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04/02/02 Colloquium
DR. M. ZUHAIR NASHED
University of Delaware
University of Delaware
VARIATIONAL INEQUALITIES, NONSMOOTH CALCULUS AND NEWTON-LIKE METHODS FOR ILL-POSED PROBLEMS: UN MÉNAGE Á TROIS
Abstract: Newton's method is one of the most widely used algorithms for finding approximate solutions of nonlinear operator equations F(x)=0. The method and the (Kantorovich) theory for its convergence require the existence and bounded invertibility of the Fréchet derivative of the operator F. The goal of this talk is to describe a theory for Newton-like methods when the derivative does not exist or when the derivative has no bounded inverse or bounded generalized inverse. Along the way we visit variational inequalities and discuss their role in minimization of nonsmooth functionals. We also introduce a new concept of "differentiability" for nonsmooth operators and use it to formulate a new Newton-like method. Finally, we give applications to bounded-variation regularization and nonsmooth ill-posed problems.