応用数理学会「計算の品質部会」
S.M.Rump氏 (Hamburg-Harburg大学教授)講演会
場所 早稲田大学理工学部51号館3階第3会議室(新宿区大久保3-4-1)
精度保証付き数値計算の話題I
日時 10月22日(月) 2:40-4:10
精度保証付き数値計算の話題I
日時 10月24日(水) 2:40-4:10
Rump氏は精度保証付き数値計算の研究分野に長く従事し，さまざまな理論やMATLAB上のTool Box INTLABの開発などで良い成果を出されています。本分野の入門から初めて，対称性のある摂動の理論など氏の最新成果まで２回にわたり講演いただきます。皆様のご参加をお待ちしております。
International Workshop on Numerical Analysis with Guaranteed Accuracy
Oct. 8 13:00-18:10
The Third Meeting Room, 51-Building, School of Science and Engineering, Waseda University, Tokyo 169-8555
Sponsored by Waseda University, IEICE, Technical Group on Self-Validating Computing Japan Society for Simulation Technology, Japan SIAM
Program: 8 Oct
(1) 13:00-13:50 Gotz Alefeld(Universitat Karlsruhe) Verification of solutions of linear and nonlinear complementarity problems
(2) 13:50-14:30 Markus Neher(Universitat Karlsruhe) Geometric Series Bounds for the Local Errors of Taylor Methods for ODEs
(3) 14:40-15:30 Jurgen P. Herzberger (Universitaet Oldenburg) Bounds for the effective rate of interest of some special cashflows
Abstrct: We first consider the cashflow of an ordinary simple annuity. Applying the principles in finance and the US-rule for the calculation of the effective rate of interest, we get as result a certain polynomial to be solved for its unique positive root. Under closer inspection, this type of polynomial has already been considered in Numerical Analysis in connection with the calculation of the $Q$-order or $R$-order of convergence of iterative numerical processes, beginning with A.~Ostrowski and J.F.~Traub. It can be shown that the bounds for the positive root in question in some later papers can indeed easily be derived by applying a simple variable transformation to such a polynomial and to the bounds given by J.F.~Traub. Next, we examine an interesting practical problem concerning the estimation or the bounding of the effective rate of interest of an annuity which we get by changing in a certain way the conditions of a given annuity with known datas. We give quite reasonable bounds for the effective rate of the changed annuities in terms of the effective rate of the original one. This question will then be reconsidered under the more general aspect of an annuity with geometrically growing payments. Applying the results of the first part of the talk we get also bounds in this case in a simple manner. As a kind of byproduct we show an interesting possibility for bounding the order of convergence of certain types of interpolatory iteration methods in Numerical Analysis and thus come back to the oberservation at the beginning.
(4) 15:30-16:00 Wolfgang Schwarz (Technische Universitaet Dresden) Some open problems form chaos theory
(5) 16:10-16:40 Yusuke Nakaya (Waseda Univ) Fast implementation of Krawczyk's method
(6) 16:40-17:10 Maruyama (Waseda Univ) Element-wise eigenvalue inclusion method
(7) 17:10-17:30 Shin'ichi Oishi (Waseda Univ) Slab: a MATLAB-like interpreter for verified computation
(8) 17:30-18:10 Lylia A'tanassova(MAN,Munich) Convex directions for complex Hurwitz stable polynomials and quasipolynomials
Workshop on Numerical Calculation with Guaranteed Accuracy
早稲田大学理工学部55館S棟２階第4会議室 (新宿区大久保3-4-1)
参加は自由，参加費無料です。
目的 Kulisch教授の来日を機会にワークショップを開催して精度保証付き数値計算の最近の話題について議論する。
5月8日(火) (午前の部) 10:40-12:10
(特別講演) Ulrich W. Kulisch (Universitaet Karlsruhe, D-76128 Karlsruhe, Germany)
Do we Need and can we Build Better Computers?
