메뉴 건너뛰기




Volumn 4, Issue 3-4, 2007, Pages 368-391

Reduced order modeling of some nonlinear stochastic partial differential equations

Author keywords

Brownian motion; Finite element methods; Monte carlo methods; Reduced order modeling; Stochastic differential equations

Indexed keywords


EID: 45449090596     PISSN: 17055105     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (28)

References (65)
  • 1
    • 0037606706 scopus 로고    scopus 로고
    • Finite Element and Difference Approximation of Some Linear Stochastic Partial
    • E. J. ALLEN AND S. J. NOVOSEL AND Z. ZHANG, Finite Element and Difference Approximation of Some Linear Stochastic Partial, Stochastics Rep. 64, 1998, 117-142.
    • (1998) Stochastics Rep , vol.64 , pp. 117-142
    • ALLEN, E.J.1    NOVOSEL, S.J.2    ZHANG, Z.3
  • 2
    • 0036189629 scopus 로고    scopus 로고
    • E. ALÓS AND S. BONACCORSI, Stochastic Partial Differential Equations with Dirichlet White-Noise Boundary Conditions, Ann. I. H. Poincaré 38, 2002, 152-154.
    • E. ALÓS AND S. BONACCORSI, Stochastic Partial Differential Equations with Dirichlet White-Noise Boundary Conditions, Ann. I. H. Poincaré 38, 2002, 152-154.
  • 3
    • 0001108405 scopus 로고
    • Preserving symmetries in the proper orthogonal decomposition
    • N. AUBRY AND W. LIAN, AND E. TITI, Preserving symmetries in the proper orthogonal decomposition, SIAM J. Sci. Comp. 14, 1993, 483-505.
    • (1993) SIAM J. Sci. Comp , vol.14 , pp. 483-505
    • AUBRY, N.1    LIAN, W.2    TITI, E.3
  • 5
    • 13844314882 scopus 로고    scopus 로고
    • Solving Elliptic Boundary Value Problems with Uncertain Coefficients by the Finite Element method: The Stochastic Formulation
    • I. BABUŚKA AND R. TEMPONE AND G. ZOURARIS, Solving Elliptic Boundary Value Problems with Uncertain Coefficients by the Finite Element method: The Stochastic Formulation, Comput. Methods Appl. Mech. Engrg. 194, 2005, 1251-1294.
    • (2005) Comput. Methods Appl. Mech. Engrg , vol.194 , pp. 1251-1294
    • BABUŚKA, I.1    TEMPONE, R.2    ZOURARIS, G.3
  • 6
    • 13844319204 scopus 로고    scopus 로고
    • Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations
    • I. BABUŚKA AND R. TEMPONE AND C. ZOURARIS, Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations, SIAM J. Numeric. Anal. 42, 2004, 800-825.
    • (2004) SIAM J. Numeric. Anal , vol.42 , pp. 800-825
    • BABUŚKA, I.1    TEMPONE, R.2    ZOURARIS, C.3
  • 7
    • 12044259644 scopus 로고
    • The proper orthogonal decomposition in the analysis of turbulent flows
    • G. BERKOOZ, P. HOLMES AND J. LUMLEY, The proper orthogonal decomposition in the analysis of turbulent flows, Ann. Rev. Fluid. Mech. 25, 1993, 539-575.
    • (1993) Ann. Rev. Fluid. Mech , vol.25 , pp. 539-575
    • BERKOOZ, G.1    HOLMES, P.2    LUMLEY, J.3
  • 8
    • 0009632228 scopus 로고
    • Galerkin projections and the proper orthogonal decomposition for equivariant equations
    • G. BERKOOZ AND E. TITI, Galerkin projections and the proper orthogonal decomposition for equivariant equations, Phys. Let. A 174 1993, 94-102.
    • (1993) Phys. Let. A , vol.174 , pp. 94-102
    • BERKOOZ, G.1    TITI, E.2
  • 9
    • 0010092088 scopus 로고
    • Monotonicity methods for white noise driven quasilinear SPDEs, in Diffusion Processes and Related Problems in Analysis
    • R. BUCKDAHN AND E. PARDOUX, Monotonicity methods for white noise driven quasilinear SPDEs, in Diffusion Processes and Related Problems in Analysis, I, M. Pinsky, ed., 1990, 219-233.
