BIF is an incomplete factorization of a square matrix into triangular factors in which standard LU or LDL^T factors (direct factors) and their inverses (inverse factors) can be obtained at the same time. This method is derived from the approach based on the Sherman-Morrison formula. Direct and inverse factors directly influence each other throughout the computation, and consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. For the symmetric positive definite case, we derive the theory and present an algorithm for computing the incomplete LDL^T factorization, we also briefly analyze the nonsymmetric case and how to apply the preconditioner to least square problems. Experimental results will be presented.

 In this talk, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method.

 What happens if the comparison matrix of a given matrix is a singular M-matrix? This question will be answered and characterizations of H-matrices with singular or nonsingular comparison matrix will be analyzed. In particular, the case of irreducible matrices will be analyzed and some insights into the reducible case will be given. The spectral radius of the Jacobi matrix of the comparison matrix and the generalized diagonal dominance property are used in the characterizations. Finally, from these characterizations, a partition of the general H-matrix set in three classes is obtained.

  In the second part we will focus on the Schur complement of H-matrices. It is well-known that the Schur complement of some H-matrices is an H-matrix. We will study the Schur complement of any general H-matrixt. In particular it is proved that the Schur complement, if it exists, is an H-matrix and it is studied to which class of H-matrix the Schur complement belongs to. In addition, results are given for singular irreducible H-matrices and for the Schur complement of nonsingular irreducible H-matrices.

FIFTH CONFERENCE ON NUMERICAL ANALYSIS
(NumAn 2012)
RECENT APPROACHES TO NUMERICAL ANALYSIS:
THEORY, METHODS AND APPLICATIONS

• 場所: 芝浦工業大学・大宮校舎 5 号館 (旧・システム工学部棟) 5241 教室


• 講演者: 山中 脩也 氏 (早稲田大学 理工学術院総合研究所)　


• 講演題目: 高精度で高可搬な精度保証付き高速初等関数計算法の構築


• 講演概要: 本講演は，高精度・高可搬・高速・精度保証付きという四つの特徴を持つ数値計算法について述べます．講演者は，初等関数や特殊関数の計算について，最近点丸めにおける倍精度浮動小数点数だけを利用して（高可搬），倍精度浮動小数点数の倍の精度を（高精度），既存手法より実行時間を早く（高速），精度保証付きで達成する，計算手法の設計を目指しています．これらが達成されれば，現在使っている計算機環境のまま，わずかなソフトウエアの変更で，従来の倍の精度の結果が 100% 間違える事なく計算できることになります．既にいくつかの初等関数に関してこれらを構築しましたので，その手法についてお話させていただきたいと思います．

（私事で恐縮です…）

日時: 2012年4月20日（金）13:00 - 16:15
場所: 東京都目黒区駒場3-8-1 東京大学駒場キャンパス
　 　 大学院数理科学研究科第2講義室

　　　　　　　プログラム

13:00-13:40 植田琢也（聖路加国際病院）
「臨床画像診断医より数学者へのメッセージ
　～ヒトの画像評価アルゴリズム理解のために～」

14:00-15:00 石橋雄一 （（株）スタットラボ ）
「画像情報とテキスト情報をもとにした病理診断支援システム」

15:15-16:15 中根和昭（大阪大学）
「組合せ不変量アルゴリズムを用いた病理画像自動診断システムについて」

主催：科学技術振興機構戦略的創造研究推進事業CREST
　　『放射線医学と数理科学の協働による高度臨床診断の実現』研究チーム

共催：東京大学数値解析セミナー (GCOEプログラム, 東京大学)

Further information

http://www.ems.okayama-u.ac.jp/suito/CREST/WS120420.html

Image deblurring, i.e., reconstruction of a sharper image from a blurred and noisy one, involves the solution of a large and very ill-conditioned system of linear equations, and regularization is needed in order to compute a stable solution. Krylov subspace methods are often ideally suited for this task: their iterative nature is a natural way to handle such large-scale problems, and the underlying Krylov subspace provides a convenient mechanism to regularized the problem by projecting it onto a low-dimensional "signal subspace" adapted to the particular problem. In this talk we consider the three Krylov subspace methods CGLS, MINRES, and GMRES. We describe their regularizing properties, and we discuss some computational aspects such as preconditioning and stopping criteria.

A fundamental imaging problem in microstructural analysis of metals is the reconstruction of local crystallographic orientations from X-ray diffraction measurements. This work develops a fast, accurate, and robust method for the computation of the 3D orientation distribution function for individual grains of the material in consideration. We study an iterative large-scale reconstruction algorithm, CGLS, and demonstrate that right preconditioning is necessary to provide satisfactory reconstructions. Our right preconditioner is not a traditional one that accelerates convergence; its purpose is to modify the smoothness properties of the reconstruction. We also show that a new stopping criterion, based on the information available in the residual vector, provides a robust choice of the number of iterations for these preconditioned methods.

Total Variation (TV) regularization is a powerful technique for image reconstruction tasks such as denoising, in-painting, and deblurring, because of its ability to produce sharp edges in the images. In this talk we discuss the use of TV regularization for tomographic imaging, where we compute a 2D or 3D reconstruction from noisy projections. We demonstrate that for a small signal-to-noise ratio, this new approach allows us to compute better (i.e., more reliable) reconstructions than those obtained by classical methods. This is possible due to the use of the TV reconstruction model, which incorporates our prior information about the solution and thus compensates for the loss of accuracy in the data. A consequence is that smaller data acquisition times can be used, thus reducing a patient's exposure to X-rays in medical scanning and speeding up non-destructive measurements in
materials science.

There are a number of applications where one wishes to know an acoustic field over an extended domain, whereas in most cases one can only perform point measurements (e.g. with a microphone). Even when few sources are active, it remains a challenging problem due to reverberation, that may be hard to characterize. This is typically a sampling problem, that raises a number of interesting questions: how many sampling points are needed, what are good distributions of sampling points, etc?
In this talk we will review a few test studies, in 2D (plates) and 3D (rooms), with numerical and experimental data, where a physicsbased sparse model for the acoustic wavefield is successfully used for its reconstruction, using significantly less measurements then would be required by classical Shannon sampling.