(abstract) Advances in computer technology are now so profound that the capability and repertoire of computers can and should be extended. At a time where millions of transistors can be placed on a single chip, where computing speed is measured in GIGAFLOPS and memory space in GIGA-words there is no need anymore to perform all computer operations by the four elementary floating-point operations with all the shortcomings of this methodology. The talk will define advanced computer arithmetic. It extends the accuracy requirement of the elementary floating-point operations - for instance, as defined by the IEEE arithmetic standard -- to all operations in the usual product spaces of computation: the real and complex vector spaces and their interval counterparts. This enhances the mathematical power of the digital computer considerably. The new expanded computational capability is gained at modest cost and even implicates a performance advantage. It is obtained by putting a methodology into modern computer hardware which was already available on old calculators before the electronic computer entered the scene. The new arithmetic increases both, the speed of a computation as well as the accuracy of the computed result. By operator overloading in a programming language a long real arithmetic (array of reals), matrix and vector arithmetic, interval arithmetic, a long interval arithmetic as well as automatic differentiation arithmetic become part of the runtime system of the compiler. This simplifies programming a great deal. For instance, derivatives, Taylor-coefficients, gradients, Jacobian and Hessian matrices or enclosures of these are directly computed out of the expression by a simple type change of the operands. Techniques are now available so that with this expanded capability, the computer itself can be used to appraise the quality and the reliability of the computed results over a wide range of applications. Problem solving routines with automatic result verification have been developed for many standard problems of numerical analysis as for linear or nonlinear systems of equations, for differential or integral equations, etc. as well as for a large number of applications in the engineering and natural sciences. The talk will discuss a variety of basic solutions and applications.
5月8日 (午後の部 オーガナイズドセッション) 14:00--17:00
(1) *Takeshi Ogita (Waseda University), Shin'ichi Oishi (Waseda University) and Yasunori Ushiro (Hitachi Co.) Fast Inclusion and Residual Iteration of Solution of Linear Systems
(2) Daisuke Oishi (University of Tokyo) *Sunao Murashige (University of Tokyo) Shin'ichi Oishi (Waseda University) Numerical verification of solutions of the Chandrasekhar integral equation
(3) Masato KAMIYAMA (University of Tokyo) *Ken HAYAMI (National Institute of Informatics) Sunao MURASHIGE (University of Tokyo) Shin'ichi OISHI (Waseda University) Validated Numerical Solution of Large Scale Eigenvalue Problems
Discussion
*印 は講演者
5月9日(水) (午前の部) 10:40-12:10
(特別講演) Ulrich W. Kulisch (Universitaet Karlsruhe, D-76128 Karlsruhe, Germany)
Advanced Arithmetic for the Digital Computer - Design of Arithmetic
(abstract) The speed of digital computers is ever increasing. While emphasis in computing was traditionally on speed, more emphasis can and must now be put on accuracy and reliability of results. Numerical mathematics has devised algorithms that deliver highly accurate and automatically verified results. This means that these computations carry their own accuracy control. However, the arithmetic available on existing processors makes these methods extremely slow. Their implementation requires suitable arithmetic support and powerful programming tools which are not generally available. The talk will define advanced computer arithmetic. It extends the accuracy requirement of the elementary floating-point operations - for instance, as defined by the IEEE arithmetic standard -- to all operations in the usual product spaces of computation: the real and complex vector spaces and their interval correspondents. The talk will discuss the design of arithmetic units for advanced computer arithmetic. The new expanded computational capability is gained at modest cost. It increases both, the speed of a computation as well as the accuracy of the computed result. A new computer operation, the scalar product, is fundamental to the development of advanced computer arithmetic. It will be shown that fixed-point accumulation of products is the fastest way to execute scalar products on the computer. This is the case for all kinds of computers (Personal Computer, Workstation, Mainframe or Super Computer). In contrast to floating-point accumulation, fixed-point accumulation is error free. Not a single bit is lost. With a fast and accurate scalar product, fast multiple precision arithmetic can easily be provided on the computer. Finally it will be shown that on superscalar processors interval operations can be made as fast as simple floating-point operations with only very modest hardware costs. A coprocessor for advanced computer arithmetic has been designed and built in CMOS VLSI-technology at the speakers institute. It speeds up vector, matrix and other operations and computes them to full accuracy or with only one final rounding. It is the first hardware implementation of the "GAMM/IMACS Proposal for Accurate Floating-point Vector Arithmetic."
Literature: Kulisch, U. W.: Advanced Arithmetic for the Digital Computer - Design of Arithmetic Units.: http//www.elsevier.nl/locate/entcs/volume24.html 72 pages.
