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, Satya N. Atluri

Research output: Contribution to journalArticle

54 Scopus citations

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.

Original languageEnglish
Pages (from-to)279-308
Number of pages30
JournalCMES - Computer Modeling in Engineering and Sciences
Volume60
Issue number3
StatePublished - 2010

Keywords

  • Hilbert matrix
  • Ill-posed linear system
  • Inversion of ill-conditioned matrix
  • Left-inversion
  • Regularization vector
  • Right-inversion
  • Tikhonov regularization
  • Vandermonde matrix

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