    • (1990) I, M. Pinsky, ed , pp. 219-233
    • BUCKDAHN, R.1    PARDOUX, E.2
  • 10
    • 50049115769 scopus 로고    scopus 로고
    • J. BURKARDT AND M. GUNZBURGER AND H. C. LEE, POD and CVT-based Reduced-order Modeling of Navier-Stokes flows, 2004, to appear
    • J. BURKARDT AND M. GUNZBURGER AND H. C. LEE, POD and CVT-based Reduced-order Modeling of Navier-Stokes flows, 2004, to appear
  • 11
    • 50049125790 scopus 로고    scopus 로고
    • J. BURKARDT AND M. GUNZBURGER AND H. C. LEE, Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems, 2003, to appear
    • J. BURKARDT AND M. GUNZBURGER AND H. C. LEE, Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems, 2003, to appear
  • 12
    • 0033295675 scopus 로고    scopus 로고
    • Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows
    • E. CHRISTENSEN AND M. BRONS, AND J. SORENSEN, Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows, SIAM J. Sci. Comp. 21, 2000, 1419-1434.
    • (2000) SIAM J. Sci. Comp , vol.21 , pp. 1419-1434
    • CHRISTENSEN, E.1    BRONS, M.2    SORENSEN, J.3
  • 13
    • 10644261980 scopus 로고    scopus 로고
    • Some Nonlinear SPDE's that are Second Order in Tiime
    • R. DALANG AND C. MUELLER, Some Nonlinear SPDE's that are Second Order in Tiime, Electronic Journal of Probability 8, 2003, 1-21.
    • (2003) Electronic Journal of Probability , vol.8 , pp. 1-21
    • DALANG, R.1    MUELLER, C.2
  • 16
    • 0035605884 scopus 로고    scopus 로고
    • Convergence of Numerical Schemes for the Solution of Parabolic Stochastic Partial Differential Equatioins
    • A. M. DAVIE AND J. G. GAINES, Convergence of Numerical Schemes for the Solution of Parabolic Stochastic Partial Differential Equatioins, Mathematics of Computation. 70, 2000, 121-134.
    • (2000) Mathematics of Computation , vol.70 , pp. 121-134
    • DAVIE, A.M.1    GAINES, J.G.2
  • 17
    • 0026067446 scopus 로고
    • Low-dimensional models for complex geometry flows: Applications to grooved channels and circular cylinders
    • A. DEANE AND I. KEVREKIDIS AND G. KARNIADAKIS AND S. ORSZAG, Low-dimensional models for complex geometry flows: Applications to grooved channels and circular cylinders, Phys. Fluids A 3, 1991, 2337-2354.
    • (1991) Phys. Fluids A , vol.3 , pp. 2337-2354
    • DEANE, A.1    KEVREKIDIS, I.2    KARNIADAKIS, G.3    ORSZAG, S.4
  • 18
    • 0035423977 scopus 로고    scopus 로고
    • Thermal fluctuations of superconducting vortices
    • J. DEANG AND Q. DU AND M. GUNZBURGER, Thermal fluctuations of superconducting vortices,, Phy. Rev. B., 64, 2001, 52506-52510.
    • (2001) Phy. Rev. B , vol.64 , pp. 52506-52510
    • DEANG, J.1    DU AND, Q.2    GUNZBURGER, M.3
  • 19
    • 0035964634 scopus 로고    scopus 로고
    • M. K. DEB AND I. BABUŚKA AND J. T. ODEN, Solution of stochastic partial differential equations using Galerkin finite element, Comput. Methods Appl. Mech. Engrg. 190, 2001, 6359-6372.
    • M. K. DEB AND I. BABUŚKA AND J. T. ODEN, Solution of stochastic partial differential equations using Galerkin finite element, Comput. Methods Appl. Mech. Engrg. 190, 2001, 6359-6372.
  • 20
    • 0038620976 scopus 로고
    • Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises
    • Q. DU AND T. ZHANG, Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises, SIAM J. Numer. Anal. 40, 1993, 1421-1445.
    • (1993) SIAM J. Numer. Anal , vol.40 , pp. 1421-1445
    • DU AND, Q.1    ZHANG, T.2
  • 23
    • 0037127128 scopus 로고    scopus 로고
    • Meshfree, probabilistic determination of points, sets and support regions for meshless computing
    • Q. DU AND M. GUNZBURGER AND L. JU, Meshfree, probabilistic determination of points, sets and support regions for meshless computing, Comput. Meth. Appl. Mech. Engrg. 191, 2002, 1349-1366.
