TY - EJOU
AU - Liu, Chein-Shan
AU - Atluri, Satya N.
TI - Double Optimal Regularization Algorithms for Solving Ill-Posed Linear Problems under Large Noise
T2 - Computer Modeling in Engineering \& Sciences
PY - 2015
VL - 104
IS - 1
SN - 1526-1506
AB - A double optimal solution of an *n*-dimensional system of linear equations Ax = b has been derived in an affine *m « n*. We further develop a *double optimal iterative algorithm* (DOIA), with the descent direction z being solved from the residual equation Az = 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 ||r||^{2} = ||b - Ax||^{2} being reduced by a positive quantity ||Az_{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.
KW - Ill-posed linear equations system
KW - Double optimal solution
KW - Affine Krylov subspace
KW - Double optimal iterative algorithm
KW - Double optimal regularization algorithm
DO - 10.3970/cmes.2015.104.001