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題名 縮基法初始值問題之數值研究
Numerical studies of reduced basis methos for initial value problems
作者 陳揚敏
貢獻者 林美佑
陳揚敏
關鍵詞 縮基法，投影法
Reduced Basis Method, Projection
日期 1990
1989
上傳時間 3-May-2016 14:17:47 (UTC+8)
摘要 縮基法(RBM) 是對參數化的曲線求逼近解的一個方法，基本上乃使用投影法將解曲線投射到解空間的一子空間中，如此一來，可將原問題轉換成一較小的系統，並經由數值計算出小系統的解，來求得大系統的一逼近解。在本篇論文中主要的乃探討RBM在常微分方程組初始值問題上的應用，並發展一套含有誤差控制的演算法。
The reduced basis method(RBM) is a scheme for approximating parametric solution curves. The basic technique of RBM is projection. By applying the method, we can find an approximate solution of the original system which satisfies a system of smaller size. In this paper, we mainly concern the applications of RBM for ODE initial value problems and develop an algorithm which contains a set of error controls.
參考文獻 [1] N. N. Abdelmalek, Roundoff Error Analysis for Gram-Schmidt Method and Solution of Linear Least Squares Problems, BIT, 11(1971), pp.945-968.
     [2] B. O. Almroth, P. Stern and F. A. Brogan, Automatic Choice of Global Shape Functions in Structural Analysis, AIAA J., 16(1978), pp. 525-528.
     [3] E. Fehlberg, Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsprobleme, Computing, 6(1970), pp. 61-71.
     [4] J. P. Fink and W. C. Rheinboldt, On the Discretization Error of Parametrized Nonlinear Equations, SIAM J. Numer. Anal., 20(1989),pp. 792-746.
     [5] J. P. Fink and W. C. Rheinboldt, On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations,Z. Angew. Math.Mech., 69(1989), pp. 21-28.
     [6] J . P. Fink and W. C. Rheinboldt, Local Error Estimates for Parametrized Nonlinear Equations, SIAM J. Numer. Anal., 22(1985),pp. 729-795.
     [7] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, 1971, Prentice-Hall inc., Englewood Cliffs, New Jersey.
     [8] G. H. Golub and C. F. Van Loan, Matrix Computations, 1989.
     [9] M. K. Gordon and L. F. Shampine, Computer Solution of Ordinary Differential Equations, The Initial Value Problem, ~974, W. H. Freeman and Company, San Francisco.
     [10] J. D. Lambert, Computational Methods in Ordinary Differential Equations, 1989 J. W. Arrowsmith Ltd. Bristol.
     [11] M. Lin Lee, Estimation of the Error in the Reduced Basis Method Solution of Differential Algebraic Equation System, SIAM J. Numer. Anal., to appear.
     [12] M. Lin Lee, The Reduced Basis Method for Differential Algebraic Equation System, Technical Report ICMA-85-85, Inst. for Compo Math. And Appl., University of Pittsburgh, Pittsburgh, PA, July, 1985.
     [13] N. A. Nagy, Model Representation of Geometrically Nonlinear Behavior by the Finite Element Method, Computers and Structures,10(1977), pp. 689-688.
     [14] A. K. Noor and J. M. Peters, Reduced Basis Technique for Nonlinear Analysis of Structures, AIAA J., 18(1980), pp. 455-462.
     [15] A. K. Noor, C. M. Andersen and J. M. Peters, Reduced Basis Technique for Collapse of Shells, AIAA J., 19(1981), pp. 999-997.
     [16] T. A. Porsching, Estimation of the Error in the Reduced Basis Method Solution of Nonlinear Equations, Math. Comp., 45(1985), pp. 487-496.
     [17] T. A. Porsching and M. Lin Lee, The Reduced Basis Method For Initial Value Problems, SIAM J. Numer. Anal., 24(1987), pp. 1277-1287.
     [18] J. R. Rice, Experiments on Gram-Schmidt Orthogonalization, Math. Compo 20(1966), pp. 925-928.
     [19] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 1980,Springer, New York, Heidelberg, Berlin.
     [20] G. W. Steward, Introduction to Matrix Computation, 1979, Academic Press, New York and London.
     [21] G. W. Steward, Perturbation Bounds for the QR Factorization of a Matrix, SIAM J, Numer. Anal., 14(1977), pp. 509-518.
     [22] J. S. Vandergraft, Introduction to Numerical Computation, 1989,Automated Sciences Group, Inc., Silver Spring, Maryland.
     [23] R. E. Williamson, Introduction to Differential Equations, ODE, PDE and Series.
