Koyama DAISUKE (小山 大介)

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: 3400000, indirect: 1020000)
Transmissions of information or energy can be performed by use of the wave propagation phenomena. In this study we treated the information transmission by spoken language and electromagnetic radiation with various frequencies from antennas. We developed the methods to reduce the problem in unbounded region into the one in a bounded region using two techniques and then discretized the problems by FEM or FDM. We also developed similar methods for energy propagation phenomena such as those in elasticity or in electromagnetism and applied the method to several problems in seismology or electromagnetic wave heating phenomena in MRI.

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: 12800000, indirect: -)
1.Main Results Obtained in the Joint Works around the Head Investigator :
Demonstration of the possibility for determination of the mapping of wing through finite element computation. Error estimate for the solutions of FSM(=Fundamental Solution Method)approximate problems to reduced wave problems in a domain exterior to a disc. Confirmation of the effectiveness of an FEM-FSM combined method applied to 2D exterior reduced wave problem, and its application to linear water wave problems in an exterior water region with constant water depth, where the abbreviation FEM stands for Finite Element Method.
2.Remarkable Progress Obtained in the Works by Investigators :
(1)Establishment of a method solving linear systems determining discrete vector potentials(by J.Watanabe).
(2)Application of multi layer neural networks to various types of inverse problems(in computer tomography, in data assimilation, in parameter evaluation, in time series prediction)(by T.Takeda).
(3)Application of finite element analysis for stationary wave transmission phenomena in unbounded domains to the problem of voice generation with successfully captured formants(by T.Kako).
(4)Development of a new method for determination of upper bounds for error estimation constants appeared in finite element computation of Poisson equations in non-convex polygonal domains(by N.Yamamoto).
(5)Theoretical study on the effect of stationary non homogeneous spatial structure to qualitative properties of solutions in the case of non-linear reaction-diffusion equations through numerical simulation and asymptotic analysis(by K.Nakamura).
(6)Mathematical and numerically experimental analysis of characteristic futures of approximation methods for various types of partial differential equations obtained through Runge-Kutta type formulas(by T.Koto).
(7)Overcome of the difficulties in finite element numerical solution methods in flow problems through upwinding technique and approximation way of characteristic curves(by M.Tabata).
(8)Finite element analysis of non-stationary field of eddy current based on moving coordinate system(by H.Kanayama).
(9)Flux free finite element method applied to two phase fluid problems(by K Ohmori).
(10)Parallel computation through mortar domain decomposition method(by S.Fujima).
(11)Purely theoretical analysis and numerical analysis of numerical instability problems arising with association of steep change of phenomena, in such as shock waves(by H.aiso).
3.Invitation of Foreign Cooperative Researcher :
(1)Professor Han Hou-de of Applied Mathematics Department, Tsinghua University, Beijing, China from July 26 to August 16,2002.
(2)Professor Yu De-hao of Institute of Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China from September 12 to October 3,2004.
4.Research Meetings :
(1)Yokohama Research Meeting was held an KKR Hotel Port Pear Yokohama from January 8 to 10,2003.
(2)Chofu Research Meeting was held at the University of Electro-Communications from February 19 to 20,2004.
(3)Chofu Symposium 2005 was held at the University of Electro-Communication from February 17 to 19,2005.

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: 4000000, indirect: -)
The purpose of this research project is to develop the approximation methods for wave propagation problems in unbounded region and as its applications we study the numerical simulation of voice generation. We formulate the problem as the exterior Helmholtz equation, and reduce the problem to the one in a bounded region by introducing the artificial boundary condition on an artificial boundary. We developed the numerical methods for this problem based on the finite element discretization method and study the application problems including the voice generation.
The results of the head investigator Kako are the followings. He found out the variational formula of the complex eigenvalues with respect to the deformation of vocal tract. The eigenvalues are related to the formants of frequency response function that is important for voice generation. He then developed the algorithm for designing the shape of vocal tract by use of the variational formula, and validated the algorithm through numerical simulations. He also studied the application of the Finite Difference Time Domain method to the acoustic problem and obtained several basic results.
For the voice problem, Yoshida developed the method to obtain the mapping from the articulation parameters to the phonetic transmission characteristics by use of the neural networks. Suito studied the shape optimization problem for minimizing the reflection of the wave propagating in a tubular region with spatially changing impedance parameters and obtained unusual numerical results.
Related to the numerical methods for the wave problem, Koyama studied the three dimensional Helmholtz problem by use of the fictitious domain method and derived the a priori error estimates for the approximation, and investigated the validity by some numerical experiences. Ushijima studied the Helmholtz problem by the collocation method based on the fundamental solutions and obtained a sufficient condition for the exponential convergence of the approximate solution and tried to validate of the theoretical results by multi-precision arithmetic computation.
As for the numerical methods for solving large linear equations appearing in the application problems including the Helmholtz equation, Zhang studied the fast and efficient iteration methods. Imamura developed the automatic tuning techniques with high actuary and stability in the implementation for the parallel computation methods for the large linear systems.