    • (2002) Comput. Meth. Appl. Mech. Engrg , vol.191 , pp. 1349-1366
    • DU AND, Q.1    GUNZBURGER, M.2    JU, L.3
  • 24
    • 0000754401 scopus 로고
    • Large Fluctuations for a Nonlinear Heat Equation with Noise
    • W. G. FARIS AND G. JONA-LASINO, Large Fluctuations for a Nonlinear Heat Equation with Noise, J. Phys. A:MAth Gen. 18, 1982, 3025-3055.
    • (1982) J. Phys. A:MAth Gen , vol.18 , pp. 3025-3055
    • FARIS, W.G.1    JONA-LASINO, G.2
  • 25
    • 0030131207 scopus 로고    scopus 로고
    • Pattern analysis and model reduction: Some alternative approaches to the Karhunen-Lóeve decomposition
    • M. GRAHAM AND I. KEVREKIDIS, Pattern analysis and model reduction: Some alternative approaches to the Karhunen-Lóeve decomposition, Comp. Chem. Engng. 20, 1996, 495-506.
    • (1996) Comp. Chem. Engng , vol.20 , pp. 495-506
    • GRAHAM, M.1    KEVREKIDIS, I.2
  • 26
    • 0042078403 scopus 로고    scopus 로고
    • Lattice Approximations of Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise II
    • I. GYONGY, Lattice Approximations of Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise II, Potential Analysis. 11, 1999, 1-37.
    • (1999) Potential Analysis , vol.11 , pp. 1-37
    • GYONGY, I.1
  • 29
    • 0031206340 scopus 로고    scopus 로고
    • Low-dimensional models of coherent structures in turbulence
    • P. HOLMES AND J. LUMLEY AND G. BERKOOZ AND J. MATTINGLY AND R. WITTENBERG, Low-dimensional models of coherent structures in turbulence, Phys. Rep. 287, 1997, 337-384.
    • (1997) Phys. Rep , vol.287 , pp. 337-384
    • HOLMES, P.1    LUMLEY, J.2    BERKOOZ, G.3    MATTINGLY, J.4    WITTENBERG, R.5
  • 30
    • 50049086922 scopus 로고
    • Numerical Solution of Partial Differential Equations by the Finite Element Method
    • C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press., 1994.
    • (1994) Cambridge University Press
    • JOHNSON, C.1
  • 32
    • 0033177060 scopus 로고    scopus 로고
    • Control of Burger's equation by a reduced order approach using proper orthogonal decomposition
    • K. KUNISCH AND S. VOLKWEIN, Control of Burger's equation by a reduced order approach using proper orthogonal decomposition, JOTA 102, 1999, 345-371.
    • (1999) JOTA , vol.102 , pp. 345-371
    • KUNISCH, K.1    VOLKWEIN, S.2
  • 35
    • 50049134885 scopus 로고    scopus 로고
    • Galerkin Methods for Linear and Nonlinear Elliptic Stochastic
    • to appear
    • H. G. MATTHIES AND A. KEESE, Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations, 2003, to appear.
    • (2003) Partial Differential Equations
    • MATTHIES, H.G.1    KEESE, A.2
  • 36
    • 0036628270 scopus 로고    scopus 로고
    • Stochastic Partial Differential Equation Driven by Stable Noise
    • L. MYTNIK, Stochastic Partial Differential Equation Driven by Stable Noise, Probab. Theory Relat. Fields. 123, 2002, 157-201.
    • (2002) Probab. Theory Relat. Fields , vol.123 , pp. 157-201
    • MYTNIK, L.1
  • 37
    • 0018503565 scopus 로고
    • Modal representation of geometrically nonlinear behavior by the finite element method
    • D. NAGY, Modal representation of geometrically nonlinear behavior by the finite element method, Comput. & Struct. 10, 683 (1979).
    • (1979) Comput. & Struct , vol.10 , pp. 683
    • NAGY, D.1
  • 38
    • 0019242096 scopus 로고
    • Recent advances in reduction methods for nonlinear problems
    • A. K. NOOR, Recent advances in reduction methods for nonlinear problems, Comput. & Struct. 13, 31 (1981).
    • (1981) Comput. & Struct , vol.13 , pp. 31
    • NOOR, A.K.1
  • 39
    • 85003373847 scopus 로고    scopus 로고
    • A. K. NOOR AND C.M. ANDERSON AND J.M. PETERS, Reduced basis technique for collapse analysis of shells, AAIA J. 19, 393 (1981).
    • A. K. NOOR AND C.M. ANDERSON AND J.M. PETERS, Reduced basis technique for collapse analysis of shells, AAIA J. 19, 393 (1981).
  • 40
    • 0019607338 scopus 로고    scopus 로고
    • A. K. NOOR AND J.M. PETERS, Tracking post-linit-paths with reduced basis technique, Comput. Methods Appl. Mech. Eng. 28, 217 (1981).
