Novel Algorithms Based on the Conjugate Gradient Method for Inverting Ill-Conditioned Matrices, and a New Regularization Method to Solve Ill-Posed Linear Systems
Chein-Shan Liu;  Hong-Ki Hong and Satya N. Atluri;

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 60, No. 3, pp. 279-308, 2010
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Keywords Ill-posed linear system, Inversion of ill-conditioned matrix, Left-inversion, Right-inversion, Regularization vector, Vandermonde matrix, Hilbert matrix, Tikhonov regularization
Abstract 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.
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