Numerical study of a parametric parabolic equation and a related inverse boundary value problem
For researchers in inverse problems, this work offers a novel computational approach that reduces complexity in Gauss-Newton iterations, but the results are incremental as they only demonstrate feasibility without quantitative comparisons to existing methods.
The paper tackles an inverse boundary value problem for a time-dependent diffusion equation, aiming to determine the diffusion coefficient from boundary observations. The proposed method uses a spectral Galerkin approach in the parameter domain, enabling parameter-independent evaluation of the forward solution, and demonstrates feasibility with reconstructions in 2D and 3D.
We consider a time-dependent linear diffusion equation together with a related inverse boundary value problem. The aim of the inverse problem is to determine, based on observations on the boundary, the non-homogeneous diffusion coefficient in the interior of an object. The method in this paper relies on solving the forward problem for a whole family of diffusivities by using a spectral Galerkin method in the high-dimensional parameter domain. The evaluation of the parametric solution and its derivatives is then completely independent of spatial and temporal discretizations. In case of a quadratic approximation for the parameter dependence and a direct solver for linear least squares problems, we show that the evaluation of the parametric solution does not increase the complexity of any linearized subproblem arising from a Gauss-Newtonian method that is used to minimize a Tikhonov functional. The feasibility of the proposed algorithm is demonstrated by diffusivity reconstructions in two and three spatial dimensions.