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.
======================================================================