• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2009 Fiscal Year Final Research Report

Establishment of Verified Numerical Computation

Research Project

  • PDF
Project/Area Number 17002012
Research Category

Grant-in-Aid for Specially Promoted Research

Allocation TypeSingle-year Grants
Review Section Science and Engineering
Engineering
Research InstitutionWaseda University

Principal Investigator

OISHI Shin'ichi  Waseda University, 理工学術院, 教授 (20139512)

Co-Investigator(Renkei-kenkyūsha) NAKAO Mitsuhiro  九州大学, 数理学研究院, 教授 (10136418)
NISHIDA Takaaki  早稲田大学, 理工学術院, 教授 (70026110)
SHIBATA Yoshihiro  早稲田大学, 理工学術院, 教授 (50114088)
YAMAMOTO Nobito  電気通信大学, 電気通信学部, 教授 (30210545)
WATANABE Yoshitaka  九州大学, 情報基盤研究開発センター, 准教授 (90243972)
Project Period (FY) 2005 – 2009
Keywords数値解析 / 数値計算 / 非線形方程式 / 情報数理 / アルゴリズム
Research Abstract

Establishment of Verified Numerical Computation We have studied verified numerical computations for partial differential equations and systems of linear equations using digital computers. Calculating sum of a vector and dot product of two vectors with guaranteed high accuracy is ubiquitous in scientific computing. We have developed such algorithms for accurate sum and dot product, which are known to be the fastest so far. As applications, we have applied the fast and accurate algorithms to sparse matrix computations, computational geometry and so forth. Moreover, we have succeeded in proving the existence and uniqueness of a solution of a partial differential equation, and in calculating an error bound of its approximate solution.

  • Research Products

    (53 results)

All 2010 2009 2008 2007 2006 2005

All Journal Article (36 results) (of which Peer Reviewed: 36 results) Presentation (17 results)

  • [Journal Article] Fast Verified Solutions of Linear Systems2009

    • Author(s)
      T. Ogita, S. Oishi
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 26:2

      Pages: 169-190

    • Peer Reviewed
  • [Journal Article] Adaptive and Efficient Algorithm for 2D Orientation Problem2009

    • Author(s)
      K. Ozaki, T. Ogita, S.M. Rump, S. Oishi
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 26:2

      Pages: 215-231

    • Peer Reviewed
  • [Journal Article] Iterative Refinement for Ill-conditioned Linear Systems2009

    • Author(s)
      S. Oishi, T. Ogita, S.M. Rump
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 26:2

      Pages: 465-476

    • Peer Reviewed
  • [Journal Article] Numerical Existence Proof of Five Solutions for Certain Two-Transistor Circuit Equations2009

    • Author(s)
      Y. Nakaya, T. Nishi, S. Oishi, M. Claus
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 26:2

      Pages: 327-336

    • Peer Reviewed
  • [Journal Article] An Application of Taylor Models to the Nakao Method on ODEs2009

    • Author(s)
      N. Yamamoto, T. Komori
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 26:2

      Pages: 365-392

    • Peer Reviewed
  • [Journal Article] Numerical Verification Method of Solutions for Elliptic Equations and Its Application to2009

    • Author(s)
      Y. Watanabe, M.T. Nakao
    • Journal Title

      the Rayleigh-Be'nard Problem, Japan Journal of Industrial and Applied Mathematics 26:2

      Pages: 443-463

    • Peer Reviewed
  • [Journal Article] A Numerical Verification Method for Two-Coupled Elliptic Partial Differential Equations2009

    • Author(s)
      Y. Watanabe
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 26:2

      Pages: 233-247

    • Peer Reviewed
  • [Journal Article] Computer assisted proofs of bifurcating solutions for nonlinear heat convection problems2009

    • Author(s)
      M.T. Nakao, Y. Watanabe, N. Yamamoto, T. Nishida, M. Kim
    • Journal Title

      Journal of Scientific Computing 10

      Pages: 1-18

    • Peer Reviewed
  • [Journal Article] Pattern formations in heat convection problems, Chinese Annals of Mathematics2009

    • Author(s)
      T. Nishida, Y. Teramoto
    • Journal Title

      Ser. B 30:6

      Pages: 769-784

    • Peer Reviewed
  • [Journal Article] Bifurcation theorems for the model system of Benard-Marangoni convection2009

