Flexible Krylov Subspace Projection Methods for Inverse Problems

Emory University, Atlanta GA

Investigators

Abstract

In many scientific and engineering applications, it is necessary to solve an inverse problem; that is, to determine quantities defining an object or a system through indirect measurements. The mathematics used to produce images in devices such as X-Ray Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are excellent examples. The mathematical models describing an inverse problem are often so complicated that it is not possible to find an exact analytical solution, and it is necessary to use an approximation based on a set of simpler equations. However, many equations may be needed; in medical imaging, for example, it is not unusual to compute approximations by solving millions of equations. Additional complications arise because an inverse problem may have infinitely many solutions, or the solutions may change dramatically if there are small errors in the indirect measurements. Thus, additional constraints and mathematical tools (often referred to as regularization) are needed to narrow the search to a set of appropriate feasible solutions, and to stabilize the computations. The type and amount of regularization is problem dependent, requiring the algorithms to be able to easily adapt to user and/or problem specifications. The aim of this project is develop a flexible and adaptable computational platform that can be used for this important class of problems, which arise in many application areas, including space object identification, geophysics, microscopy and medical imaging. Thus advancements made from this project have the potential to benefit national defense, energy exploration, and human health. Efficient iterative methods to compute regularized solutions of large-scale discrete ill-posed inverse problems will be developed. The approach will use flexible Krylov subspace iterative methods, and will exploit low-rank structure of data, solutions and operators. The approach will allow to incorporate a variety of regularization techniques, including sparse and low-rank constraints on the solution. The methods developed in this project can be used as tools to obtain approximate solutions of ill-posed inverse problems, and optimized implementations will be included in a new MATLAB package called IR Tools (Iterative Regularization Tools). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.