Computational algorithms for solving the coefficient inverse problem for parabolic equations

Abstract

Among inverse problems for partial differential equations, we distinguish coefficient inverse problems, which are associated with the identification of coefficients and/or the right-hand side of an equation using some additional information. When considering time-dependent problems, the identification of the coefficient dependences on space and on time is usually separated into individual problems. In some cases, we have linear inverse problems (e.g. identification problems for the right-hand side of an equation); this situation essentially simplify their study. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a non-linear inverse problem, linearized approximations in time are constructed using standard finite difference approximations in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems.

Keywords:

• inverse problem
• identification of the coefficient
• parabolic partial differential equation
• finite difference scheme

AMS Subject Classifications:

• 65M06
• 65M32
• 80A23

Acknowledgements

This research was supported by the Russian Foundation for Basic Research (project 14-01-00785). We express our deep gratitude to Prof. George Dulikravich for constructive suggestion about work text.