Workshop on Numerical Calculation with Guaranteed Accuracy
主催 早稲田大学，日本シミュレーション学会，電子情報通信学会，日本応用数理学会
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)。
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最終更新 2001/5/11