Mathematical Foundation of Computer Science Laboratory, Annual Review 2005, The University of Aizu

Annual Review 2005 > Mathematical Foundation of Computer Science Laboratory

Mathematical Foundation of Computer Science Laboratory


Hidesada Kanda Professor	Nobuyoshi Asai Assistant Professor

We conduct research works extensively related to the fundamental theories and advanced applications of mathematics, especially in numerical computation.

Prof. Kanda's research interests are mainly in numerical analysis of partial differential equations, and emphasis is placed on the development of program codes for Navier-Stokes Equation in complex-shaped flow fields. Differential geometry, tensor analysis, and grid generation are also studied as needed.

In 1883, Osborne Reynolds observed the transition between laminar and turbulent flows in circular pipes; the critical values of Rc were 12,830 for the color-dye method and 2030 for the pressure method. These values have yet to be theoretically obtained. Hence, Prof. Kanda succeeded in the calculation of the minimum critical Reynolds number, Rc(min), of 2040 in 1999 for the first time in history. Then, he wanted to confirm the Rc of 12,830 experimentally before starting calculations, but it is very difficult to reproduce Rc. Therefore, he has been carrying out experiments similar to Reynolds color-dye experiment everyday and every weekends since March 2003 to fix physical constants for Rc. Thus, he realized in 2004 that studies of science take three steps of analysis: verification of real phenomena, building of physical model, and calculation based on mathematical model.

Prof. Asai's major research fronts may be described as follows:

Matrix-theoretic high-performance algorithm construction, with application to the numerical computation of critically important quantities associated with difficult special functions occurring from 3-D wave equations, such as regular Coulomb wave functions, Mathieu functions, spheroidal wave functions, Lame functions, ellipsoidal wave functions. The Bessel function of the first kind, an easier function serves as a model case, where two types of computational problems were solved in the complex domain as well as real domain: given a complex order Bessel function, compute a given number of zeros closest to the origin to a specified accuracy; and given a complex number find a specified number of orders of Bessel functions closest to the origin, to a specified accuracy. There are no comparable algorithms known elsewhere which give as accurate an error estimate as ours.
Construction of hyper English word study dictionary: it is available at http://hidic.u-aizu.ac.jp/. This represents a long-range project, aiming at compiling a database for the study of English words for Japanese students. The main feature is that the dictionary is organized to bring out the best fruits of comparative linguistics in the past century: the usefulness of IndoEuropean roots as presented in the wonderful book by Calvert Watkins, The American Heritage Dictionary of Indo-European Roots, Second Edition, HoughtonMifflin, 2000. Example: The organizing principle of the dictionary will tell you that the following words are best studied as a group: riddle, garble, crime, decree, discern, secret, hypocrisy (they are all derivatives from the same Indo-European root krei-, meaning "to sieve", indicating that a group of seemingly unrelated words come from the same notion "to sieve"). Another example: the usage of the so-called synonyms "anger, rage and wrath" can be clearly explained by tracing them back to their respective Indo-European root: anger to angh-, meaning painful, rage to rebh-, meaning violent, impetuous, and wrath to wreit-, meaning to turn, twist (anger is akin to angst, angina, rage is akin to rabid, rabies, and wrath is akin to wreath, writhe).

Refereed Journal Papers

[kanda-01:2005]	K. Shimomukai and H. Kanda. Numerical study of normal pressure distribution in entrance flow between parallel plates, I. Finite difference calculation. ETNA, 2006.
[nasai-01:2005]	Y. Miyazaki, N. Asai, D. Cai, and Y. Ikebe. Numerical Computation of the Eigenvalues for the Spheroidal Wave Equation with Accurate Error Estimation by Matrix Method. ETNA(Electronic Transactions on Numerical Analysis), 2005.
	(to appear) A method to compute the eigenvalues of the spheroidal wave equations is proposed, as an application of a theorem on eigenvalues of certain classes of infinite matrices. The computation of its inverse problem (namely, solving another parameter c2 for given eigenvalue 兩) is likewise given. As a result, precise and explicit error estimates are obtained for the approximated eigenvalues.

Referred Proceeding Papers

[kanda-02:2005]	K. Shimomukai and H. Kanda. Accurate study of normal pressure distribution in entrance region of channel. In Proc. of 2005 ASME International Mechanical Engineering Congress & Exposition, pages IMECE2005-80353. ASME, ASME, 2005.
[kanda-03:2005]	H. Kanda and T. Yanagiya. Experimental conditions for minimum critical Reynolds number in pipe flow. In Proc. of 2005 ASME International Mechanical Engineering Congress & Exposition, pages IMECE2005-80637. ASME, ASME, 2005.
[kanda-04:2005]	K. Shimomukai and H. Kanda. Accurate pressure distribution in the y-direction of the entrance region of parallel plates. In Malaysia Institute of Engineers, editor, Proc. of the Eleventh Asian Congress of Fluid Mechanics. Asian Fluid Mechanics Committee, 2006.
[nasai-02:2005]	Asai N, Y. Miyazaki, D. Cai, and Y. Ikebe. The Error Estimate for the Computation of Eigenvalues of Lame Equation. In SciCADE05(2005 International Conference on Scientific Computation And Differential Equation), pages 13-14, 2005.
	Consider the eigenvalue problem for Lame equation. We show the convergence of the eigenvalues of the truncated matrices taken appropriately and propose the error estimate for the numerical computation in equality from theoretical aspect and practical numerical experiment.

Unreferred Papers

[kanda-05:2005]

H. Kanda. New Model for Laminar-Turbulent Transition in Pipe Flow (in Japaenese). Tohoku University Information Synergy Center Super Computing System , SENAC, 38(4):13-25, 2005.

Ph.D. and Other Theses

[kanda-06:2005]	Haruki Saito. Graduation Thesis: Drawing of polyhedrons using Wing-Edge Structure, University of Aizu, 2005. Thesis Advisor: Kanda, H.
[kanda-07:2005]	Yuji Inaguma. Graduation Thesis: Water Surface Simulations using Finite Difference Solution in Cartesian and Cylyndrical Coordinate Systems, University of Aizu, 2005. Thesis Advisor: Kanda, H.
[kanda-08:2005]	Takashi Kashimura. Graduation Thesis: Experimental Formulas for Grid generation in a Bellmouth Region, University of Aizu, 2005. Thesis Advisor: Kanda, H.
[kanda-09:2005]	Ayumi Katori. Graduation Thesis: An Elliptic Grid Generation System Based on Grid Control Maps, University of Aizu, 2005. Thesis Advisor: Kanda, H.
[kanda-10:2005]	Nao Kushida. Graduation Thesis: Process of Transfinite Interplolation (TFI) Generation System and its Visualization, University of Aizu, 2005. Thesis Advisor: Kanda, H.
[kanda-11:2005]	Masaaki Saitou. Graduation Thesis: Evaluation of Laplace and Poisson Equations in Bellmouth Region, University of Aizu, 2005. Thesis Advisor: Kanda, H.
[kanda-12:2005]	Shin Sato. Graduation Thesis: Approximated grid lines to the experimental data by using a length generator system, University of Aizu, 2005. Thesis Advisor: Kanda, H.