    • A. K. NOOR AND J.M. PETERS, Tracking post-linit-paths with reduced basis technique, Comput. Methods Appl. Mech. Eng. 28, 217 (1981).
  • 41
    • 0019242096 scopus 로고
    • Recent advances in reduction methods for nonlinear problems
    • A. K. NOOR, Recent advances in reduction methods for nonlinear problems, Comput. & Struc. 13, 1981, 31-44.
    • (1981) Comput. & Struc , vol.13 , pp. 31-44
    • NOOR, A.K.1
  • 42
    • 50049108123 scopus 로고    scopus 로고
    • B. ØKSENDAL, Stochastic Differential Equations. An Introduction with Applications, Spring-Verlag., fifth edition, 1998.
    • B. ØKSENDAL, Stochastic Differential Equations. An Introduction with Applications, Spring-Verlag., fifth edition, 1998.
  • 43
    • 0030297465 scopus 로고    scopus 로고
    • H. M. PARK AND D. CHO, Low dimensional modeling of flow reactors, Int. J. Heat Mass Transf. 39, 1996, 3311-3323.
    • H. M. PARK AND D. CHO, Low dimensional modeling of flow reactors, Int. J. Heat Mass Transf. 39, 1996, 3311-3323.
  • 44
    • 0036608807 scopus 로고    scopus 로고
    • H. M. PARK AND J. S. CHUNG A sequential method of solving inverse natural convection problems. Inverse Problems 18 (2002), no. 3, 529-546.
    • H. M. PARK AND J. S. CHUNG A sequential method of solving inverse natural convection problems. Inverse Problems 18 (2002), no. 3, 529-546.
  • 45
    • 0033906352 scopus 로고    scopus 로고
    • H. M. PARK AND Y. D. JANG Control of Burgers equation by means of mode reduction. Internat. J. Engrg. Sci. 38 (2000), no. 7, 785-805
    • H. M. PARK AND Y. D. JANG Control of Burgers equation by means of mode reduction. Internat. J. Engrg. Sci. 38 (2000), no. 7, 785-805
  • 46
    • 0034340154 scopus 로고    scopus 로고
    • H. M. PARK AND J. H. LEE Solution of an inverse heat transfer problem by means of empirical reduction of modes. Z. Angew. Math. Phys. 51 (2000), no. 1, 17-38
    • H. M. PARK AND J. H. LEE Solution of an inverse heat transfer problem by means of empirical reduction of modes. Z. Angew. Math. Phys. 51 (2000), no. 1, 17-38
  • 47
    • 0032031194 scopus 로고    scopus 로고
    • H. M. PARK AND M. W. LEE An efficient method of solving the Navier-Stokes equations for flow control. Internat. J. Numer. Methods Engrg. 41 (1998), no. 6, 1133-1151
    • H. M. PARK AND M. W. LEE An efficient method of solving the Navier-Stokes equations for flow control. Internat. J. Numer. Methods Engrg. 41 (1998), no. 6, 1133-1151
  • 48
    • 0037051068 scopus 로고    scopus 로고
    • H. M. PARK AND W. J. LEE A new numerical method for the boundary optimal control problems of the heat conduction equation. Internat. J. Numer. Methods Engrg. 53 (2002), no. 7, 1593-1613
    • H. M. PARK AND W. J. LEE A new numerical method for the boundary optimal control problems of the heat conduction equation. Internat. J. Numer. Methods Engrg. 53 (2002), no. 7, 1593-1613
  • 49
    • 0000166867 scopus 로고
    • The reduced basis method for incompressible viscous flow calculations
    • J. S. PETERSON, The reduced basis method for incompressible viscous flow calculations, SIAM J. Sci. Stat. Comput. 10. 777 (1989).
    • (1989) SIAM J. Sci. Stat. Comput , vol.10 , pp. 777
    • PETERSON, J.S.1
  • 50
    • 33947244131 scopus 로고    scopus 로고
    • A new look at proper orthogonal decomposition
    • to appear
    • M. RATHINAM AND L. PETZOLD, A new look at proper orthogonal decomposition, to appear.
    • RATHINAM, M.1    PETZOLD, L.2
  • 51
    • 0004948711 scopus 로고    scopus 로고
    • Dynamic iteration using reduced order models: A method for simulation of large scale modular systems
    • to appear
    • M. RATHINAM AND L. PETZOLD, Dynamic iteration using reduced order models: A method for simulation of large scale modular systems, to appear.