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: -)
By using squared residuals of a differential equation for the object function of a neural network we can solve the differential equation as the network itself becomes the solution after training. We investigated this method by applying it to Navier-Stokes equation, Poisson equation, Lorenz equation and so on, and obtained satisfactory results. We found that similar method is applied to CT image reconstruction problem with small amount of projection data. We applied it to model CT image reconstruction problem where an integral equation is solved and obtained satisfactory results. By generalizing the form of the object function this method can be applied to very wide range of problems. Defining the object function by the squared residuals of differential equations, integral equations, and algebraic equations multiplied by some appropriate penalty coefficients various kinds of problems can be solved owing to the excellent expressivity of the multi-layer neural network comparatively easily. One of the important application is the solution of data assimilation problem which is very important in the field of the meteorological and oceanological numerical simulations. We have also performed model numerical experiment of the data assimilation and obtained satisfactory results. This method is also applicable to the generalized Abel inversion which appears in various scientific and engineering researches.
For comparing the new method with the conventional standard methods we studied these methods. As the mathematical basis of the CT image reconstruction we studied the inverse Radon transform. As for the decay Radon transform we derived an inverse formula, which we proved mathematically and numerically. We studied the Poisson equation and the Helmholtz equation in the infinite domain. We introduced an artificial boundary condition and studied the validity of the condition mathematically and numerically.

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: 2400000, indirect: -)
1. Main Results Obtained in the Joint Works around the Head Investigator :
(1) Discretization method through charge simulation method for Steklov operator associated with Laplace problems in exterior domain of a disc. An FEM-CSM (Finite Element Method - Charge Simulation Method) combined method for 2D exterior Laplace problems was proposed in 1998 and its mathematical analysis has been studied during whole the term of project. Relation between continuous and discrete fourier coefficients, which is a key for the study, has been investigated.
(2) Confirmation of the effectiveness of the FEM-FSM combined method through numerical computation (jointly with Masuda, S.).
(3) Determination of the conformal mapping of wing through precise finite element computation for the potential function and stream function.
(4) Proposal of the fundamental solution method (FSM, in short) for the determination of Steklov operator associated with the reduced wave problems in exterior domain of a disc.
2. Remarkable Progress Obtained in the Works by Investigators :
(1) Numerical method for partial differential equations through neural network. Boundary condition fitting with the use of shape factor in the form of approximate function. Neural network solver for MHD equilibrium equation and Napier-Stokes equation. Acceleration of the neural net solution method with the use of Gauss-Newton method. (Obtained by Takeda and Fukuhara)
(2) An iterative method for Helmholtz equation in domain decomposition computation. Numerical analysis and computation of the wave propagation in a bounded domain with open boundary portion connected with exterior infinite region. Proof of stability and convergence for Newmark's method for second order problem (obtained by Kako, T.).
(3) Stability analysis for Runge-Kutta method through linear system theory. Stability analysis for Runge-Kutta method applied to delay-differential equations (obtained by Koto).
(4) Mathematical justification for a wave equation method solving exterior Helmholtz problems. Proposal and justification of a new artificial boundary condition to the wave equation solver for exterior Helmholtz problem (obtained by Koyama).
3. Keeping the UEC-NA-Seminar (University of Electro-Communications Numerical Analysis Seminar). In each academic year of 1998 and 1999, 14 seminars were held in the Friday morning, generally. Within seminars, talks given by Professor Kamitani, A. of Yamagata University, Professor Sakajo, T. of Nagoya University, Professor Yamamoto, N. of then Kyushu university, Professor Matsuo, T. of Nagoya University, Professor Kanayama, H. of Kyushu University were supported by this grant. In addition to these regular seminar, special one-day and two-day workshops were held to discuss the subject of the project and related topics.

