A Bidiagonalization-Regularization Procedure for Large Scale Discretizations of Ill-Posed Problems

In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices $K$ of high dimensions. The algorithm proposed uses $K$ only as an operator and does not need to explicitly store or modify it. A method related to one of Lanczos is used to project the problem onto a subspace for which $K$ is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given.