New Perturbation Results For Regularized Tikhonov Inverses And Pseudo-Inverses.

Abstract

. Consider the Tikhonov regularized linear least squares problem minx kJx \Gamma bk 2 2 + ¯ 2 kL(x \Gamma c)k 2 2 , where J 2 ! m\Thetan ; b 2 ! m and L 2 ! p\Thetan . The interesting part of the solution to this problem (attained by putting c = 0) is J # L b; J # L = (J T J + ¯ 2 L T L) \Gamma1 J T . As ¯ ! 0 the solution of the regularized problem tends to the solution, J + L b, of minx kL(x \Gamma c)k2 subject to the constraint that kJx \Gamma bk2 is minimized. The main result of this paper is perturbation identities for J + L . However, in order to attain this result perturbation identities for J # L are derived first and then the fact that J # L ! J + L b is used. The perturbation identities for J + L and J # L are useful for ill-posed, ill-conditioned and rank-deficient problems. Key words. Tikhonov regularization, GSVD, perturbation theory, rank deficiency, pseudoinverses, filter factors, numerical rank AMS subject classifications. 65 K ..