描述 碩士
國立政治大學
應用數學系
資料來源 http://thesis.lib.nccu.edu.tw/record/#B2002005457
資料類型 thesis
dc.contributor.advisor 林美佑zh_TW
dc.contributor.author (Authors) 陳揚敏zh_TW
dc.creator (作者) 陳揚敏zh_TW
dc.date (日期) 1990en_US
dc.date (日期) 1989en_US
dc.date.accessioned 3-May-2016 14:17:47 (UTC+8)-
dc.date.available 3-May-2016 14:17:47 (UTC+8)-
dc.date.issued (上傳時間) 3-May-2016 14:17:47 (UTC+8)-
dc.identifier (Other Identifiers) B2002005457en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/90192-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description.abstract (摘要) 縮基法(RBM) 是對參數化的曲線求逼近解的一個方法，基本上乃使用投影法將解曲線投射到解空間的一子空間中，如此一來，可將原問題轉換成一較小的系統，並經由數值計算出小系統的解，來求得大系統的一逼近解。在本篇論文中主要的乃探討RBM在常微分方程組初始值問題上的應用，並發展一套含有誤差控制的演算法。zh_TW
dc.description.abstract (摘要) The reduced basis method(RBM) is a scheme for approximating parametric solution curves. The basic technique of RBM is projection. By applying the method, we can find an approximate solution of the original system which satisfies a system of smaller size. In this paper, we mainly concern the applications of RBM for ODE initial value problems and develop an algorithm which contains a set of error controls.en_US
dc.description.tableofcontents 1. Introduction ......................... 1
     2. Reduced Basis Approximation . ......................... 2
     3. Ordinary Differential Equation Solver ......................... 3
     3.1 Operation Count of GS-solver ......................... 8
     4. Implementing RBM.......................... 9
     4.1 Selecting Subspaces ......................... 10
     4.2 Algorithm of Gram-Schmidt .................................... 14
     4.3 Roundoff Error......................... 16
     4.4 Order Control ......................... 22
     4.5 Operation Count of Overhead of RBM......................... 22
     5. Error Control ......................... 24
     5.1 A posteriori Error Estimate......................... 24
     5.2 A priori Error Estimate......................... 25
     6. Numerical Studies ......................... 26
     References......................... 30		zh_TW
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#B2002005457en_US
dc.subject (關鍵詞) 縮基法，投影法zh_TW
dc.subject (關鍵詞) Reduced Basis Method, Projectionen_US
dc.title (題名) 縮基法初始值問題之數值研究zh_TW
dc.title (題名) Numerical studies of reduced basis methos for initial value problemsen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] N. N. Abdelmalek, Roundoff Error Analysis for Gram-Schmidt Method and Solution of Linear Least Squares Problems, BIT, 11(1971), pp.945-968.
     [2] B. O. Almroth, P. Stern and F. A. Brogan, Automatic Choice of Global Shape Functions in Structural Analysis, AIAA J., 16(1978), pp. 525-528.
     [3] E. Fehlberg, Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsprobleme, Computing, 6(1970), pp. 61-71.
     [4] J. P. Fink and W. C. Rheinboldt, On the Discretization Error of Parametrized Nonlinear Equations, SIAM J. Numer. Anal., 20(1989),pp. 792-746.
     [5] J. P. Fink and W. C. Rheinboldt, On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations,Z. Angew. Math.Mech., 69(1989), pp. 21-28.
     [6] J . P. Fink and W. C. Rheinboldt, Local Error Estimates for Parametrized Nonlinear Equations, SIAM J. Numer. Anal., 22(1985),pp. 729-795.
     [7] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, 1971, Prentice-Hall inc., Englewood Cliffs, New Jersey.
     [8] G. H. Golub and C. F. Van Loan, Matrix Computations, 1989.
     [9] M. K. Gordon and L. F. Shampine, Computer Solution of Ordinary Differential Equations, The Initial Value Problem, ~974, W. H. Freeman and Company, San Francisco.
     [10] J. D. Lambert, Computational Methods in Ordinary Differential Equations, 1989 J. W. Arrowsmith Ltd. Bristol.
     [11] M. Lin Lee, Estimation of the Error in the Reduced Basis Method Solution of Differential Algebraic Equation System, SIAM J. Numer. Anal., to appear.
     [12] M. Lin Lee, The Reduced Basis Method for Differential Algebraic Equation System, Technical Report ICMA-85-85, Inst. for Compo Math. And Appl., University of Pittsburgh, Pittsburgh, PA, July, 1985.
     [13] N. A. Nagy, Model Representation of Geometrically Nonlinear Behavior by the Finite Element Method, Computers and Structures,10(1977), pp. 689-688.
     [14] A. K. Noor and J. M. Peters, Reduced Basis Technique for Nonlinear Analysis of Structures, AIAA J., 18(1980), pp. 455-462.
     [15] A. K. Noor, C. M. Andersen and J. M. Peters, Reduced Basis Technique for Collapse of Shells, AIAA J., 19(1981), pp. 999-997.
     [16] T. A. Porsching, Estimation of the Error in the Reduced Basis Method Solution of Nonlinear Equations, Math. Comp., 45(1985), pp. 487-496.
     [17] T. A. Porsching and M. Lin Lee, The Reduced Basis Method For Initial Value Problems, SIAM J. Numer. Anal., 24(1987), pp. 1277-1287.
     [18] J. R. Rice, Experiments on Gram-Schmidt Orthogonalization, Math. Compo 20(1966), pp. 925-928.
     [19] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 1980,Springer, New York, Heidelberg, Berlin.
     [20] G. W. Steward, Introduction to Matrix Computation, 1979, Academic Press, New York and London.
     [21] G. W. Steward, Perturbation Bounds for the QR Factorization of a Matrix, SIAM J, Numer. Anal., 14(1977), pp. 509-518.
     [22] J. S. Vandergraft, Introduction to Numerical Computation, 1989,Automated Sciences Group, Inc., Silver Spring, Maryland.
     [23] R. E. Williamson, Introduction to Differential Equations, ODE, PDE and Series.		zh_TW