Regularization of linear inverse problems

Abstract

Inverse problems arise in many applications in science and engineering. They are characterized by the fact that directly computing a solution to an inverse problem via a well-defined operator is generally not possible. We have measured data that are generated by an (approximately) known model, the forward model, that depends on some input. This forward model generates simulated data, and the solution to the inverse problem is the input that matches the simulated data and the measured data. Limitations in the measurement setup, noise in the data, et cetera, add extra difficulty to obtaining a solution. Multiple solutions may lead to roughly the same data, and one has to choose which solution is best. Moreover, we may not want the input to match the data exactly, since we know the data are corrupted. Selecting the best among many possible solutions is done through a technique called regularization. Regularization is prior information about the solution that we want to incorporate when solving the inverse problem. We now have two elements that we have to rely on when solving the inverse problem: how well the simulated data fits the measured data and how well the solution is in accordance with our prior knowledge. The balance between these two terms is determined by the regularization parameter, which has to be specified by the user. For certain types of regularization there exist parameter selection rules that can be evaluated to get an estimate of the regularization parameter, but evaluating them is as expensive as solving the inverse problem. We generally have to solve the inverse problem for multiple regularization parameters and select the best solution. On top of that, the type of regularization has a large influence on how the inverse problem is solved. This is due to the fact that we have to use different mathematical tools for different regularization methods. The goal of this thesis has been to develop fast algorithms for inverse problems and to investigate the estimation of the regularization parameter. We have worked on a number of different regularization methods for linear inverse problems. For the simplest one, Tikhonov regularization, we have compared two algorithms that can be used to estimate the regularization parameter efficiently. Moreover, we have developed a new algorithm where the dimension of the low-dimensional surrogate model is automatically determined. We have developed a mathematical framework for a problem arising in geophysics, called Multi-Dimensional Deconvolution. We have discussed the ill-posedness of the problem and show how to incorporate constraints induced by the laws of physics. Moreover, we have discussed additional regularization that is needed to obtain a stable solution. Finally, we have extended the analysis on SR3, which is a fast algorithm for solving inverse problems with a certain type of regularization. Additionally, we have shown how it may be used to estimate the regularization parameter and proposed a novel implementation to make it suitable for large-scale problems.