The solution of linear discrete ill-posed problems is very sensitive to perturbations in the data. Confidence intervals for solution coordinates provide insight into the sensitivity. This paper presents an efficient method for computing confidence intervals for large-scale linear discrete ill-posed problems. The method is based on approximating the matrix in these problems by a partial singular...

Abstract. For large-scale symmetric discrete ill-posed problems, MINRES and MR-II are commonly used iterative solvers. In this paper, we analyze their regularizing effects. We first prove that the regularized solutions by MINRES have filtered SVD forms. Then we show that (i) a hybrid MINRES that uses explicit regularization within projected problems is generally needed to compute a best possibl...

with a large matrix A of ill-determined rank. Thus, A has many “tiny” singular values of different orders of magnitude. In particular, A is severely ill-conditioned. Some of the singular values of A may be vanishing. We allow m ≥ n or m < n. The right-hand side vector b̃ is not required to be in the range of A. Linear systems of equations of the form (1) with a matrix of ill-determined rank are ...

Many problems of physical chemistry belong to the class of inverse problems, in which from known experimental data of the object we need to determine some of its properties based on a certain model connecting these properties with measured characteristics. Inverse problems typically lead to mathematical models that are not well-posed in the sense of Hadamard, i.e. to the ill-posed problems. Thi...