Adjoint approach to parameter identification with application to the Richards equation

Abstract

The inverse problem for the unknown coefficient ingredient of a class of quasilinear parabolic partial differential equations is considered. An approach based on utilizing adjoint versions of the direct problems to derive integral equations explicitly relating changes in inputs (coefficients) to changes in outputs (measured data) is presented. Using the integral equations it is possible to demonstrate properties of these maps. In the first problem, we show that the coefficient to data mappings are continuous, strictly monotone and injective. In the second, the mapping is shown to be explicitly invertible. The equations are further exploited to construct an approximate solution to the inverse problem. In the first problem, these equations are also used to analyze the error in the approximation. These equations are then used to construct a numerical recovery algorithm, which we call the integral identity method. Finally, numerical experiments are presented which explore the recovery process.