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

Abstract

Abstract. 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.

A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems

“…The Newton update (17) for finding the regularization parameter based on root finding for the χ 2 -curve provides an alternative approach to standard techniques such as the L-curve, GCV and UPRE, which are also based on the use of the GSVD, for estimating the single variable regularization parameter. These methods are well-described in a number of research monographs and therefore no derivation is provided here.…”

“…Instead, relevant methods include projection-based techniques, which seek to project the noise out of the system and lead to solutions of reduced problems, [17], and algorithms in which a regularization term is introduced. Both directions introduce complications, not least of which are stopping criteria for the projection iterations in the first case and in the second case finding a suitable regularization parameter which trades-off a regularization term relative to the data-fitting or fidelity term (1).…”

Iterative Methods for Image Restoration

“…Thus, various authors have considered computational and implementation issues, such as robust approaches to choose regularization parameters and stopping iterations; see for example, [13,19,25,44,56,57,60,75]. The biggest disadvantage of the hybrid approach is that all y k vectors must be kept throughout the iteration process, and thus the storage requirements grow as the iteration proceeds.…”

“…Hybrid methods were first proposed by O'Leary and Simmons in 1981 [75], and later by Björck in 1988 [11]. The basic idea is to regularize the projected least squares problem (20) involving B k , which can be done very cheaply because of the smaller size of B k .…”

Deconvolution for Acoustic Emission

“…We have previously developed a singular-value matrix method (SVD) for better solving the deconvolution problem, and this method is quite powerful [1]. However, it requires selecting a best guess filtered answer and frequently that is difficult to do.…”