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 analyze the error in the approximation. Finally, results of some numerical experiments are displayed.

Semigroup approach for identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation

“…Denote by T (t) the semigroup of linear operators generated by the operator −A [5,6]. Note that we can easily find the eigenvalues and eigenfunctions of the differential operator A.…”

Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem

“…Similar orthogonality relations and adjoint problems have been used for onedimensional equations before. In [10], the identifiability of a is established by controlling the sign of u 1x and λ x , which is possible with monotonicity arguments in the one-dimensional case. This argument is however not applicable in the multi-dimensional case.…”

“…In this context, let us refer to the work of Alessandrini [1] and also to [24,25]. Uniqueness results for other types of problems, e.g., of parabolic type or in nonlinear elasticity can be found in [7,9,10,17,18,23,31] and [22,30]. A broad overview over inverse problems for partial differential equations and many more results and references can be found in the book of Isakov [20].…”

Identification problems for degenerate parabolic equations

“…When some of the required inputs are not available we may be able to determine the missing inputs from outputs that are measured rather than computed by formulating and solving an appropriate inverse problem. In particular, when the missing inputs are one or more unknown coefficients in the partial differential equation, the problem is called a coefficient identification problem, and when a source term is missing it is a source identification problem (see [4], [10], [14], [21], [22]). …”