Hidenori OGATA (緒方 秀教)

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: competitive_research_funding, Overall Grant Amount: - (direct: 2000000, indirect: 600000)
以下の2点がこれまでの研究実績である．
（１．佐藤超函数論に基づく関数近似、数値微分および数値不定積分）佐藤超函数論は複素関数論に基づく一般化関数の理論であり，超函数とよばれる一般化関数を定義関数とよばれる複素解析関数の実軸上の境界値の差で表す．そして，通常の関数も，標準定義関数という定義関数を構成することにより超函数として表すことができる．本研究ではこのことに着目して関数近似・数値微分・数値不定積分を行う方法を考案した．具体的には，近似の対象とする関数に対し，標準定義関数を数値積分および連分数を用いることにより数値的に求め，それを用いて関数近似を超関数として与え，さらに，標準定義関数の導関数・原始関数をもちいて数値微分・数値不定積分を与える．そして，数値実験により本方法の有効性を確かめた．本研究の成果は国内学会で口頭発表し，和文誌に論文投稿した．
（２．IMT型変数変換を用いた数値不定積分および第2種Volterra型積分方程式の数値解法）数値計算において変数変換を用いた技法は数値積分でよく用いられてきた．ところで近年DE変換とよばれる変数変換は，Sinc近似という関数近似の技法と組み合わせて，数値積分以外の数値計算（積分方程式、微分方程式など）に用いられるようになった．それに対し本研究では，数値積分で用いられているもうひとつの変数変換「IMT型変換」について，周期関数に対するSinc近似と組み合わて数値不定積分および1次元第2種Volterra積分方程式に応用する数値計算法を考案した．そして，数値実験および理論誤差解析によりその計算法の有効性を示した．本研究の成果について，数値不定積分法は国内学会で口頭発表し和文誌に論文投稿して採録された．積分方程式の数値解法は国内学会で口頭発表した．

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: competitive_research_funding, Overall Grant Amount: - (direct: 3500000, indirect: 1050000)
The aim of this study is the development and theoretical analysis of numerical computation method based on the hyperfunction theory. The hyperfunction theory is a theory of generalized functions based on the complex analysis, and describes singularities of functions such as poles, discontinuities and the delta functions. In this study, we aim the applications of the hyperfunction theory to numerical computations, namely, we approach to problems involving singularities which make computation difficult by using analytic functions, which are ultimately smooth functions, from the viewpoint of the hyperfunction theory. To be specific, we develop numerical computation methods for integrations with singularities, etc. and consider problems related to numerical computations based on the hyperfunction theory such as the numerical expressions of hyperfunctions.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3500000, indirect: 1050000)
We have investigated in computer program library including validated computation techniques which give the basis of validated computation of time evolution equations, and derived a number of new approaches to numerical verification of ODEs which appear as semi-discretized equations of PDEs. The results were published in academic journals.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (B), Fund Type: -, Overall Grant Amount: - (direct: 5000000, indirect: 1500000)
Using the charge simulation method, we proposed the method of numerical conformal mappings of unbounded multiply connected domains onto the following canonical slit domains: (a) the parallel and the colinear slit domains, (b) the linear slit domain and (c) the circular and radial slit domain; and showed the effectiveness of our method. We studied the property of the charge simulation method, and proposed a fundamental solution method for periodic Stokes flow problems. These are important in scientific and engineering applications. Basic studies on numerical methods for singular integral equations and ill-conditioned linear equations are progressed, which are closely related to the subject of this study.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3300000, indirect: 480000)
無限領域における波動現象で現れる周波数応答問題や共鳴現象に対して、有限要素法やFDTD法による数値計算手法の開発と関連する研究を行った。その中で、外部ディリクレ・ノイマン写像の離散化手法の開発とその応用について数値実験を含む検証を行って成果を得た。応用として、音声生成では声道設計問題に対してアルゴリズムを考案して数値実験により有効性を確認した。また騒音低減問題などに関係した構造音場連成問題の数値解法の応用研究を展開した。DtN(Dirichlet-to-Neumann)境界条件を課した問題にたいする有限要素法については、3次元水面波動問題へも適用し数値解法の事前誤差評価を導出すると共にヘルムホルツ問題については修正DtN境界条件や多重DtN境界条件を課した場合についても事前誤差評価を導出した。さらに、アンテナからの電磁波の放射問題とその応用に関してPML技法を用いた無限領域における電磁場の非定常現象の数値解法の開発に取り組み3次元計算コードの基本的な部分を完成させ、幾つかの応用についても成果を得た。また、偏微分方程式の数値解法である基本解解法(代用電荷法)についても、従来同法の適用が難しいとされた周期的構造を持つストークス流問題や弾性問題への拡張をおこない、さらに境界要素法、およびその複素関数論的拡張である複素変数境界要素法について従来適用困難とされた空間周期的2次元ポテンシャル問題への拡張を行った。関連する問題としては、量子多体系で現れる行列の対角化問題においてLOBPCGの効率的な並列実装を行い世界記録となる1600億次元の計算に成功した。さらに、来るべきペタスケール計算機時代に備えた大規模固有値計算ついて、段階的な収束をブロックレベルでコントロールする手法を提案するとともに、密行列の対称行列についてマルチコア向けのブロック化手法により帯行列化の系統のアルゴリズムの研究で成果を得た。

