Novel Error Measures and Source Conditions of Regularization Methods

Inverse problems try to determine the cause for an observation. The most prominent example in this field is tomography where the ability of a medium to absorb X-rays (the cause) is visualized from measurements of the attenuation of the X-rays after penetrating through the body. Typically, inverse problems are ill-posed, in the sense that small errors in the observation may have a significant influence on the errors for the cause, when direct computation is applied. Regularization methods are designed to limit the reconstruction errors for the cause. The basic principle of regularization is to limit ourselves in the reconstruction process to causes which respect certain a-priori information, such as a maximal and minimal magnitude, smoothness, or certain conservation principles. The project investigates new areas of applications, such as the reconstruction of tensors, displacements, and color data. This has for instance applications in Computational Elastography. To quantitatively evaluate new regularization methods for such applications, we need to develop new efficiency measure, and develop a new convergence analysis. This is the overall topic of this proposal.

Inverse problems try to determine the cause for an observation. The most prominent example in the area of inverse problems is tomography, where the ability of a medium to absorb X-rays (the cause) is visualized from measurements of the attenuation of the X-rays after penetrating through the body. Typically inverse problems are ill-posed, in the sense that small errors in the observation may have a significant influence on the errors in the numerically computed cause, when direct computation methods are applied. Regularization methods are designed to limit the reconstruction errors for the cause. The basic principle of regularization is to limit ourselves in the reconstruction process to causes which respect certain a-priori information, such as a maximal and minimal magnitude, smoothness, or certain conservation principles. The project investigates the analysis of regularization methods in new areas of applications, such as the reconstruction of tensors, displacements, and color data. This has for instance applications in Computational Elastography and Megnetic Resonance Imaging. To quantitatively evaluate new regularization methods for such applications, we need to develop new efficiency measure and develop a new convergence analysis of regularization methods. This has been the overall topic of this project.