GKB-FP: an algorithm for large-scale discrete ill-posed problems

Abstract

We describe an algorithm for large-scale discrete ill-posed problems, called GKB-FP, which combines the Golub-Kahan bidiagonalization algorithm with Tikhonov regularization in the generated Krylov subspace, with the regularization parameter for the projected problem being chosen by the fixed-point method by Bazán (Inverse Probl. 24(3), 2008). The fixed-point method selects as regularization parameter a fixed-point of the function r λ 2 / f λ 2 , where f λ is the regularized solution and r λ is the corresponding residual. GKB-FP determines the sought fixed-point by computing a finite sequence of fixed-points of functions rapproximates f λ in a k-dimensional Krylov subspace and r (k) λ is the corresponding residual. Based on this and provided the sought fixed-point is reached, we prove that the regularized solutions f (k) λ remain unchanged and therefore completely insensitive to the number of iterations. This and the performance of the method when applied to well-known test problems are illustrated numerically.

Some results on the regularization of LSQR for large-scale discrete ill-posed problems

“…To be definitive, for simplicity we assume that these coefficients satisfy a widely used model in the literature, e.g., [20, p. 81, 111 and 153] and [22, p. 68 is the minimum-norm least squares solution of the perturbed problem that replaces A in (1.1) by its best rank k approximation A k , and the best possible TSVD solution of (1.1) by the TSVD method is x T SV D k0 [22, p. 98]. A number of approaches have been proposed for determining k 0 , such as discrepancy principle, discrete L-curve and generalized cross validation; see, e.g., [1,2,20,27,35] for comparisons of the classical and new ones. In our numerical experiments, we use the L-curve criterion in the TSVD method and hybrid LSQR.…”

Métodos para Problemas Inversos de Grande Porte

“…al. [49], bem como FP-tools, uma proposta desenvolvida recentemente pelos autores [5]. Resultados numéricos usando problemas teste disponíveis na toolbox RegularizationTools tais como phillips, foxgood, shaw, heat, etc, também são incluídos.…”

“…O algoritmo LANC-FP foi introduzido recentemente por Bazán e Borges [5] com intuito de implementar o algoritmo de ponto-fixo para problemas de grande porte. Objetivamente, LANC-FP determina a sequência finita {λ…”

Extension of GKB‐FP algorithm to large‐scale general‐form Tikhonov regularization

“…/, D 1g D 0.0116. The relative errors in x optimal and x 320 F. S. V. BAZÁN, M. C. C. CUNHA AND L. S. BORGES on large-scale problems reported in [24] show that the largest convex FP of . / is quickly captured.…”

“…/ is quickly captured. For a detailed description of GKB-FP, see [24] again; here, we summarize the main steps of GKB-FP, because, as we will see later, they will be also followed by the methods proposed in this work.…”