A globally convergent numerical method for a coefficient inverse problem for a parabolic equation

Abstract

a b s t r a c tIn this work a Multidimensional Coefficient Inverse Problem (MCIP) for a parabolic PDE with the data resulting from a single measurement event is considered. This measured data is obtained using a single position of the point source. The most important property of the method presented here is that even though we do not need any advanced knowledge of a small neighborhood of the solution, we still obtain points in this neighborhood. This is the reason why the numerical algorithm for this method is called globally convergent. In the literature a globally convergent numerical method for the MCIP with single measurement data has inclusively been considered in the book and some other publications of Beilina and Klibanov. In those publications a globally convergent algorithm for MCIP for a hyperbolic PDE was developed. Here we modify their technique to prove the global convergence property of their method for the parabolic case.

Reduction an Inverse Problem to a System of Second Kind Fredholm Integral Equations with No Singularities

“…There are many numerical methods to solve these problems. Among them, we cite the method of fundamental solutions (MFS) by Marin and Lesnic [6] and Wang et al [7] and by Chen et al [8] and Sun and He [9] for the twodimensional and three-dimensional inverse problems, respectively, the boundary function method (BFM) by Wang et al [10], the boundary particle method (BPM) by Chen and Fu [11], the variational iteration method (VIM) by Canon and Tatari [12], the globally convergent numerical method by Baysal [13], the weighted homotopy analysis method (WHAM) by Shidfar et al [14], and the conjugate gradient method (CGM) by Lu et al [15]. In some special cases, these problems are solved by the analytical methods by Liua and Tatar [16] or Liu [17].…”