Double Optimal Regularization Algorithms for Solving Ill-Posed Linear Problems under Large Noise
Chein-Shan Liu and Satya N. Atluri

doi:10.3970/cmes.2015.104.001
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 104, No. 1, pp. 1-39, 2015
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Keywords Ill-posed linear equations system, Double optimal solution, Affine Krylov subspace, Double optimal iterative algorithm, Double optimal regularization algorithm.
Abstract

A double optimal solution of an \(n\)-dimensional system of linear equations \({\bf A}{\bf x}={\bf b}\) has been derived in an affine \(m\)-dimensional Krylov subspace with \(m\ll n\). We further develop a double optimal iterative algorithm (DOIA), with the descent direction \({\bf z}\) being solved from the residual equation \({\bf A}{\bf z}={\bf r}_0\) 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 \(\|{\bf r}\|^2=\|{\bf b}-{\bf A}{\bf x}\|^2\) being reduced by a positive quantity \(\|{\bf A}{\bf z}_k\|^2\) 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.

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