Sparsity regularization of the diffusion coefficient problem: well-posedness and convergence rates

In this paper, we investigate sparsity regularization for the diffusion coefficient identification problem. Here, the regularization method is incorporated with the energy functional approach. The advantages of our approach are to deal with convex minimization problems. Therefore, the well-posedness of the problem is obtained without requiring regularity property of the parameter. The convexity of regularized problems also allows to use the fast algorithms developed recently. Furthermore, the convergence rates of the method are obtained under a simple source condition. The main results of the paper are the well-posedness and convergence rates of sparsity regularization. We also obtain some new results of the continuity and the differentiability of related operators.