A local regularization method using multiple regularization levels

Engineering problems often involve the solution and analysis of inverse problems. Inverse problems are those types of problems where we need to obtain some knowledge about a system through measurement of some other features of the system. Examples of inverse problems include areas such as signal processing, finance, geophysics, image reconstruction and helioseismology. For many practical cases, the inverse problems that are dealt with in engineering, are ill-posed, in the sense of Hadamard definition of well-posedness. To deal with the implications of ill-posedness of such problems, different types of regularization methods are used to replace the original ill-posed problem with a nearby well-posed one. The focus of this work is on regularization of discrete ill-posed problems where the concept of "numerical rank" cannot be applied. More specifically we focus on the "Tikhonov regularization" method, for which the major aspects are choosing regularization operator and also regularization parameter as a measure to balance residual vs. the smoothing norm. For the classical Tikhonov regularization method, most frequently "constant" regularization operator and parameter are used. There are problematic consequences as a result of using this strategy. One major difficulty for many cases of inverse problems is that the solution obtained by these standard regularization techniques is oversmoothed. This is due to the fact that the solution is smoothed (regularized) uniformly with no regard for its specific properties of its different subdomains. This problem is even more pronounced when the actual/exact solution displays sharp changes in different subsections of its domain and regularization with uniform smoothing destroys these features. In this thesis, the concept of multiple regularization is developed. This approach of multiple regularization allows different levels of regularization within different subdomains of the solution. To choose the regularization parameters, the method of generalized cross validation (GCV) is extended to multiparameter cases, so that the regularization parameter is variable throughout the domain of the solution. This method would allow different levels of smoothing based on the properties of different subdomains (i.e. more smoothing (regularization) where the solution is more smooth and less regularization where the solution is less smooth).