TY - JOUR

T1 - Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems

AU - Liu, Chein Shan

AU - Hong, Hong Ki

AU - Atluri, Satya N.

PY - 2010

Y1 - 2010

N2 - We propose novel algorithms to calculate the inverses of ill-conditioned matrices, which have broad engineering applications. The vector-form of the conjugate gradient method (CGM) is recast into a matrix-form, which is named as the matrix conjugate gradient method (MCGM). The MCGM is better than the CGM for finding the inverses of matrices. To treat the problems of inverting ill-conditioned matrices, we add a vector equation into the given matrix equation for obtaining the left-inversion of matrix (and a similar vector equation for the right-inversion) and thus we obtain an over-determined system. The resulting two modifications of the MCGM, namely the MCGM1 and MCGM2, are found to be much better for finding the inverses of ill-conditioned matrices, such as the Vandermonde matrix and the Hilbert matrix. We propose a natural regularization method for solving an ill-posed linear system, which is theoretically and numerically proven in this paper, to be better than the well-known Tikhonov regularization. The presently proposed natural regularization is shown to be equivalent to using a new preconditioner, with better conditioning. The robustness of the presently proposed method provides a significant improvement in the solution of ill-posed linear problems, and its convergence is as fast as the CGM for the well-posed linear problems.

AB - We propose novel algorithms to calculate the inverses of ill-conditioned matrices, which have broad engineering applications. The vector-form of the conjugate gradient method (CGM) is recast into a matrix-form, which is named as the matrix conjugate gradient method (MCGM). The MCGM is better than the CGM for finding the inverses of matrices. To treat the problems of inverting ill-conditioned matrices, we add a vector equation into the given matrix equation for obtaining the left-inversion of matrix (and a similar vector equation for the right-inversion) and thus we obtain an over-determined system. The resulting two modifications of the MCGM, namely the MCGM1 and MCGM2, are found to be much better for finding the inverses of ill-conditioned matrices, such as the Vandermonde matrix and the Hilbert matrix. We propose a natural regularization method for solving an ill-posed linear system, which is theoretically and numerically proven in this paper, to be better than the well-known Tikhonov regularization. The presently proposed natural regularization is shown to be equivalent to using a new preconditioner, with better conditioning. The robustness of the presently proposed method provides a significant improvement in the solution of ill-posed linear problems, and its convergence is as fast as the CGM for the well-posed linear problems.

KW - Hilbert matrix

KW - Ill-posed linear system

KW - Inversion of ill-conditioned matrix

KW - Left-inversion

KW - Regularization vector

KW - Right-inversion

KW - Tikhonov regularization

KW - Vandermonde matrix

UR - http://www.scopus.com/inward/record.url?scp=77956115510&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77956115510

VL - 60

SP - 279

EP - 308

JO - CMES - Computer Modeling in Engineering and Sciences

JF - CMES - Computer Modeling in Engineering and Sciences

SN - 1526-1492

IS - 3

ER -