Matrix Coefficient Identification in an Elliptic Equation with the Convex Energy Functional Method
This work provides a theoretical and numerical framework for a specific inverse problem in PDEs, but it is incremental as it applies known methods (convex energy functional, Tikhonov regularization) to a particular problem.
The paper addresses the inverse problem of identifying the diffusion matrix in an elliptic PDE from measurements, using the convex energy functional method with Tikhonov regularization. They prove convergence, error bounds, and strong convergence of a gradient-projection algorithm, with a numerical experiment illustrating the results.
In this paper we study the inverse problem of identifying the diffusion matrix in an elliptic PDE from measurements. The convex energy functional method with Tikhonov regularization is applied to tackle this problem. For the discretization we use the variational discretization concept, where the PDE is discretized with piecewise linear, continuous finite elements. We show the convergence of approximations. Using a suitable source condition, we prove an error bound for discrete solutions. For the numerical solution we propose a gradient-projection algorithm and prove the strong convergence of its iterates to a solution of the identification problem. Finally, we present a numerical experiment which illustrates our theoretical results.