(立教大学)

場所：立教大学 ４号館３階共同研究室

共催 「数値／数式ハイブリッド計算に基づくロバスト最適化プラットフォームの 構築」 研究代表者 穴井宏和 富士通 CREST, JAPAN SCIENCE & TECHNOLOGY AGENCY 10:00-11:00 Prof. V. Gerdt (Joint Institute for Nuclear Research, Russia) Linear Difference Ideals and Generation of Difference Schemes for PDEs 11:15-12:15 Prof. R.Liska (Czech Technical University in Prague, Czech) Using Computer Algebra in Numerical Treatment of Partial Differential Equations 14:00-15:00 Dr. A. Dolzmann (University of Passau, Germany) Combined Quantifier Elimination 15:15-16:15 Dr. A.Seidl (University of Passau, Germany) Speeding up Cylindrical Decomposition ======================================================================= Andreas Dolzmann University of Passau, Germany Title: Combined Quantifier Elimination Abstract: Practical real quantifier elimination began with the work of George Collins and his efforts to implement all necessary algorithms in the SACLIB library. His vision finally results in the famous QEPCAD project. Since then many theoretical research was done in the area of effective quantifier elimination. A small number of persons and research groups, however, pursue the ideal of Collins to realize the found methods in implementation. In the group of Volker Weispfenning, much theoretical and practical efforts was spent to build an efficient extension, called REDLOG, of the computer algebra system REDUCE augmenting computer algebra with computer logic, in particular with quantifier elimination. In this talk we present some quantifier elimination procedures implemented in REDLOG and a application of combining them. We focus on the gain of combining several main algorithms of computer algebra and computer logic. ======================================================================= Vladimir Gerdt Laboratory of Information Technologies, Joint Institute for Nuclear Research, Russia Title: Linear Difference Ideals and Generation of Difference Schemes for PDEs Abstract: In this talk we present an algorithmic approach to generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations for unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations for the unknown functions and their partial derivatives is derived. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Groebner basis of the linear differential ideal generated by the polynomials in the discrete system. For this computation we use a Maple package implementing our Groebner bassis algorithm and illustrate the method by some examples. ======================================================================= Richard Liska Czech Technical University in Prague, Czech Republic Title: Using Computer Algebra in Numerical Treatment of Partial Differential Equations Abstract: Most practical partial differential equations (PDEs) cannot be solved analytically and have to be treated numerically. The analysis of PDEs and the development of the numerical method solving them often require tedious algebra which can be preformed by computer using computer algebra (CA) tools. Advantages of using the CA tools on different stages of the PDEs analysis and numerical finite difference method development will be demonstrated. These stages include well-posedness analysis of PDEs, truncation error, modified equation and stability analysis of finite difference schemes together with automatic numerical source code generation. Advanced quantifier elimination algorithms are employed for well-posedness and stability analysis. ====================================================================== Andreas Seidle University of Passau, Germany Title: Speeding up Cylindrical Decomposition Abstract: Cylindrical algebraic decomposition on its own and as a method to perform real quantifier elimination is a powerful tool to solve a wide range of interesting problems. To successfully put real world problems into the reach of this method, however, there is need to improve the practical complexity. Luckily, significant past and current progress has been made to meet this goal. In this talk we will look into mathematical improvements that can be made, methods to prepare the input that can be deployed, degrees of freedom that can be exploited and application-oriented paradigms that can be adopted. ======================================================================

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The computer algebra seminar is co-organized by

M.Noro, noro@math.kobe-u.ac.jp

N.Takayama, takayama@math.kobe-u.ac.jp

K.Yokoyama, yokoyama@rkmath.rikkyo.ac.jp

For more info see http://www.math.kobe-u.ac.jp/seminar.html/compalg.html

http://www.math.kobe-u.ac.jp/seminars.html