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Hirokazu Anai

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Computer Algebra Theory and its Applications

Hirokazu Anai
Degree: Doctor of Information Science and Engineering (Information Science and Technology, the University of Tokyo)
Research Interests: Computer Algebra, Symbolic-Numeric Hybrid Computation
Unit:Number & Equation

Report

Figure 1 Model-based design flow

My research interest lies in computational algebra, so called ``computer algebra”, i.e., the theory and application of algebraic and symbolic computation. We investigate efficient algorithms of computer algebra in order to develop new methodologies to resolve various kinds of issues in science, engineering, and industry. Our research results have been implemented as some tools for practical engineering and industrial applications, where the applications range from system and control theory to physics and biology. In particular, our current focus is on “Monozukuri (manufacturing)”. We have been developing new simulation techniques based on algebraic computations (including the joint use of numerical computations) and moreover advanced Monozukuri methodologies (e.g., systematic design and verification methods) utilizing these simulation techniques. Our aim is to establish a new paradigm of Monozukuri through building a new system and control theory based on mathematics.

Figure 2 Real algebraic geometry and hybrid optimization

Recently, in manufacturing design, model-based design and development have attracted much attention. Analysis, design and verification problems in Monozukuri can often be treated as mathematical constraints via mathematical models of the target systems. Therefore, developments in design processes are closely related to the available computational methods (optimization methods) at that time. In fact, a lot of research on design methods based on numerical optimization has been done in various fields. In particular, methods which solve analysis and design problems using a numerical convex optimization method enable us to get globally optimum solutions to problems that cannot be solved analytically up to then. These design methods are becoming more practical due to enhanced computer performance and the development of algorithms with superior accuracy and efficiency. However, some hurdles still remain in these numerically computed design methods. Demand for higher quality, better performance, high-value added, and manufacturing small quantities of a wide variety of products require more accurate global optimum solutions of non-convex problems as well as parametric solutions to the problems (sets of feasible solutions as possible regions together with optimal solutions with respect to the parameters)

As an effective and promising method, we focus on constraint solving/optimization problems based on symbolic and algebraic computation, and we are trying to develop efficient computational algorithms. Specifically, our research includes methods using Quantifier Elimination (QE) and Groebner Bases (GB), which are algebraic algorithms based on theories of algebraic geometry and real algebraic geometry. However, to realize a practically effective method using the QE algorithm, the method must be sped up. For the purpose, we have been developing innovative optimization methods based on symbolic-numeric computation by combining it with numerical optimization methods.

Figure 3 Research objectives and goal

We strive to discover mathematical principles in these activities, including practical applications, such as Monozukuri, so that we can find their synergy to applied(computational) mathematics. Then we also aim at creating new mathematical principles necessary for practical applications and conducting research which generates innovative mathematical concepts and theoretical development. To this end, we are currently doing collaborative research with researchers from diverse fields, as shown in Fig. 3, such as practical applications, system and control theory, applied mathematics, and pure mathematics.

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