Reconstruction of the absorption coefficient in a model of stationary reaction–convection–diffusion

A. I. Korotkii   ; Yu. V. Starodubtseva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 3, pp. 166-181

Résumé

Direct and inverse problems for the stationary reaction–convection–diffusion model are studied. The direct problem is to find a generalized or strong solution to the corresponding boundary value problems for all given model parameters. Conditions for generalized or strong solvability of the direct problem are given, a priori estimates for solutions are presented, and a continuous dependence of a solution to the direct problem on a number of parameters is established in various metrics. The inverse problem consists of finding the a priori unknown absorption coefficient of a medium, which characterizes the absorption of some substance (or heat sink) in a chemical process. Additional information for solving the inverse problem is the result of measuring the substance (or heat) flow on the accessible part of the boundary of the region where the process takes place. It is proved that the inverse problem is ill-posed. Examples are given showing that the inverse problem is unstable under the disturbance of the measured quantity and may have several solutions. To solve the inverse problem, a variational method based on the minimization of some suitable residual functional (objective functional) is proposed. The extremal properties of the problem of minimizing the residual functional are studied. An explicit analytical formula is found for calculating the gradient of the residual functional, and the corresponding adjoint system and optimality system are written. Several stable iterative methods for minimizing the residual functional are proposed. Numerical modeling of the solution to the inverse problem is carried out.

Détail

Mots-clés : reaction–convection–diffusion equation
Keywords: direct problem, inverse problem, residual functional, functional gradient, adjoint system, variational method, gradient minimization methods.

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