Multi-parameter regularization techniques for ill-conditioned linear systems

Abstract

When a system of linear equations is ill-conditioned, regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties: the ill-conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen. In this paper, we consider the case of several regularizations terms added simultaneously, thus overcoming the problem of the best choice of the regularization matrix. The error of this procedure is analyzed and numerical results prove its efficiency

The structure of iterative methods for symmetric linear discrete ill-posed problems

“…where Q (1) k+1 ∈ R (k+1)×(k+1) is orthogonal and R (1) k+1,k ∈ R (k+1)×k has a leading k × k upper triangular submatrix, R (1) k , and a vanishing last row. The matrix Q (1) k+1 can be expressed as a product of k Givens rotations, 12) where G j ∈ R (k+1)×(k+1) is a rotation in the planes j and j +1.…”

“…The matrix Q (1) k+1 can be expressed as a product of k Givens rotations, 12) where G j ∈ R (k+1)×(k+1) is a rotation in the planes j and j +1. Thus, G j is the identity matrix except for a 2 × 2 block in the rows and columns j and j + 1.…”

“…Thus, G j is the identity matrix except for a 2 × 2 block in the rows and columns j and j + 1. The representation (2.12) shows that Q (1) k+1 is upper Hessenberg. Moreover, the matrix R…”

“…Since both the matrices V k+1 and Q (1) k+1,k have orthonormal columns, so does W (1) k . We obtain from (2.6) and (2.11) that 14) which shows that R(…”

“…We will use this stopping criterion in the computed examples of Section 4; however, other stopping rules also can be applied in conjunction with the iterative method discussed, such as the quasioptimality principle, generalized cross validation, and the L-curve; see, e.g., [1,13,14,19] for discussions and references.…”