    • Author(s)
      T. Nishida, Y. Teramoto
    • Journal Title

      Journal of Mathematical Fluid Mechanics 11:3

      Pages: 383-406

    • Peer Reviewed
  • [Journal Article] On a resolvent estimate of the Stokes system in a half-space arising from a free boundary problem for the Navier-Stokes equations2009

    • Author(s)
      Y. Shibata, S. Shimizu
    • Journal Title

      Math. Nachr 282:3

      Pages: 482-499

    • Peer Reviewed
  • [Journal Article] On the L2 a priori error estimates to the finite element solution of elliptic problems with singular adjoint operator2009

    • Author(s)
      T. Kinoshita, K. Hashimoto, M.T. Nakao
    • Journal Title

      Numerical Functional Analysis and Optimization 30

      Pages: 289-305

    • Peer Reviewed
  • [Journal Article] Tight Enclosures of Solutions of Linear Systems2009

    • Author(s)
      T. Ogita, S. Oishi
    • Journal Title

      International Series of Numerical Mathematics 157

      Pages: 167-178

    • Peer Reviewed
  • [Journal Article] A Parallel Algorithm for Accurate Dot Product2008

    • Author(s)
      N. Yamanaka, T. Ogita, S.M. Rump, S. Oishi
    • Journal Title

      Parallel Computing 34:6-8

      Pages: 392-410

    • Peer Reviewed
  • [Journal Article] Accurate Floating-Point Summation Part I: Faithful Rounding2008

    • Author(s)
      S.M. Rump, T. Ogita, S. Oishi
    • Journal Title

      SIAM Journal on Scientific Computing 31:1

      Pages: 189-224

    • Peer Reviewed
  • [Journal Article] Accurate Floating-Point Summation Part II: Sign2008

    • Author(s)
      S.M. Rump, T. Ogita, S. Oishi
    • Journal Title

      K-fold Faithful and Rounding to Nearest, SIAM Journal on Scientific Computing 31:2

      Pages: 1269-1302

    • Peer Reviewed
  • [Journal Article] Guaranteed error bounds of finite element approximations for non-coercive elliptic problems and its applications2008

    • Author(s)
      M.T. Nakao, K. Hashimoto
    • Journal Title

      Journal of Computational and Applied Mathematics 218:1

      Pages: 106-115

    • Peer Reviewed
  • [Journal Article] On the Oseen semigroup with rotating effect, Functional Analysis and Evolution Equations2008

    • Author(s)
      Y. Shibata, H. Amann, W. Arendt, M. Hieber, F., Neubrander, S. Nicaise, J. von Below (eds)
    • Journal Title

      The Gunter Lumer Volume

      Pages: 595-611

    • Peer Reviewed
  • [Journal Article] A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Benard problems2008

    • Author(s)
      M. Kim, M.T. Nakao, Y. Watanabe, T. Nishida
    • Journal Title

      Numerische Mathematik 111

      Pages: 389-406

    • Peer Reviewed
  • [Journal Article] A Method of Obtaining Verified Solutions for Linear Systems Suited for Java2007

    • Author(s)
      K. Ozaki, T. Ogita, S. Miyajima, S. Oishi, S.M. Rump
    • Journal Title

      Journal of Computational and Applied Mathematics 199:2

      Pages: 337-344

    • Peer Reviewed
  • [Journal Article] Convergence of Rump's Method for Inverting Arbitrarily Ill-conditioned Matrices2007

    • Author(s)
      S. Oishi, K. Tanabe, T. Ogita, S.M. Rump
    • Journal Title

      Journal of Computational and Applied Mathematics 205:1

      Pages: 533-544

    • Peer Reviewed
  • [Journal Article] Numerical verification of stationary solutions for Navier-Stokes problems2007

    • Author(s)
      K. Nagatou, K. Hashimoto, M.T. Nakao
    • Journal Title

      Journal of Computational and Applied Mathematics 199

      Pages: 424-431

    • Peer Reviewed
  • [Journal Article] Numerical method for verifying the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed Gelfand equation2007

    • Author(s)
      T. Minamoto, M.T. Nakao
    • Journal Title

      Journal of Computational and Applied Mathematics 202

      Pages: 177-185

    • Peer Reviewed
  • [Journal Article] Decay properties of the Stokes semigroup in exterior domains with Neumann condition2007

