Vector extrapolation applied to truncated singular value decomposition and truncated iteration
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This paper is concerned with the computation of accurate approximate solutions of linear systems of equations and linear least-squares problems with a very ill-conditioned matrix and error-contaminated data. The solution of these kinds of problems requires regularization. Common regularization methods include truncated singular value decomposition (TSVD) and truncated iteration with a Krylov subspace method. It can be difficult to determine when to truncate. Recently, it has been demonstrated that extrapolation of approximate solutions determined by TSVD gives a new sequence of approximate solutions that is less sensitive to errors in data than the original approximate solutions. The present paper describes a novel approach to determining a suitable truncation index by comparing the original and extrapolated approximate solutions. Applications to TSVD and the LSQR iterative method are presented.
KeywordsDiscrete ill-posed problem LSQR Truncation criterion Truncated iteration Truncated singular value decomposition Vector extrapolation
This research was supported in part by National Science Foundation Grant DMS-1115385. The research was also supported by the Jiangsu Oversea Research & Training Program for Prominent Young & Middle-Aged University Teachers and Presidents and the Fundamental Research Funds for Central Universities Grant NZ2012307.
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