Flexible Krylov Subspace Projection Methods for Inverse Problems

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In many scientific and engineering applications, it is necessary tosolve an inverse problem; that is, to determine quantities defining anobject or a system through indirect measurements. The mathematicsused to produce images in devices such as X-Ray Computed Tomography(CT) and Magnetic Resonance Imaging (MRI) are excellent examples. Themathematical models describing an inverse problem are often socomplicated that it is not possible to find an exact analyticalsolution, and it is necessary to use an approximation based on a setof simpler equations. However, many equations may be needed; inmedical imaging, for example, it is not unusual to computeapproximations by solving millions of equations. Additionalcomplications arise because an inverse problem may have infinitelymany solutions, or the solutions may change dramatically if there aresmall errors in the indirect measurements. Thus, additionalconstraints and mathematical tools (often referred to asregularization) are needed to narrow the search to a set ofappropriate feasible solutions, and to stabilize the computations.The type and amount of regularization is problem dependent, requiringthe algorithms to be able to easily adapt to user and/or problemspecifications. The aim of this project is develop a flexible andadaptable computational platform that can be used for this importantclass of problems, which arise in many application areas, includingspace object identification, geophysics, microscopy and medicalimaging. Thus advancements made from this project have the potential tobenefit national defense, energy exploration, and human health. Efficient iterative methods to compute regularized solutions oflarge-scale discrete ill-posed inverse problems will be developed.The approach will use flexible Krylov subspace iterative methods, andwill exploit low-rank structure of data, solutions and operators. Theapproach will allow to incorporate a variety of regularizationtechniques, including sparse and low-rank constraints on the solution.The methods developed in this project can be used as tools to obtainapproximate solutions of ill-posed inverse problems, and optimizedimplementations will be included in a new MATLAB package called IRTools (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.