Details of a Researcher - Nodera, Takashi

Papers - Nodera, Takashi

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Block symplectic Gram-Schmidt metho

Yoichi MATSUO, NODERA TAKASHI

ANZIAM J. 66 C416 - C430 2016.03

Research paper (scientific journal), Joint Work, Accepted

A note on Lanczos algorithm for computing PageRank

#HKAZUMA TERAMOTO, TAKSHI NODERA</>

Springer Proceedings in Mathematics & Statistics (Springer) 124 25 - 33 2016.01

Joint Work, Accepted

Block Symplectic Gram-Schmidt Method

MATSUO YOUICHI, NODERA TAKASHI

KSTS/RR-15/004 (Department of Mathematics, Keio University) 2015.03

Joint Work

The L-BFGS Method for Nonlinear GMRES Acceleration

YOKOTA YUTO, NODERA TAKASHI

KSTS/RR-15/005 (Department of Mathematics, Keio University) 2015.03

Joint Work

Lanczos Type Method for Computing PageRank

ERAMOTO KAZUMA, NODERA TAKASHI

KSTS/RR-15/003 (Department of Mathematics, Keio University) 2015.03

An Efficient Implementation of the Block Gram-Schmidt Method

NODERA TAKASHI, YOICHI MATUO

The ANZIAM J. 54 C394 - C409 2013.06

Research paper (scientific journal), Joint Work, Accepted

Two-level Parallel Preconditioning derived from an Approximate Inverse based on the Sharman-Morrison Formula

ZHANG LINJIE, MORIYA KENTARO, NODERA TAKASHI

The ANZIAM J. 54(E) ( 1 ) E1-E25 2012.10

Research paper (scientific journal), Joint Work, Accepted, ISSN 1445-8735

適切なblock-sizeの決定法を用いたBlock Gram-Schmidt法の並列

MATSUO YOUICHI, NODERA TAKASHI

情報処理学会HPC電気通信大学研究会 (情報処理学会) HPC-134 ( 2 ) 1-4 2012.06

Research paper (conference, symposium, etc.), Joint Work

A New Stopping Rule: GMRES for Linear Discrete Ill-posed Problems

KUROIWA NAO, NODERA TAKASHI

KSTS 2012.03

Research paper (other academic), Joint Work

The Optimal Block-size for the Block Gram-Schmidt Orthogonalization

MATUO YOUICHI、NODERA TAKASHI

Journal of Science and Technology 49 ( 4A ) 348-354 2011.10

Research paper (scientific journal), Joint Work, ISSN 0866 708X

We have proposed an adaptive procedure for determining the optimal block-size for the Block Gram-Schmidt orthogonalization, which is a process for computing the QR factorization of matrix X.Accurate block-size is different for each numerous problem. In this paper, we have explored two new schemes for searching and finding accurate
block-size under one execution of the QR factorization.
Some numerical experiments will be used to illustrate the effectiveness of the proposed algorithm.

離散型線形悪条件問題に対する修正GMRES法

KUROIWA NAO, NODERA TAKASHI

情報処理学会論文誌 52 ( 9 ) 2812-2821 2011.09

Research paper (scientific journal), Joint Work

近年，第一種Fredholm積分方程式の離散化から導かれる連立$1$次方程式の近似解法としてGMRES法が注目を集めている．この連立1次方程式は離散型線形悪条件問題に分類され，悪条件問題としての困難な性質を持つため，克服すべき点は多い．今回提案する反復終了条件は，悪条件問題に対する古典的な手法の1つであるTikhonov正則化に基づく値を利用したものである．通常，GMRES法の反復終了条件には，相対残差ノルムを利用することが多い．しかし，離散型線形悪条件問題に対して用いる場合には，適切な反復終了条件として働かないため，他の基準が必要となる．そこで，適当なタイミングで反復終了を可能とするSimplified Tikhonov値を
GMRES法の新たな反復終了条件として提案する．

最小二乗問題に対するblock Gram-Schmidt法の適用

MATSUO YOUICHI，NODERA TAKASHI

2011-HPC-130(SWoPP2011) (情報処理学会) 1-4 2011.07

Research paper (conference, symposium, etc.), Joint Work

downdating を含む最小二乗解を求める問題においても，Gram-Schmidt法は有用とされている．block Gram-Schmidt法は，$Q$のデータを使う回数を減らしキャッシュメモリを効率的に使用することにより，直交化をより効率的に行うことができる．これによって，block downdating のQR分解をより高速に求めることができる．block Gram-Schmidt法を用いたblock downdating に対する手法を提案し，数値実験でその有用性を示す．

A GMRES(m) Method with Two Stage Deflated Preconditioners

SHIROISHI JUNGO、NODERA TAKASHI

ANZIAM J. 52 C222-C236 2011.05

Research paper (scientific journal), Joint Work

The GMRES($m$) method is often used to compute Krylov subspace solutions oflarge sparse linear systems of equations. Morgan developed a new procedure, called Deflated GMRES($m$, $k$)[SIAM J. Sci. Comput. 24 (2002), pp. 20-37].
This method deflates $k$, the smallest eigenvalue, and improves eigenvalue distribution.

