Smoothing methods and semismooth methods for nondifferentiable operator equations

Xiaojun Chen, Zuhair Nashed, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

159 Citations (Scopus)

Abstract

We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally Lipschitzian operator in Rnis based on Rademacher's theorem which does not hold in function spaces. We introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework. We show that a function is slantly differentiable at a point if and only if it is Lipschitz continuous at that point. An application to the Dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed.
Original languageEnglish
Pages (from-to)1200-1216
Number of pages17
JournalSIAM Journal on Numerical Analysis
Volume38
Issue number4
DOIs
Publication statusPublished - 1 Dec 2001

Keywords

  • Nondifferentiable operator equation
  • Nonsmooth elliptic partial differential equations
  • Semismooth methods
  • Smoothing methods
  • Superlinear convergence

ASJC Scopus subject areas

  • Numerical Analysis

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