The effect of the nonlinearity on GCV applied to conjugate gradients in computerized tomography

Abstract

Abstract.We study the effect of the nonlinear dependence of the iterate x k of Conjugate Gradients method (CG) from the data b in the GCV procedure to stop the iterations. We compare two versions of using GCV to stop CG. In one version we compute the GCV function with the iterate x k depending linearly from the data b and the other one depending nonlinearly. We have tested the two versions in a large scale problem: positron emission tomography (PET). Our results suggest the necessity of considering the nonlinearity for the GCV function to obtain a reasonable stopping criterion.Mathematical subject classification: 62G05, 92C55, 65F10, 65C05.

Conjugate Gradient Method for the Solution of Inverse Problems: Application in Linear Seismic Tomography

“…We employed k max = N . In analogy with [15], the error is normalized so that its scale is comparable with the scale of the stopping criterion functions. The minimal product (MP) function is the most economical criterion, but did not predict early enough the error increase when the perturbations were given by α = 10 −3 and α = 10 −2 .…”

“…We consider the classical conjugate gradient method for normal equations to solve the leastsquares linear system (2.9), which is often referred to as CGLS [3,10,15]. Given a initial guess s 0 and a maximum number of iterations k max , r 0 ← t − Gs 0 , z 0 ← G T r 0 , and p 0 ← z 0 ; repeat…”

“…In this work, we consider two stopping criteria. The first criterion is based on an estimate of the GCV functional that requires a small computational cost [15]. The second criterion, which has an even lower cost, depends on the product of the norm of the residual and the norm of the current iterate [3, Sec.…”

An inner-outer regularizing method for ill-posed problems

“…The discrepancy principle, widely used as a stopping rule, does not produce sufficiently accurate estimates of the stopping index. The use of the Generalized Cross Validation rule (GCV) [5,12,14] gives better results, even if in some cases it is not fully reliable.…”

“…i . To obtain the same regularizing effect we would get by applying Tikhonov method with the optimal parameter λ T , a value for λ in (12) should verify ψ…”