Kulisch, U. W.: Advanced Arithmetic for the Digital Computer - Interval Arithmetic revisited.: ftp://ftp.iam.uni-karlsruhe.de/pub/documents/kulisch/advarith.ps.gz 63 pages.
5月9日 (午後の部 オーガナイズドセッション) 14:00--17:00
(1) (特別講演) Tetsuro Yamamoto (Ehime University) Sharp error estimates for finite difference method with non-equidistant nodes applied to two-point boundary value problems
(2) Nobito Yamamoto (The University of Electro-Communications) Error estimates of finite element solutions by Spectrum Method with Verified Computation
(3) Masahide Kashiwagi and *Takatomi Miyata (Waseda University) On the Range of Evaluation of Polynomials Using Affine Arithmetic
*印 は講演者
終了後 懇親会を開催します(懇親会のみ会費あり)。
(プログラムは現在編成中ですので変更の可能性があります。)
事務局 :早稲田大学理工学部情報学科大石研究室 事務局へのメール
設立 1992年7月21日 理事会承認
目的 「計算のやりっぱなし」の時代から脱却するための工学的方法論を確立すること。
- 精度保証付き計算，自動微分，丸め誤差の制御，区間計算，無限精度計算，等々のような，かなり高価につくが理論的に厳密性の高い方法（これらにも真に厳密なものといくらか近似的なものとがありうる）と，より伝統的，経験的な方法との比較評価（理論的，モデル実験的，および実規模計算によるもの）を中心とする。
- 特に，費用・効果 の観点を重視する。（”効果”の評価は立場により大きく異なるはず。）
初代主査 伊理正夫先生の趣意書より
主査 大石進一（早大理工情報)
部会員：
日本応用数理学会会員あるいは同会員になる意志のある者。会合ごとに，話題によって，ゲストは随時招待できる。
founding member
伊理正夫（初代主査） 大石進一，川合敏雄，久保田光一，杉原正顕，築山修治，戸川隼人，鳥居達生，中尾充宏，牧野光則，平野管保，三井たけ友，室田一雄，森正武，山本哲朗
活動の目的を具体的に述べた伊理先生の構想
1. 諸手法の評価，比較
(a) 視点
a-1 品質評価手法
(i) 最も素朴な（恣意的パラメタも含む）”やりっぱなし”の計算
(ii) 離散化のパラメタ（刻み幅，分割数，等）を系統的に変化させながら数値解をかんさつすることによって，離散か誤差（打ち切り誤差）とそのパラメタ依存性を推定し，その推定値を利用して”品質管理”をする。(Richardson,Romberg等の系統。）
(iii) 計算桁数を変化させて何遍か計算することによって，丸め誤差とその計算桁数依存性を推定し，それを利用して”品質管理”をする。(データの摂動をする場合としない場合がある。）
(iv) 以下の(v)の線形近似版---すなわち，線形近似の”感度解析”で離散か誤差や丸め誤差の影響を推定して，それにもとずいて”品質管理”をする。
(v) 区間計算（そぼくなものからかなり凝ったものまでいろいろある）を用いる。
(vi) 無限精度計算を用いる。（可能な場合は限られるし，また，上記(v)との組み合わせも多くの場合必要であろう。）
a-2 評価基準
(i) 計算時間
(ii) 所用記憶領域
(iii) プログラムの難易度
a-3 計算の種類
(i) 数値積分，数値微分
(ii) 代数方程式
(iii) 線形方程式
(iv) 非線形方程式
(v) 常微分方程式
(vi) 偏微分方程式(有限要素法，境界要素法)
(b) 作業方針 略
2001年度の活動予定
5月 Kulisch教授講演会とワークショップ(5月8日，９日於早稲田大学)
10月 山本哲朗先生のシンポジウムを共催 その前後にワークショップ開催予定
Recent Advances in Computational Mathematics :
International Conference on "Recent Advances in Computational Mathematics"
-- organized by Ehime university, -- co-organaized by Japan SIAM, Kochi university, Shimane university, Yamaguchi university
2001 年 10 月 10 -- 13 日 at Hotel JAL CITY Matsuyama, Matsuyama, JAPAN.
関連リンク
応用数理学会 (2001/4/5 応用数理学会からこのページにリンクが貼られた)