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (B), Fund Type: -, Overall Grant Amount: - (direct: 7900000, indirect: -)
The numerical conformal mapping has been an important subject in computational and applied mathematics. Our major concern is to develop new methods of numerical conformal mappings by the charge simulation method (or the fundamental solution method) and apply them to potential flow problems.
1.We constructed approximate mapping functions of the conformal mapping w= f(z) of an unbounded multiply connected domains D onto the unbounded canonical slit domains of Nehari (Mc-Graw Hill, 1952) under the condition f(v) = ∞, where v is a finite point given in the problem domain. They were applied to the problem of potential flows past obstacles caused by a dipole source, a pair of positive and negative vortexes or a pair of point source and sink.
2.We constructed by the charge simulation method approximate mapping functions of the conformal mapping of bounded multiply connected domains onto all the unbounded and bounded canonical slit domains of Nehari.
3.We proposed a new technique to apply the charge simulation method to a nonlinear compressible fluid flow problem. We also proposed a fundamental solution method for viscous flow problems with obstacles in a periodic array, which gives an approximate solution by a linear combination of periodic fundamental solutions.
4.We proved the convergence of the approximate mapping function obtained by the charge simulation method.
Many other interesting results were obtained in relation to methods of numerical computation and thier application to fluid mechanics.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (B), Fund Type: -, Overall Grant Amount: - (direct: 8200000, indirect: -)
The numerical conformal mapping has been an important subject in computational mathematics. On the other hand, the charge simulation method is a simple accurate solver for the Dirichlet problem of the Laplace equation.
1.We have proposed a simple method of numerical conformal mappings of multiply-connected domains onto the canonical domains of Nehari (Mc-Graw Hill, 1952), i.e., (a) the parallel slit domain, (b) the circular slit domain, (c) the radial slit domain, (d) the circle with concentric circular slits and (e) the circular ring with concentric circular slits. The method uses the charge simulation method on the complex plane, i.e., a linear combination of complex logarithmic functions, and gives approximate mapping functions with high accuracy if boundary curves and boundary data are analytic.
2.We have successfully applied the numerical conformal mapping to potential flow analysis, and presented simple methods to find the stagnation points around obstacles and to compute the forces on obstacles.
3.We have presented some techniques to apply the charge simulation method to domains with corners or slits. We also presented new types of the charge simulation method applicable to periodic domains, which use periodic fundamental solutions of the problem instead of logarithmic functions.
4.We proved the unique solvability of the linear systems appearing in the invariant scheme of the charge simulation method.
These results revive the classical method using fundamental solutions as a modern method in the computer age.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (B), Fund Type: -, Overall Grant Amount: - (direct: 12100000, indirect: -)
This project aims at developing numerical methods based on Sine functions incorporated with the double exponential transformation technique. The following results have been obtained.
1. A Sinc method using the double exponential transformation technique is developed for computing indefinite integrals. Two Sinc methods using the conventional single exponential transformation technique are well-known : one is due to Kearfott, and the other due to Haber. While these well-known methods converge at the rate exp(-c-√) (n : the number of function evaluations), our method converges at the rate exp(-c'n/ log n).
2. A Sinc-Galerkin method incorporated with the double exponential transformation technique for two-point boundary value problems is developed. While the rate of the convergence of the original Sinc-Galerkin method due to Stenger is exp(-c-√)(n: the number of basis functions), that of our method is exp(-c'n/ logn), which is a remarkable improvement.
3. A Sinc-collocation method combined with the double exponential transformation technique for Sturm-Liouville eigenvalue problems is developed. Our method enjoyes the convergence rate O(exp(-c'n/ logn)) (n : the number of basis functions), whereas the original Sine-collocation method proposed by Lund et al. does the convergence rate O(exp(-c-√)).
4. Three spectral methods using the double exponential transformation technique are developed for solving the Poisson equation on a fan-shaped domain. One employes the Sinc functions as basis functions, another does the Legendre polynomials, and the other does the Chebyshev polynomials. All the methods converge at the rate exp(-c√/logn), where n is the number of basis functions.