    • RATHINAM, M.1    PETZOLD, L.2
  • 52
    • 0034327551 scopus 로고    scopus 로고
    • Proper orthogonal decomposition in optimal control of fluids
    • S. RAVINDRAN, Proper orthogonal decomposition in optimal control of fluids, Int. J. Numer. Meth. Fluids 34, 2000, 425-448.
    • (2000) Int. J. Numer. Meth. Fluids , vol.34 , pp. 425-448
    • RAVINDRAN, S.1
  • 53
    • 0034435557 scopus 로고    scopus 로고
    • Reduced-order adaptive controllers for fluid flows using POD
    • S. RAVINDRAN, Reduced-order adaptive controllers for fluid flows using POD, J. Sci. Comput. 15 (2000), no. 4, 457-478.
    • (2000) J. Sci. Comput , vol.15 , Issue.4 , pp. 457-478
    • RAVINDRAN, S.1
  • 54
    • 0002367721 scopus 로고
    • Low-dimensional dynamics for the complex Ginzburg-Landau equations
    • J. RODRÍGUEZ AND L. SIROVICH, Low-dimensional dynamics for the complex Ginzburg-Landau equations, Physica D 43, 1990, 77-86.
    • (1990) Physica D , vol.43 , pp. 77-86
    • RODRÍGUEZ, J.1    SIROVICH, L.2
  • 55
    • 0141827555 scopus 로고    scopus 로고
    • Non-equilibrium Ginzburg-Landau model driven by colored noise
    • J. SANCHO, J. GARCIA-OJALVO AND H. GUO, Non-equilibrium Ginzburg-Landau model driven by colored noise, Phys. D. 113, 1998, 331-337.
    • (1998) Phys. D , vol.113 , pp. 331-337
    • SANCHO, J.1    GARCIA-OJALVO, J.2    GUO, H.3
  • 56
    • 10644245188 scopus 로고    scopus 로고
    • Numerical Simulation of Stochastic PDEs for Excitable Media
    • T. SHARDLOW, Numerical Simulation of Stochastic PDEs for Excitable Media, Journal of Computational and Applied Mathematics. 175, 2005, 429-446.
    • (2005) Journal of Computational and Applied Mathematics , vol.175 , pp. 429-446
    • SHARDLOW, T.1
  • 57
    • 4043178349 scopus 로고    scopus 로고
    • Weak Convergence of a Numerical Method For a Stochastic Heat Equation
    • T. SHARDLOW, Weak Convergence of a Numerical Method For a Stochastic Heat Equation, BIT Numerical Mathematics. 43, 2003, 179-193.
    • (2003) BIT Numerical Mathematics , vol.43 , pp. 179-193
    • SHARDLOW, T.1
  • 58
    • 17644388473 scopus 로고    scopus 로고
    • Nucleation of Waves in Excitable Media by Noise
    • T. SHARDLOW, Nucleation of Waves in Excitable Media by Noise, Multiscal Model Simul. 3, 2004, 151-167.
    • (2004) Multiscal Model Simul , vol.3 , pp. 151-167
    • SHARDLOW, T.1
  • 59
    • 0000246863 scopus 로고
    • Turbulence and the dynamics of coherent structures, I-III
    • L. SIROVICH, Turbulence and the dynamics of coherent structures, I-III, Quart. Appl. Math. 45, 1987, 561-590.
    • (1987) Quart. Appl. Math , vol.45 , pp. 561-590
    • SIROVICH, L.1
  • 60
    • 0031215981 scopus 로고    scopus 로고
    • Symmetry and the Karhunen-Loève analysis
    • N. SMAOUI AND D. ARMBRUSTER, Symmetry and the Karhunen-Loève analysis, SIAM J. Sci. Comput. 18, 1997, 1526-1532.
    • (1997) SIAM J. Sci. Comput , vol.18 , pp. 1526-1532
    • SMAOUI, N.1    ARMBRUSTER, D.2
  • 61
    • 50049135324 scopus 로고    scopus 로고
    • I. M. SOBOL, A Primer for Monte Carlo Method, CRC Press. 1994.
    • I. M. SOBOL, A Primer for Monte Carlo Method, CRC Press. 1994.
  • 62
    • 33747791725 scopus 로고    scopus 로고
    • Optimal control of a phase field model using the proper orthogonal decomposition
    • S. VOLKWEIN, Optimal control of a phase field model using the proper orthogonal decomposition, ZAMM 81, 2001, 83-97.
    • (2001) ZAMM , vol.81 , pp. 83-97
    • VOLKWEIN, S.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.