A Quasi-Newton Method in Infinite-Dimensional Spaces and Its Application for Solving a Parabolic Inverse Problem

Abstract

A Quasi-Newton method in Infinite-dimensional Spaces (QNIS) for solving operator equations is presented and the convergence of a sequence generated by QNIS is also proved in the paper. Next, we suggest a finite-dimensional implementation of QNIS and prove that the sequence defined by the finite-dimensional algorithm converges to the root of the original operator equation providing that the later exists and that the Fréchet derivative of the governing operator is invertible. Finally, we apply QNIS to an inverse problem for a parabolic differential equation to illustrate the efficiency of the finite-dimensional algorithm.