Pivoted Cauchy-Like Preconditioners for Regularized Solution of Ill-Posed Problems

Abstract

Abstract. Many ill-posed problems are solved using a discretization that results in a least squares problem or a linear system involving a Toeplitz matrix. The exact solution to such problems is often hopelessly contaminated by noise, since the discretized problem is quite ill conditioned, and noise components in the approximate null-space dominate the solution vector. Therefore we seek an approximate solution that does not have large components in these directions. We use a preconditioned conjugate gradient algorithm to compute such a regularized solution. A unitary change of coordinates transforms the Toeplitz matrix to a Cauchy-like matrix, and we choose our preconditioner to be a low rank Cauchy-like matrix determined in the course of Gu's fast modified complete pivoting algorithm. We show that if the kernel of the ill-posed problem is smooth, then this preconditioner has desirable properties: the largest singular values of the preconditioned matrix are clustered around one, the smallest singular values, corresponding to the lower subspace, remain small, and the upper and lower spaces are relatively unmixed. The preconditioned algorithm costs only O(n lg n) operations per iteration for a problem with n variables. The effectiveness of the preconditioner for filtering noise is demonstrated on three examples.

A multilevel augmentation method for solving ill-posed operator equations

“…We next review several results of this area closely related to the research presented in this paper. Hanke and Vogel [11,12] developed a two-level preconditioner for solving the discretized operator equation and Riley [29] extended it to a general case. In [13][14][15] by Hämarik, [24] by Plato and [27] by Plato and Vainikko, the regularized projection methods were studied with a priori and a posteriori choices of the regularization parameter, whereas results in [13][14][15] include a posteriori parameter choice in the Lavrentiev regularization for the Galerkin method.…”

A Trust-Region Approach to the Regularization of Large-Scale Discrete Forms of Ill-Posed Problems

“…However, in recent years interesting methods for large-scale ill-posed problems have been proposed. Among those are Golub and von Matt [14], Björck, Grimme and van Dooren [4], Calvetti, Reichel and Zhang [8], Rojas, Santos and Sorensen [35], as well as several variants of the Conjugate Gradient Method on the Normal Equations (CGLS), including the use of preconditioners chosen according to the structure of the problem [22], [24], [29]. In spite of these developments, the efficient solution of large-scale discrete forms of ill-posed problems remains a challenge.…”

Linearized solution to electrical impedance tomography based on the Schur conjugate gradient method

“…Hanke and Vogel proposed the Schur CG algorithm for illposed problems [13,17]. Riley investigated the computational cost of the Schur CG algorithm [18]. The numerical details were extensively analysed in [19].…”