Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient

Abstract

An inverse problem for the identification of an unknown coefficient in a quasilinear parabolic partial differential equation is considered. We present an approach based on utilizing adjoint versions of the direct problem in order to derive equations explicitly relating changes in inputs (coefficients) to changes in outputs (measured data). Using these equations it is possible to show that the coefficient to data mappings are continuous, strictly monotone and injective. The equations are further exploited to construct an approximate solution to the inverse problem and to analyse the error in the approximation. Finally, the results of some numerical experiments are displayed.