| 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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| 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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