TY - JOUR

T1 - Double optimal regularization algorithms for solving ill-posed linear problems under large noise

AU - Liu, Chein Shan

AU - Atluri, Satya N.

N1 - Publisher Copyright:
Copyright © 2015 Tech Science Press.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - A double optimal solution of an n-dimensional system of linear equations Ax = b has been derived in an affine m-dimensional Krylov subspace with mn. We further develop a double optimal iterative algorithm (DOIA), with the descent direction z being solved from the residual equation Az = r0 by using its double optimal solution, to solve ill-posed linear problem under large noise. The DOIA is proven to be absolutely convergent step-by-step with the square residual error r2 = b-Ax2 being reduced by a positive quantity Azk2 at each iteration step, which is found to be better than those algorithms based on the minimization of the square residual error in an m-dimensional Krylov subspace. In order to tackle the ill-posed linear problem under a large noise, we also propose a novel double optimal regularization algorithm (DORA) to solve it, which is an improvement of the Tikhonov regularization method. Some numerical tests reveal the high performance of DOIA and DORA against large noise. These methods are of use in the ill-posed problems of structural health-monitoring.

AB - A double optimal solution of an n-dimensional system of linear equations Ax = b has been derived in an affine m-dimensional Krylov subspace with mn. We further develop a double optimal iterative algorithm (DOIA), with the descent direction z being solved from the residual equation Az = r0 by using its double optimal solution, to solve ill-posed linear problem under large noise. The DOIA is proven to be absolutely convergent step-by-step with the square residual error r2 = b-Ax2 being reduced by a positive quantity Azk2 at each iteration step, which is found to be better than those algorithms based on the minimization of the square residual error in an m-dimensional Krylov subspace. In order to tackle the ill-posed linear problem under a large noise, we also propose a novel double optimal regularization algorithm (DORA) to solve it, which is an improvement of the Tikhonov regularization method. Some numerical tests reveal the high performance of DOIA and DORA against large noise. These methods are of use in the ill-posed problems of structural health-monitoring.

KW - Affine Krylov subspace

KW - Double optimal iterative algorithm

KW - Double optimal regularization algorithm

KW - Double optimal solution

KW - Ill-posed linear equations system

UR - http://www.scopus.com/inward/record.url?scp=84925745030&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84925745030

VL - 104

SP - 1

EP - 39

JO - CMES - Computer Modeling in Engineering and Sciences

JF - CMES - Computer Modeling in Engineering and Sciences

SN - 1526-1492

IS - 1

ER -