Research on Krylov Subspace Type Iterative Solvers for Large Scale Systems of Equations and Least Squares Problems
| Project/Area Number |
17560056
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Engineering fundamentals
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| Research Institution | National Institute of Informatics |
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Principal Investigator |
HAYAMI Ken National Institute of Informatics, Principles of Informatics Research Division, Professor (20251358)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥1,750,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥150,000)
Fiscal Year 2007: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
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| Keywords | least squares problems / iterative method / preconditioning / Krylov subspace method / GMRES method / CGLS method / generalized inverse / large sparse matrices / 正則化 / 特異系 / RIF / 優決定問題 / 劣決定問題 / 収束解析 / 反復法 / 連立代数方程式 / ホモトピー法 / 脳磁界逆問題 |
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| Research Abstract |
Research was done on applying the Generalized Minimal Residual (GMRES) method, which is a Krylov subspace iterative method for solving systems of linear equations, to large scale least squares problems by using mapping matrices. More specifically, let A be the coefficient matrix of the original least squares problem, and let B be the mapping matrix. We proposed methods applying the GMRES method to systems with square coefficient matrix AB or BA.
Next, we derived sufficient conditions on A and B such that the proposed methods converge to the solution of the least squares problem without break-down for the general case including the over-determined, under-determined and rank-deficient cases.
For the preconditioning (mapping) matrix B, we tested diagonal scaling, incomplete modified Gram-Schmidt method, incomplete Givens method and the Robust Incomplete Factorization (RIF) method by numerical experiments on various matrices. As a result, it was found that the RIF method converged best and was the fastest By applying this RIF method, we verified by numerical experiments that the proposed method is faster than the previous preconditioned conjugate gradient least squares (CGLS) method for ill-conditioned large scale problems.
Moreover, we performed theoretical analysis on the convergence properties of the proposed methods using singular values of matrices.
On the other hand, we also proposed a different preconditioning method for least squares problems using an approximate generalized inverse for rectangular matrices. This was done by generalizing the method of constructing the approximate inverse of square matrices using the steepest descent method. We verified its usefulness by theoretical analysis and numerical experiments.
Further, we proposed a preconditioning method for rank deficient least squares problems and are evaluating its usefulness.
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Report
(4 results)
Research Products
(61 results)
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[Journal Article] Convergence Analysis of GMRES Methods for Least Squares Problems,(Invited paper)2007
Author(s)
Hayami, K.
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Journal Title
Y.Kaneda, H.Kawamura and M.Sasai eds., Frontiers of Computational Science, Proceedings of the International Symposium on Frontiers of Computational Science 2005(FCS2005), Nagoya, Japan, Dec.12-13, 2005, Springer-Verlag
Pages: 181-187
Description
「研究成果報告書概要(和文)」より
Related Report
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[Journal Article] Convergence Analysis of GMRES Methods for Least Squares Problems, (Invited paper)2007
Author(s)
Hayami, K., Ito, T.
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Journal Title
Frontiers of Computational Science, Proceedings of the International Symposium of Frontiers of Computational Science 2005 (FCS2005), Nagoya, Japan, Dec. 12-13, 2005(Y. Kaneda, H. Kawamura and M. Sasai eds.)(Springer-Verlag)
Pages: 181-187
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