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_{L 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, namelyH^s space, that stays between L² and H¹. The H^s seminorm regularization is designed to minimize the undesirable aspects of the existing L² and H¹ regularization functionals. The H^s seminorm regularization also allows discontinuities and stabilizes the perturbations.

We study theH^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 theH^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 L² and H¹ regularization methods. Furthermore, we also apply theH^s seminorm regularization method to a fluorescence optical tomography problem.

In summary, we study and implement theH^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 theH^s seminorm regularization method produces the closest approximation of the exact solution than the L² norm and H ¹ seminorm regularization methods for the model problem.