    • Author(s)
      Y. Shibata, S. Shimizu
    • Journal Title

      Journal of the Mathematical Society of Japan 59

      Pages: 1-35

    • Peer Reviewed
  • [Journal Article] A Mathematical Theory for Numerical Treatment of Nonlinear Two-Point Boundary Value Problems2006

    • Author(s)
      T. Yamamoto, S. Oishi
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 23:1

      Pages: 31-62

    • Peer Reviewed
  • [Journal Article] 実対称行列の各固有対の精度保証付き数値計算法2006

    • Author(s)
      宮島信也, 荻田武史, 大石進一
    • Journal Title

      日本応用数理学会論文誌 16:4

      Pages: 535-552

    • Peer Reviewed
  • [Journal Article] 点と平面との位置関係を判定する高速かつロバストなアルゴリズム2006

    • Author(s)
      尾崎克久, 荻田武史, S.M. Rump, 大石 進一
    • Journal Title

      日本応用数理学会論文誌 16:4

      Pages: 553-562

    • Peer Reviewed
  • [Journal Article] 高精度内積計算アルゴリズムを用いた連立一次方程式の精度保証付き数値計算法2006

    • Author(s)
      大石進一, 荻田武史, 太田貴久
    • Journal Title

      シミュレーション 25:5

      Pages: 170-178

    • Peer Reviewed
  • [Journal Article] 連立一次方程式のメモリ量を低減した精度保証付き数値計算法2006

    • Author(s)
      荻田武史, 大石進一
    • Journal Title

      シミュレーション 25:5

      Pages: 179-184

    • Peer Reviewed
  • [Journal Article] 悪条件連立一次方程式の精度保証付き数値計算法2005

    • Author(s)
      太田貴久, 荻田武史, S.M. Rump, 大石 進一
    • Journal Title

      日本応用数理学会論文誌 15:3

      Pages: 269-287

    • Peer Reviewed
  • [Journal Article] Accurate Sum and Dot Product2005

    • Author(s)
      T. Ogita, S.M. Rump, S. Oishi
    • Journal Title

      SIAM Journal on Scientific Computing 26:6

      Pages: 1955-1988

    • Peer Reviewed
  • [Journal Article] 実対称行列の各固有値に対する精度保証付き数値計算法2005

    • Author(s)
      宮島信也, 荻田武史, 大石進一
    • Journal Title

      日本応用数理学会論文誌 15:3

      Pages: 253-268

    • Peer Reviewed
  • [Journal Article] 大規模連立一次方程式のための高速精度保証法2005

    • Author(s)
      荻田武史, 大石進一
    • Journal Title

      情報処理学会論文誌:数理モデル化と応用 46

      Pages: 10-18

    • Peer Reviewed
  • [Journal Article] On the Stokes and Navier-Stokes equation in a perturbed half-space Advances2005

    • Author(s)
      Y. Shibata, T. Kubo
    • Journal Title

      in Differential Equations 10:6

      Pages: 695-720

    • Peer Reviewed
  • [Journal Article] On the Stokes and Navier-Stokes flows in a perturbed half space2005

    • Author(s)
      Y. Shibata, T. Kubo
    • Journal Title

      Regularity and other aspects of the Navier-Stokes equations, Banach Center Publications 70

      Pages: 157-167

    • Peer Reviewed
  • [Journal Article] Hopf bifurcation in viscous incompressible flow down an inclined plane2005

    • Author(s)
      T. Nishida, Y. Teramoto, H. Yoshihara
    • Journal Title

      J. Math. Fluid Mechanics Vol.7

      Pages: 29-71

    • Peer Reviewed
  • [Presentation] 常微分方程式の精度保証付き計算の技法と利用法2010

    • Author(s)
      山本野人
    • Organizer
      天体力学n体力学研究会
    • Place of Presentation
      千葉大学
    • Year and Date
      2010-03-19
  • [Presentation] Error free transformations of floating point numbers and its applications to constructing error free algorithms2009

    • Author(s)
      S. Oishi
    • Organizer
      Symbolic Numeric Computation (SNC 2009)
    • Place of Presentation
      Kyoto, Japan
    • Year and Date
      20090803-20090805
  • [Presentation] Numerical existence theorem for solutions of fixed point type equations and its applications2009

    • Author(s)
      S. Oishi
    • Organizer
      International Conference on Engineering and Computational Mathematics (ECM2009)
    • Place of Presentation
      The Hong-Kong Polytechnic University, Hong-Kong
    • Year and Date
      20090527-20090529
  • [Presentation] An eigenvalue excluding method for the Orr-Sommerfeld problem2009

