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- 저자 : Faezeh Toutounian (검색결과 4 건)
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NUMERICAL IMPLEMENTATION OF THE QMR ALGORITHM BY USING DISCRETE STOCHASTIC ARITHMETIC
TOUTOUNIAN, FAEZEH,KHOJASTEH SALKUYEH, DAVOD,ASADI, BAHRAM 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.17 No.1
In each step of the quasi-minimal residual (QMR) method which uses a look-ahead variant of the nonsymmetric Lanczos process to generate basis vectors for the Krylov subspaces induced by A, it is necessary to decide whether to construct the Lanczos vectors $v_{n+l}\;and\;w{n+l}$ as regular or inner vectors. For a regular step it is necessary that $D_k\;=\;W^{T}_{k}V_{k}$ is nonsingular. Therefore, in the floating-point arithmetic, the smallest singular value of matrix $D_k$, ${\sigma}_min(D_k)$, is computed and an inner step is performed if $\sigma_{min}(D_k)<{\epsilon}$, where $\epsilon$ is a suitably chosen tolerance. In practice it is absolutely impossible to choose correctly the value of the tolerance $\epsilon$. The subject of this paper is to show how discrete stochastic arithmetic remedies the problem of this tolerance, as well as the problem of the other tolerances which are needed in the other checks of the QMR method with the estimation of the accuracy of some intermediate results. Numerical examples are used to show the good numerical properties.
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BILUS: a block version of ILUS factorization
Davod K. Salkuyeh,Faezeh Toutounian 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.-
ILUS factorization has many desirable properties such as itsamenability to the skyline format, the ease with which stability may bemonitored, and the possibility of constructing a preconditioner with sym-metric structure. In this paper we introduce a new preconditioning tech-nique for general sparse linear systems based on the ILUS factorizationstrategy. The resulting preconditioner has the same properties as the ILUSpreconditioner. Some theoretical properties of the new preconditioner arediscussed and numerical experiments on test matrices from the Harwell-Boeing collection are tested. Our results indicate that the new precondi-tioner is cheaper to construct than the ILUS preconditioner.
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A PRECONDITIONER FOR THE LSQR ALGORITHM
Karimi, Saeed,Salkuyeh, Davod Khojasteh,Toutounian, Faezeh Korean Society of Computational and Applied Mathem 2008 Journal of applied mathematics & informatics Vol.26 No.1
Iterative methods are often suitable for solving least squares problems min$||Ax-b||_2$, where A $\epsilon\;\mathbb{R}^{m{\times}n}$ is large and sparse. The well known LSQR algorithm is among the iterative methods for solving these problems. A good preconditioner is often needed to speedup the LSQR convergence. In this paper we present the numerical experiments of applying a well known preconditioner for the LSQR algorithm. The preconditioner is based on the $A^T$ A-orthogonalization process which furnishes an incomplete upper-lower factorization of the inverse of the normal matrix $A^T$ A. The main advantage of this preconditioner is that we apply only one of the factors as a right preconditioner for the LSQR algorithm applied to the least squares problem min$||Ax-b||_2$. The preconditioner needs only the sparse matrix-vector product operations and significantly reduces the solution time compared to the unpreconditioned iteration. Finally, some numerical experiments on test matrices from Harwell-Boeing collection are presented to show the robustness and efficiency of this preconditioner.
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A Preconditioner for the LSQR algorithm
Saeed Karimi,Davod Khojasteh Salkuyeh,Faezeh Toutounian 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.1
Iterative methods are often suitable for solving least squares problems min∥Ax − b∥₂, where A ∈ Rm×n is large and sparse. The well known LSQR algorithm is among the iterative methods for solving these problems. A good preconditioner is often needed to speedup the LSQR convergence. In this paper we present the numerical experiments of applying a well known preconditioner for the LSQR algorithm. The preconditioner is based on the ATA-orthogonalization process which furnishes an incomplete upper-lower factorization of the inverse of the normal matrix ATA. The main advantage of this preconditioner is that we apply only one of the factors as a right preconditioner for the LSQR algorithm applied to the least squares problem min∥Ax − b∥₂. The preconditioner needs only the sparse matrix-vector product operations and significantly reduces the solution time compared to the unpreconditioned iteration. Finally, some numerical experiments on test matrices from Harwell-Boeing collection are presented to show the robustness and efficiency of this preconditioner. Iterative methods are often suitable for solving least squares problems min∥Ax − b∥₂, where A ∈ Rm×n is large and sparse. The well known LSQR algorithm is among the iterative methods for solving these problems. A good preconditioner is often needed to speedup the LSQR convergence. In this paper we present the numerical experiments of applying a well known preconditioner for the LSQR algorithm. The preconditioner is based on the ATA-orthogonalization process which furnishes an incomplete upper-lower factorization of the inverse of the normal matrix ATA. The main advantage of this preconditioner is that we apply only one of the factors as a right preconditioner for the LSQR algorithm applied to the least squares problem min∥Ax − b∥₂. The preconditioner needs only the sparse matrix-vector product operations and significantly reduces the solution time compared to the unpreconditioned iteration. Finally, some numerical experiments on test matrices from Harwell-Boeing collection are presented to show the robustness and efficiency of this preconditioner.
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