Landweber iteration. The main result of this article is a proof of weak and strong convergence under new nonlinearity conditions that generalize the classical tangential cone conditions.

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Mathematics Subject Classification: Primary: 65J20, 65J15; Secondary: 47J06.

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