In this paper, we explore a new deflated GMRES($m$, $k$, $l$) which uses a two stage deflation technique. In addition, the results of the numerical experiments for test matrices are tabulated to illustrate that our approach is effective in solving a wide range of problems.

A GMRES(m) Method with Two Stage Deflated Preconditioners

NODERA TAKASHI

ANZIAM J. 52 C222-C236 2011.05

Research paper (scientific journal), Joint Work

悪条件問題に対するGMRES法の反復終了条件

NODERA TAKASHI,　KUROIWA NAO

情処研報/SWoPP2010 (情報処理学会) 1-7 2010.08

Research paper (conference, symposium, etc.), Joint Work

Efficient Approximate Inverse Preconditioning Techniques for Reduced Systems on Parallel Computers

NODERA TAKASHI, MORIYA KENTARO, Zhang Rinjie

SUBSTRUCTURING TECHNIQUES AND DOMAIN DECOMPOSITION METHODS (United　Kingdam・Saxe-Coburg Publications) 203-228 2010.04

Research paper (conference, symposium, etc.), Joint Work

A note on the GMRES method for Linear Discrete ill-posed Problems

NODERA TAKASHI, KUROIWA NAO

Adv. Appl. Math. and Mech. 1 ( 6 ) 816-829 2009.12

Research paper (scientific journal), Joint Work, Accepted

In this paper, we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES. It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems. In this paper we have focused on the residual norm and have come-up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance respectively. Our numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.

An Approximate matrix inversion procedure by parallelization of the Sherman-Morrison formula

NODERA TAKASHI,　MORIYA　KENTARO, Zhang Linjie

The ANZIAM Journal (Australia) 51 ( 1 ) 1-9 2009.10

Research paper (scientific journal), Joint Work, Accepted

The Sherman–Morrison formula is one scheme for computing the approximate inverse preconditioner of a large linear system of equations. However, parallelizing a preconditioning approach is not straightforward as it is necessary to include a sequential process in the matrix factorization. In this paper, we propose a formula that improves the performance of the Sherman–Morrison preconditioner by partially parallelizing the matrix factorization. This study shows that our parallel technique implemented on a PC cluster system of eight processing elements significantly reduces the computational time for the matrix factorization compared with the time taken by a single processor. Our study has also verified that the Sherman–Morrison preconditioner performs better than ILU or MR preconditioners.

Usage of the Convergence Test of the Residual Norm in the Tsuno-Nodera version of the GMRES algorithm

NODERA TAKASHI,　MORIYA　KENTARO

The ANZIAM Journal (Australia) 49 ( 2 ) 293-308 2009.10

Research paper (scientific journal), Joint Work, Accepted

Tsuno and Nodera proposed a new variant of the GMRES(m) algorithm. Their algorithm is referred to as the GMRES(mmax) algorithm and performs the restart process adaptively, considering the distribution of the zeros of the residual polynomial. However, unless the zeros of the residual polynomial are distributed uniformly, mmax is always chosen and their algorithm becomes almost the same as the GMRES(m) algorithm with m=mmax.

In this paper, we include a convergence test for the residual norm in the GMRES(mmax) algorithm and propose a new restarting technique based on two criteria. Even if the distribution of zeros does not become uniform, the restart can be performed by using the convergence test of the residual norm. Numerical examples simulated on a Compaq Beowulf computer demonstrate that the proposed technique accelerates the convergence of the GMRES(mmax) algorithm.

Adaptive Augmented GMRES method for Solving Ill-posed Problems

NODERA TAKASHI, KUROIWA NAO

The ANZUAM J. (AUSTRALIA) 50 C567-C677 2009.01

Research paper (scientific journal), Single Work, Accepted

The GMRES method is an iterative method that provides better solutions when dealing with large linear systems of equations with a non-symmetric coefficient matrix. The GMRES method generates a Krylov subspace for the solution, and the augmented GMRES method allows augmentation of the Krylov subspaces by a user supplied subspace which represents certain known features of the desired solution. The augmented GMRES method performs well with suitable augmentation, but performs poorly with unsuitable augmentation. The adaptive augmented GMRES method automatically selects a suitable subspace from a set of candidates supplied by the user. This study shows that this method maintains the performance level of augmented GMRES and lightens the burden it puts on its users. Numerical experiments compare robustness as well as the efficiency of various heuristic strategies.

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