    • Author(s)
      Y. Watanabe
    • Organizer
      International Conference on Engineering and Computational Mathematics (ECM2009)
    • Place of Presentation
      The Hong Kong Polytechnic University, Hong Kong, China
    • Year and Date
      20090527-20090529
  • [Presentation] 精度保証付き数値計算と多倍長演算2009

    • Author(s)
      山本野人
    • Organizer
      デジタル解析学セミナー
    • Place of Presentation
      早稲田大学
    • Year and Date
      2009-10-19
  • [Presentation] Maxwell方程式によるナノ領域電磁界計算の精度保証2009

    • Author(s)
      大石進一
    • Organizer
      2009年春季第56回応用物理学関係連合講演会(シンポジウム:ナノフォトニクスにおけるナノ加工の最前線と理論基盤)
    • Place of Presentation
      筑波大学
    • Year and Date
      2009-03-30
  • [Presentation] (Waseda Univ. /JST): Accurate and Fast Sum of Floating Point Numbers and Applications to Verification Algorithms2008

    • Author(s)
      S. Oishi
    • Organizer
      The 13th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerical Computations
    • Place of Presentation
      The University of Texas at El Paso
    • Year and Date
      20080929-20081003
  • [Presentation] On resolvent problems for generalized Stokes equation arising from free boundary problem, International Conference on Parabolic and Navier-Stokes Equations 20082008

    • Author(s)
      Y. Shibata
    • Organizer
      Banach center
    • Place of Presentation
      porland
    • Year and Date
      20080901-20080905
  • [Presentation] Iterative Refinement for Ill-Conditioned Linear Systems2008

    • Author(s)
      S. Oishi
    • Organizer
      International Workshop on Numerical Validation in Current Hardware Architectures at Dagstuhl Seminar
    • Place of Presentation
      Germany
    • Year and Date
      20080107-20080111
  • [Presentation] Pattern formations in heat convection problems, Parabolic and Navier-Stokes Equations2008

    • Author(s)
      T. Nishida
    • Organizer
      Banach Center
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2008-09-04
  • [Presentation] Some remarks on the Aubin-Nitsche trick of FEM solutions for elliptic problems with singular adjoint operator2008

    • Author(s)
      M.T. Nakao
    • Organizer
      6th International Conference on Scientific Computing and Applications(SCA2008), Pusan National University
    • Place of Presentation
      Busan, Korea
    • Year and Date
      2008-06-03
  • [Presentation] 高速で高精度な浮動小数点数の内積計算法と精度保証付き数値計算2008

    • Author(s)
      大石進一
    • Organizer
      日本数学会(企画特別講演)
    • Place of Presentation
      大阪府立大学
    • Year and Date
      2008-03-22
  • [Presentation] Fast and Accurate Dot Product Algorithm and its Applications to Verified Numerical Computation2007

    • Author(s)
      S. Oishi
    • Organizer
      The 3rd East Asia SIAM Conference
    • Place of Presentation
      Xiamen, China
    • Year and Date
      20071102-20071105
  • [Presentation] Numerical Uniqueness and Existence Theorem for Solution of Lippmann-Schwinger Equation2007

    • Author(s)
      S. Oishi
    • Organizer
      九州大学数理学府21世紀COEシンポジウム特別講演
    • Year and Date
      20071001-20071004
  • [Presentation] 疎連立一次方程式の解の反復解法を用いた数値的精度保証法2005

    • Author(s)
      大石進一
    • Organizer
      日本応用数理学会環瀬戸内応用数理研究部会第9回シンポジウム
    • Year and Date
      2005-11-12
  • [Presentation] Numerical Verification for Solutions of Linear Systems, Algebraic and Numerical Algorithms and Computer-assisted Proofs2005

    • Author(s)
      S. Oishi
    • Organizer
      Dagstuhl Seminar
    • Place of Presentation
      Dagstuhl, Germany
    • Year and Date
      2005-09-29
  • [Presentation] 数値線形計算の精度保証に関する最新の発展-任意に精度を変更でき、大規模な問題を扱え、高速に計算するには-2005

    • Author(s)
      大石進一
    • Organizer
      力学系の研究-トポロジーと計算機による新展開
    • Place of Presentation
      京大数理解析研究所
    • Year and Date
      2005-06-22

URL: 

Published: 2011-06-18   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi