We present a new method for regularization of ill-conditioned problems that extends the traditional trust-region approach. Ill-conditioned problems arise, for example, in image restoration or mathematical processing of medical data, and involve matrices that are very ill-conditioned. The method makes use of the L-curve and L-curve maximum curvature criterion as a strategy recently proposed to find a good regularization parameter. We describe the method and show its application to an image restoration problem. We also provide a MATLAB code for the algorithm. Finally, a comparison to the CGLS approach is given and analyzed, and future research directions are proposed.

We present a new method for regularization of ill-conditioned problems that extends the traditional trust-region approach. Ill-conditioned problems arise, for example, in image restoration or mathematical processing of medical data, and involve matrices that are very ill-conditioned. The method makes use of the L-curve and L-curve maximum curvature criterion as a strategy recently proposed to find a good regularization parameter. We describe the method and show its application to an image restoration problem. We also provide a MATLAB code for the algorithm. Finally, a comparison to the CGLS approach is given and analyzed, and future research directions are proposed.

<p>This thesis gives a partial answer to the question: “Can type systems detect modeling errors in scientific computing, particularly for inverse problems derived from physical models?” by considering, in detail, the major aspects of inverse problems in Magnetic Resonance Imaging (MRI). We define a type-system that can capture all correctness properties for MRI inverse problems, including many properties that are not captured with current type-systems, e.g., frames of reference. We implemented a type-system in the Haskell language that can capture the errors arising in translating a mathe- matical model into a linear or nonlinear system, or alternatively into an objective function. Most models are (or can be approximated by) linear transformations, and we demonstrate the feasibility of capturing their correctness at the type level using what is arguably the most difficult case, the (discrete) Fourier transformation (DFT). By this, we mean that we are able to catch, at compile time, all known errors in ap- plying the DFT. The first part of this thesis describes the Haskell implementation of vector size, physical units, frame of reference, and so on required in the mathemat- ical modelling of inverse problems without regularization. To practically solve most inverse problems, especially those including noisy data or ill-conditioned systems, one must use regularization. The second part of this thesis addresses the question of defining new regularizers and identifying existing regularizers the correctness of which (in our estimation) can be formally verified at the type level. We describe such Bayesian regularization schemes based on probability theory, and describe a novel simple regularizer of this type. We leave as future work the formalization of such regularizers.</p> / Master of Science (MSc)