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Abstract Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p -norm, and a regularization term, that is defined in terms of a q -norm. We allow 0 < p , q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed.

Abstract Bregman-type iterative methods have received considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they deliver. However, these iterative methods may require a large number of iterations and this reduces their usefulness. This paper develops a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. The projection reduces the computational effort required for each iteration. A variant of this solution method, in which nonnegativity of each computed iterate is imposed, also is described. Extensive numerical examples illustrate the performance of the proposed methods.

Abstract Randomized methods can be competitive for the solution of problems with a large matrix of low rank. They also have been applied successfully to the solution of large-scale linear discrete ill-posed problems by Tikhonov regularization (Xiang and Zou in Inverse Probl 29:085008, 2013). This entails the computation of an approximation of a partial singular value decomposition of a large matrix A that is of numerical low rank. The present paper compares a randomized method to a Krylov subspace method based on Golub–Kahan bidiagonalization with respect to accuracy and computing time and discusses characteristics of linear discrete ill-posed problems that make them well suited for solution by a randomized method.