Inverse problems arise from many practical applications whenever one searches for unknown causes based on observation of their effects. From computational point of view, a characteristic property of inverse problems is their ill-posedness in the sense that their solutions do not depend continuously on the data. Due to errors in the measurements, in practical applications one never has the exact data; instead only noisy data are available. Therefore, how to use the noisy data to produce a stable approximate solution is an important topic.

Due to the ill-posedness of inverse problems, one has to employ regularization methods which transform the problem into a family of well-posed problems indexed by the so called regularization parameter. This brings several important questions; for instance, how to design an efficient regularization method? how to choose the regularization parameter to obtain good approximate solutions? The answers to these questions depend on how much information we have on the problem, the special feature of the unknown exact solution, and the noise type in the noisy data. Our project is to use functional analysis, convex analysis and optimization to introduce and analyze various regularization methods with suitable penalty terms, including the sparsity promoting functions and the total variational function. We also seek applications in imaging and industry.