他組織イベント案内

2017-03-08 Reichel教授による数値線形代数の講演

投稿者:  速水 謙(国立情報学研究所)
会場: 国立情報学研究所 17th floor, Room 1716
概要: 非適切問題および行列関数の有理近似について(Lectures on Numerical Linear Algebra by Prof. Reichel)

日程 2017年 3月 8日 10:00 - 12:00
会場 国立情報学研究所(National Institute of Informatics),
17th floor, Room 1716
(access)
Speaker Professor Lothar Reichel
Department of Mathematical Sciences
Kent State University, Kent, OH, USA
Talk 1 Time: 10:00-11:00 am

Title: Adaptive Cross Approximation for Ill-Posed Problems

Abstract:
Integral equations of the first kind with a smooth kernel and perturbed right-hand side, which represents available contaminated data, arise in many applications. Discretization gives rise to linear systems of equations with a matrix whose singular values cluster at the origin. The solution of these systems of equations requires regularization, which has the effect that components in the computed solution connected to singular vectors associated with small singular values are damped or ignored. In order to compute a useful approximate solution typically approximations of only a fairly small number of the largest singular values and associated singular vectors of the matrix are required. The present paper explores the possibility of determining these approximate singular values and vectors by adaptive cross approximation. This approach is particularly useful when a fine discretization of the integral equation is required and the resulting linear system of equations is of large dimensi
ons, because adaptive cross approximation makes it possible to compute only fairly few of the matrix entries.
This talk presents joint work with T. Mach, M. Van Barel, and R. Vandebril.
Talk 2 Time: 11:00-12:00 am

Title: Convergence Rates for Inverse-Free Rational Approximation of Matrix Functions

Abstract:
Many applications in Science and Engineering require the evaluation of matrix functions, such as the matrix exponential or matrix logarithm, of a large matrix. We are concerned with the situation when one is interested in computing a matrix function of the form f(A)v, where A is a large square matrix and v is a vector. In situations when it is impractical or impossible to evaluate f(A) explicitly, one often approximates f(A)v by first reducing A by a Krylov subspace method.
Standard Krylov methods deliver polynomial approximations of f(A), while rational Krylov subspace methods give rational approximations with predetermined poles.
The former methods generally require more Krylov steps and a Krylov subspace of larger dimension than the latter to yield approximations of comparable accuracy.
Therefore, rational Krylov methods often yield approximations that are faster to evaluate than standard Krylov methods.
It follows that if many evaluations of f(A)v are required (e.g., because f depends on a parameter that is varied), then it may be advantageous to use a rational Krylov method instead of a standard one. However, the solution of linear systems of equations with shifted matrices A required to construct an orthogonal basis for a rational Krylov subspace may create numerical difficulties and/or require excessive computing time.
It therefore may be attractive to use inverse-free rational Krylov methods, which require less storage space and yield simpler approximations of f(A)v than standard Krylov methods, and avoid the solution of linear systems of equations with shifted matrices A. We derive geometric convergence rates for approximating matrix functions by using inverse-free rational Krylov methods.
This talk presents joint work with C. Jagels, T. Mach, M. Van Barel, and R. Vandebril.
問い合わせ先 速水 謙
〒101-8430 東京都 千代田区 一ツ橋 2-1-2 国立情報学研究所
e-mail: hayamiseparatornii.ac.jp
詳細 web http://www.nii.ac.jp/en/event/list/0308-2017