An inverse problem for a steady-state hydrodynamic diffusion-advection-reaction model is considered. The sought-after quantity in the inverse problem is the thermal diffusivity of the model. To find this quantity, the trace of the normal derivative of the model state must be used on an accessible part of the boundary of the model’s operating domain. The solution to the inverse problem is reduced to an extremal (variational) problem for minimizing some appropriate residual functional with minimization over a certain set of possible sought-after coefficients. The paper focuses on the numerical modeling of the inverse problem. The problem is considered under conditions where the trace of the normal derivative of the model state is not an ordinary regular function, but a generalized function (functional). To solve the minimization problem, it is proposed to use gradient methods. The gradient of the residual functional is found explicitly and analytically from the solution of the optimality system, which consists of the direct and adjoint problems. For the practical implementation of gradient methods, various options for choosing the descent step and gradient approximation are proposed. The results of numerical solutions for several model examples are presented, demonstrating the feasibility of the proposed approach to solving the inverse problem.
Keywords: diffusion–advection–reaction equation, thermal diffusivity coefficient, inverse problem, residual functional, extremal problem, variational method, minimum point
Received January, 23 2026
Revised February, 9 2026
Accepted February, 16 2026
Alexander Illarionovich Korotkii, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: korotkii@imm.uran.ru
Igor Anatolievich Tsepelev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: tsepelev@imm.uran.ru
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Cite this article as: Korotkii A.I., Tsepelev I.A. Numerical simulation of the problem of recovering model parameters based on irregular data. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 122–130.
