Some Recent Developments and Open Problems in Solution Methods for Mathematical Inverse Problems

The area of mathematical inverse problems is quite broad and involves the qualitative and quantitative analysis of a wide variety of physical models. Applications include, for example, the problem of inverse heat conduction, image reconstruction, tomography, the inverse scattering problem, and the determination of unknown coefficients or boundary parameters appearing in partial differential equation models of physical phenomena. We will survey some recent developments in the area of regularization methods for mathematical inverse problems and indicate where further contributions are needed. We will discuss methods for which a primary goal is to avoid the oversmoothing of solutions that typically occurs when classical regularization schemes (such as Tikhonov regularization) are used. Oversmoothing is particularly troublesome in imaging and tomographic applications, where it is of high priority to recover sharp or fine features of solutions. We will also explore the trend to design localized regularization methods which complement the qualitative nature of a particular inverse problem (as opposed to the application of a generic regularization method such as Tikhonov’s method). Local regularization methods tend to be more attractive in terms of numerical costs and implementation, but a number of open questions remain in their theoretical analysis. Finally, we will discuss current work in the area of iterative solution methods, regularization schemes which have been successfully applied to a number of important nonlinear inverse problems. Key-words: mathematical inverse problems, regularization methods, iterative regularization, local regularization methods, inverse heat conduction, imaging.