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In this paper, we propose new randomization based algorithms for large scalelinear discrete ill-posed problems with general-form regularization: ${\min}\|Lx\|$ subject to ${\min} \|Ax - b\|$, where $L$ is a regularization matrix.Our algorithms are inspired by the modified truncated singular valuedecomposition (MTSVD) method, which suits only for small to medium scaleproblems, and randomized SVD (RSVD) algorithms that generate good low rankapproximations to $A$. We use rank-$k$ truncated randomized SVD (TRSVD)approximations to $A$ by truncating the rank-$(k+q)$ RSVD approximations to$A$, where $q$ is an oversampling parameter. The resulting algorithms arecalled modified TRSVD (MTRSVD) methods. At every step, we use the LSQRalgorithm to solve the resulting inner least squares problem, which is provedto become better conditioned as $k$ increases so that LSQR converges faster. Wepresent sharp bounds for the approximation accuracy of the RSVDs and TRSVDs forseverely, moderately and mildly ill-posed problems, and substantially improve aknown basic bound for TRSVD approximations. We prove how to choose the stoppingtolerance for LSQR in order to guarantee that the computed and exact bestregularized solutions have the same accuracy. Numerical experiments illustratethat the best regularized solutions by MTRSVD are as accurate as the ones bythe truncated generalized singular value decomposition (TGSVD) algorithm, andat least as accurate as those by some existing truncated randomized generalizedsingular value decomposition (TRGSVD) algorithms.

inverse problems

Modified Truncated Randomized Singular Value Decomposition (MTRSVD)  Algorithms for Large Scale Discrete Ill-posed Problems with General-Form  Regularization