Local Regularization Methods for Ill-Posed Inverse Problems: Fast Algorithms and Adaptive Parameter Selection

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Local regularization methods show great promise for the solution of a number of different classes of inverse problems, often retaining the special structure of the original problem as well as leading to very fast algorithms (especially in the case of Volterra problems). In addition, numerical tests show that local regularization methods can work well to resolve sharp features of solutions without having to rely on nondifferentiable or nonquadratic optimization schemes. To date, the convergence theory for the local regularization of Volterra problems has been limited to only mildly ill-posed inverse problems. In the case of more severely ill-posed problems, there is numerical evidence that certain local methods may suffer from lack of stability and/or convergence. Because of the cost efficiencies of local regularization methods, an important question is therefore whether new variations of these methods can be developed for which stability/convergence can be proven in the general case. The PI proposes to develop two new variations of local regularization methods which show promise in numerical tests and for which there is hope of establishing a general stability/convergence theory. The PI also proposes to develop adaptive schemes for the selection of variable regularization parameters in local regularization methods. Variable parameters are of use in applying more smoothing in some parts of the domain and less in others. In numerical tests adaptive local regularization techniques have been shown to be effective in determining the variable regularization parameter at the same time that local parts of the solution are recovered. Because no convergence theory exists at the present time for such an approach, the PI proposes to study such adaptive schemes and develop a theory which will be useful in making recommendations for adaptive parameter selection methods. The PI also proposes to extend these ideas to nonlinear Volterra problems and to linear non-Volterra problems.Inverse problems occur widely in many applications, including problems of biomedical imaging (CT scans and X-rays), image reconstruction (from satellites or other sources), and geophysical exploration. The Volterra class of inverse problems arises in the determination of the surface temperature of a space vehicle as it re-enters the earth's atmosphere; additionally, Volterra inverse problems appear as models for remote sensing problems. While classical methods exist for for solving such problems, classical methods are often very inefficient and lead to overly expensive solution techniques. A second disadvantage of classical solution methods is readily seen in imaging applications where reconstructed images may have blurred edges and inadequately detailed features. The PI proposes to address both of these difficulties with the development of new solution methods based on the ideas of local regularization. The use of these newer methods can lead to a significant decrease in cost for the solution of a wide class of practical inverse problems, with improved resolution of detailed features of solutions.