Verlag der Österreichischen Akademie der Wissenschaften

Abstract:
We devise efficient methods for dynamic inverse problems, where both the quantities of interest and the forward operator (measurement process) may change in time. Our goal is to solve for all the quantities of interest simultaneously. We consider large-scale ill-posed problems made more challenging by their dynamic nature and, possibly, by the limited amount of available data per measurement step. To alleviate these difficulties, we apply a unified class of regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each time instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy. More precisely, we develop iterative methods based on a majorization-minimization (MM) strategy with quadratic tangent majorant, which allows the resulting least-squares problem with a total variation regularization term to be solved with a generalized Krylov subspace (GKS) method; the regularization parameter can be determined automatically and efficiently at each iteration. Numerical examples from a wide range of applications, such as limited-angle computerized tomography (CT), space-time image deblurring, and photoacoustic tomography (PAT), illustrate the effectiveness of the described approaches.