Choice of Finite-Difference Schemes in Solving Coefficient Inverse Problems
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L NUMERICAL METHODS
Choice of Finite-Difference Schemes in Solving Coefficient Inverse Problems A. F. Albua, Yu. G. Evtushenkoa, and V. I. Zubova,* a
Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, Moscow, 119333 Russia *e-mail: [email protected] Received January 31, 2020; revised March 21, 2020; accepted June 9, 2020
Abstract—Various choices of a finite-difference scheme for approximating the heat diffusion equation in solving a three-dimensional coefficient inverse problem were studied. A comparative analysis was conducted for several alternating direction schemes, such as locally one-dimensional, Douglas– Rachford, and Peaceman–Rachford schemes, as applied to nonlinear problems for the three-dimensional heat equation with temperature-dependent coefficients. Each numerical method was used to compute the temperature distribution inside a parallelepiped. The methods were compared in terms of the accuracy of the resulting solution and the computation time required for achieving the prescribed accuracy on a computer. Keywords: nonlinear problems, three-dimensional heat equation, numerical methods, alternating direction schemes 10.1134/S0965542520100048
INTRODUCTION The coefficient inverse problem of identifying a temperature-dependent thermal conductivity was investigated in [1–7]. The consideration was based on the Dirichlet boundary value problem for the oneand two-dimensional nonstationary heat equation. The objects under study were a rod and a rectangular plate. The coefficient inverse problems were reduced to variational problems. An efficient algorithm for their numerical solution based on the fast automatic differentiation (FAD) technique (see [8–10]) was proposed. The one- and two-dimensional versions of the considered coefficient inverse problems are of great mathematical interest. In practice, however, experimental data are collected on three-dimensional objects. Accordingly, the problem of identifying thermal conductivity should be solved in the threedimensional formulation. A major element of the algorithm for solving the coefficient inverse problem is a direct problem (determination of the temperature field at any point of the object at any time). The efficiency of its solution affects the efficiency of the solution of the whole inverse problem. When the variational problem is solved numerically using the FAD technique, the approximation of the adjoint problem is uniquely determined by the approximation chosen for the direct problem. Moreover, the study of the coefficient inverse problem in two dimensions has shown that this choice has a larger effect in the transition from the one- to multidimensional case. For example, if the initial-boundary value problem in two dimensions is approximated by an implicit scheme with weights (see [3]), then the adjoint problem is approximated by a linear system of algebraic equations, which has to be solved iteratively (due to the two-dimensionality of the system). In this context, an economical finite-diff
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