Regularizing Inverse Problems

Abstract

An inverse problem reconstructs the unknown internal parameters of a subject based on collected data derived synthetically or from real measurements. Inverse problems often lack the well-posedness defined by J. Hadamard; in other words, solutions of inverse problems, namely the reconstructions of the parameters, may not exist, may not be unique or may be unstable. Regularization is a technique that deals with such situations. The well-known Tikhonov regularization method translates the original inverse problem to optimization problems of minimizing the norm of the data misfit plus a weighted regularization functional that incorporates the a priori information we may have about the original problem. The choices of the regularization functional r(q) include ?q??(2@L^(2 ) )??q??(2@H^(1) ), |q|BV and |q|TV. However, each of these has its limitations. In this work, we develop a novel H^(s) seminorm regularization method and present numerical results for model problems. This method relies on the evaluation of the seminorms of an intermediary Hilbert space, namely H^(s) space, that stays between L^(2) and H^(1). The H^(s) seminorm regularization is designed to minimize the undesirable aspects of the existing L^(2) and H^(1) regularization functionals. The H^(s) seminorm regularization also allows discontinuities and stabilizes the perturbations. We study the H^(s) seminorm regularization method both theoretically and numerically. We consider the theoretical analysis of this new regularization method based on a model problem. We show that a stable solution can be achieved with some conditions. In addition, we prove the convergence and guarantee a convergence rate provided additional conditions for the model problem when the considered domain is 1D. Numerically, we produce an approximated discretization of the H^(s) seminorm regularization that can be applied to 1D, 2D or 3D examples. We also provide reconstructions of both continuous and discontinuous parameters from synthetic data and a comparison of these solutions to the ones based on existing L^(2) and H^(1) regularization methods. Furthermore, we also apply the H^(s) seminorm regularization method to a fluorescence optical tomography problem. In summary, we study and implement the H^(s) seminorm regularization method for inverse problems, which can provide a stable solution to the model problem. The numerical results indicate the robustness of the new method and suggests that the H^(s) seminorm regularization method produces the closest approximation of the exact solution than the L^(2) norm and H^(1) seminorm regularization methods for the model problem.