Computational algorithms for solving the coefficient inverse problem for parabolic equations

Petr Vabishchevich

Oct. 6, 2014, 10:15 a.m. S2 416

In the theory and practice of inverse problems for partial differential equations (PDEs), much attention is paid to the problem of the identification of coefficients from some additional information. A theoretical study includes the fundamental questions of uniqueness of the solution and its stability from the viewpoint of the theory of differential equations. Many inverse problems are formulated as non-classical problems for PDEs. To solve approximately these problems, emphasis is on the development of stable computational algorithms that take into account peculiarities of inverse problems. Much attention is paid to the problem of determining the right-hand side, lower and leading coefficients of a parabolic equation of second order, where, in particular, the right-hand side and the coefficients depends on time only. An additional condition is most often formulated as a specification of the solution at an interior point or as the average value that results from integration over the whole domain. Numerical methods for solving problems of the identification of the right-hand side, lower and leading coefficients for parabolic equations are considered in many works. In view of the practical use, we highlight separately the studies dealing with numerical solving inverse problems for multidimensional parabolic equations. Approximation in space is performed using the standard finite differences or finite elements. In our report, for a multidimensional parabolic equation, we consider the problem of determining the right-hand side of an equation that depends on time only. Non-classical problems at every time level are solved on the basis of a special decomposition into two standard elliptic problems. We also study the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a nonlinear inverse problem, linearized approximations in time are constructed using standard finite difference approximations in space.