Error estimates for the regularization of least squares problems

Abstract

The a posteriori estimate of the errors in the numerical solution of ill-conditioned linear systems with contaminated data is a complicated problem. Several estimates of the norm of the error have been recently introduced and analyzed, under the assumption that the matrix is square and nonsingular. In this paper we study the same problem in the case of a rectangular and, in general, rank-deficient matrix. As a result, a class of error estimates previously introduced by the authors (Brezinski et al., Numer Algorithms, in press, 2008) are extended to the least squares solution of consistent and inconsistent linear systems. Their application to various direct and iterative regularization methods are also discussed, and the numerical effectiveness of these error estimates is pointed out by the results of an extensive experimentation

Simplified GSVD computations for the solution of linear discrete ill-posed problems

“…There are many methods available for trying to determine such a value of µ, including the discrepancy principle, the L-curve criterion, generalized cross validation, and extrapolation; see [4,5,15,16] for examples and discussions. All of these methods typically require the evaluation of Ax µ − b and possibly also of x µ for several values of µ.…”

Error estimates for large-scale ill-posed problems

“…We would like to determining an estimate of the norm of the difference d = x −x. Brezinski et al [6] recently derived the following family of estimates, 9) and showed that η 2 ν1 < η 2 ν2 for ν 1 < ν 2 .…”

“…Moreover, Brezinski et al [5] discussed the application of estimates similar to (1.8) to the determination of a suitable value of the regularization parameter µ in Tikhonov regularization when the matrix A is square and nonsingular. Error estimates that are applicable to overdetermined linear systems (1.1) are presented in [6], where both solution by the singular value decomposition and the conjugate gradient method applied to the normal equations (1.5) are discussed. The present paper considers the solution of large-scale problems by Tikhonov regularization using partial Lanczos bidiagonalization and exploits the connection of the latter with Gauss quadrature.…”

“…In the application of the error estimates (1.8) to Tikhonov regularization described by Brezinski et al [5,6], the vector x in (1.7) is chosen to be the solution (1.6) of the Tikhonov minimization problem (1.4) with L = I. Since the noise-free vectorb is assumed not to be available,b is replaced by b in (1.7).…”

“…For a fixed ν ∈ R, Brezinski et al [5,6] determine a value of the regularization parameter µ that minimizes µ → η ν (µ). The residual vector (1.11) and the quantities (1.9) are inexpensive to evaluate for several values of the parameter µ when the singular value decomposition of A is available.…”