Offer Organization: 日本学術振興会, System Name: 科学研究費助成事業, Category: 一般研究(C), Fund Type: -, Overall Grant Amount: - (direct: 1200000, indirect: -)
本研究では、磁気流体波、音波、弾性波などの振動・波動現象を始めとする自然現象を、偏微分方程式の初期値境界値問題あるいはスペクトル固有値問題として定式化し、有限要素法を用いた数値計算アルゴリズムを与え、その妥当性の数学的解析と実際の数値計算結果の検討を行うことを研究目標とした。
研究を進めるに当たっては、研究者の個別研究に基礎をおきつつ、研究の相互交流にも力を注いだ。特に、週一開催された「数値解析研究会」や統計数理研究所主催の「MHD研究会」を始めとする年間を通しての各種の研究会での相互交流につとめた。以下、個別の研究実績について述べる。
1.抵抗性線形化磁気流体作用素のスペクトル近似と、系の時間発展問題の関連につき研究し、その過程でNewmarkのβ法について興味有る事実を見いだした(加古、千葉)。
2.構造体と音場の連成振動における固有値問題の解析的性質の解明と、それを利用した有限要素近似を研究し、近似スキームの収束証明と誤差評価を得た(加古、田吉、DENG Li)。
3.散乱問題の高次放射条件を利用した数値解析手法の研究を進め、領域分割法による反復計算法の有効性を理論的に明らかにすると共に、有限要素法による近似計算を実行し有効性を確認した(加古、劉)。
4.無限領域等の特異性を持つ領域における数値計算法の開発を行い、完全流体における一様流中の物体回り流れの数値計算法を考案し、流体中の振動・波動計算の基礎を築いた(小山、牛島)。
5.移流拡散問題の差分近似の安定性と近似誤差評価を研究し、散逸と分散が同時に存在する系の数値計算法に対し新たな知見を加えた(名古屋、牛島)。
6.計算機トモグラフィーにおけるラドン変換の、フィルター補正逆投影法で高周波成分を考慮した新しい計算法を考案し、数値実験と誤差解析を行った(室屋、渡辺: Numerical analysis of a reconstruction algorithm in computed tomography, Journal of Inverse and Ill posed Problem に投稿中)。
7.高分子二相分離問題の数学的解析に取り組み、モデルの妥当性を明らかにした(大西)。

Offer Organization: 日本学術振興会, System Name: 科学研究費助成事業, Category: 一般研究(C), Fund Type: -, Overall Grant Amount: - (direct: 1700000, indirect: -)
本研究では、微分方程式の解の構造の解析と、そこで得られる情報を適切に考慮した数値計算法の開発と妥当性の数学的検証に取り組み、いくつかの成果が得られた。研究を進めるに当たっては、各分担者が独自に分担課題の研究を進めると共に、関係する研究者との研究連絡、研究交流を行った。特に、電気通信大学で定期的に開催された数値解析研究会での研究交流と、日中数値数学セミナーにおける、中国からの参加者を交えた研究交流に力を注いだ。以下、個別の研究実績について述べる。
1.磁気流体作用素のスペクトルの絶対連続性についての性質と抵抗性作用素の特異極限について、作用素解析、WKB法、有限要素計算の各方向から研究を進めた(千葉文浩、加古孝)。また、物体による散乱問題の高次放射条件を用いた有限要素計算を行い有効性の確認を行った(劉小進、加古孝)。さらに、音場・構造体連成系において現れる標準的ではない固有値問題の数理解析と、有限要素計算法の定式化と数値解析を行った(DENG Li、加古孝)。
2.角(カド)のある2次元図形の定義関数を、ラドン変換から再構成する問題で、計算値の収束と角(カド)における角(カク)の大きさの関係を明らかにした(渡辺二郎、室屋泰三)。
3.水の表面波の線形化解析におけるモード解析手法の妥当性を明らかにした(牛島照夫、小山大介)。スチュクロフ作用素を利用した計算法と数値解析で成果を得た(牛島照夫、小山大介、加古孝)。移流拡散問題に対し、ε-安定性の概念を導入し、その有効性を検証した(牛島照夫、名古屋靖一郎)。
4.非圧縮性粘性流体の分岐問題に係わる固有値問題の定式化と有限要素計算を実行した(海津聡、富森叡)。
5.遅延連立微分方程式に対する陰的ルンゲ・クッタ法の安定性を解析し、新たな知見を得た(小藤俊幸)。