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Volumn 55, Issue 1, 2008, Pages 31-40

Application of stochastic finite element methods to study the sensitivity of ECG forward modeling to organ conductivity

Author keywords

Electrocardiographic forward problem; Polynomial chaos; Stochastic finite elements; Stochastic Galerkin; Stochastic processes; Uncertainty quantification

Indexed keywords

BIOLOGICAL ORGANS; COMPUTER SIMULATION; ELECTRIC CONDUCTIVITY; FINITE ELEMENT METHOD; RANDOM PROCESSES; SENSITIVITY ANALYSIS;

EID: 37349053530     PISSN: 00189294     EISSN: None     Source Type: Journal    
DOI: 10.1109/TBME.2007.900563     Document Type: Article
Times cited : (50)

References (65)
  • 1
    • 0001562505 scopus 로고
    • The forward problem of electrocardiography
    • P. W. Macfarlane and T. D. Veitch Lawrie, Eds. Oxford, U.K, Pergamon
    • R. M. Gulrajani, F. A. Roberge, and G. E. Mailloux, "The forward problem of electrocardiography," in Comprehensive Electrocardiology, P. W. Macfarlane and T. D. Veitch Lawrie, Eds. Oxford, U.K.: Pergamon, 1989, pp. 197-236.
    • (1989) Comprehensive Electrocardiology , pp. 197-236
    • Gulrajani, R.M.1    Roberge, F.A.2    Mailloux, G.E.3
  • 2
    • 15144354136 scopus 로고    scopus 로고
    • The forward and inverse problems of electrocardiography
    • R. M. Gulrajani, "The forward and inverse problems of electrocardiography," EMBS Mag., vol. 17, no. 5, pp. 84-101, 1998.
    • (1998) EMBS Mag , vol.17 , Issue.5 , pp. 84-101
    • Gulrajani, R.M.1
  • 3
    • 0031907090 scopus 로고    scopus 로고
    • Recent progress in inverse problems in electrocardiology
    • Jan
    • R. S. MacLeod and D. H. Brooks, "Recent progress in inverse problems in electrocardiology," IEEE Eng. Med. Biol. Mag., vol. 17, no. 1, pp. 73-83, Jan. 1998.
    • (1998) IEEE Eng. Med. Biol. Mag , vol.17 , Issue.1 , pp. 73-83
    • MacLeod, R.S.1    Brooks, D.H.2
  • 5
    • 0029954935 scopus 로고    scopus 로고
    • The dielectric properties of biological tissue: I. literature survey
    • C. Gabriel, S. Gabriel, and E. Corthout, "The dielectric properties of biological tissue: I. literature survey," Phys. Med. Biol., vol. 41, pp. 2231-2249, 1996.
    • (1996) Phys. Med. Biol , vol.41 , pp. 2231-2249
    • Gabriel, C.1    Gabriel, S.2    Corthout, E.3
  • 6
    • 0032742068 scopus 로고    scopus 로고
    • The electric resistivity of human tissues (100 Hz - 10 Mhz): A meta-analysis of review studies
    • T. J. Faes, H. A. van der Meij, J. C. de Munck, and R. M. Heethaar, "The electric resistivity of human tissues (100 Hz - 10 Mhz): A meta-analysis of review studies," Physiol. Meas., vol. 20, pp. R1-R10, 1999.
    • (1999) Physiol. Meas , vol.20
    • Faes, T.J.1    van der Meij, H.A.2    de Munck, J.C.3    Heethaar, R.M.4
  • 7
    • 0029428588 scopus 로고
    • The effects of inhomogeneities and anisotropies on electrocardiographic fields: A three-dimensional finite elemental study
    • R. N. Klepfer, C. R. Johnson, and R. S. MacLeod, "The effects of inhomogeneities and anisotropies on electrocardiographic fields: A three-dimensional finite elemental study," in Proc. 17th Annu. IEEE Engineering Medicine Biology Soc. Int. Conf., 1995, pp. 233-234.
    • (1995) Proc. 17th Annu. IEEE Engineering Medicine Biology Soc. Int. Conf , pp. 233-234
    • Klepfer, R.N.1    Johnson, C.R.2    MacLeod, R.S.3
  • 9
    • 0026775782 scopus 로고
    • A computer model for the study of electrical current flow in the human thorax
    • C. R. Johnson, R. S. MacLeod, and P. R. Ershler, "A computer model for the study of electrical current flow in the human thorax," Comput. Biol. Med., vol. 22, no. 3, pp. 305-323, 1992.
    • (1992) Comput. Biol. Med , vol.22 , Issue.3 , pp. 305-323
    • Johnson, C.R.1    MacLeod, R.S.2    Ershler, P.R.3
  • 10
    • 0042664133 scopus 로고    scopus 로고
    • The effect of torso impedance on epicardial and body surface potentials: A modeling study
    • Jul
    • M. L. Buist and A. J. Pullan, "The effect of torso impedance on epicardial and body surface potentials: A modeling study," IEEE Trans. Biomed. Eng., vol. 50, no. 7, pp. 816-824, Jul. 2003.
    • (2003) IEEE Trans. Biomed. Eng , vol.50 , Issue.7 , pp. 816-824
    • Buist, M.L.1    Pullan, A.J.2
  • 11
    • 33846899838 scopus 로고
    • The effect of torso inhomogeneities on body surface potentials
    • G. J. M. Huiskamp and A. van Oosterom, "The effect of torso inhomogeneities on body surface potentials," J. Electrocardiol., vol. 22, pp. 1-20, 1989.
    • (1989) J. Electrocardiol , vol.22 , pp. 1-20
    • Huiskamp, G.J.M.1    van Oosterom, A.2
  • 12
    • 0024586455 scopus 로고
    • The effect of torso inhomogeneities on body surface potentials quantified using "tailored" geometry
    • A. van Oosterom and G. J. Huiskamp, "The effect of torso inhomogeneities on body surface potentials quantified using "tailored" geometry," J. Electrocardiol., vol. 22, pp. 53-72, 1989.
    • (1989) J. Electrocardiol , vol.22 , pp. 53-72
    • van Oosterom, A.1    Huiskamp, G.J.2
  • 13
    • 0013395974 scopus 로고    scopus 로고
    • The inverse problem of electrocardiography: Modeling, experimental, and clinical issues
    • supplement
    • A. Pullan, "The inverse problem of electrocardiography: Modeling, experimental, and clinical issues," Biomed. Technik, vol. 46, pp. 197-198, 2001, supplement.
    • (2001) Biomed. Technik , vol.46 , pp. 197-198
    • Pullan, A.1
  • 14
    • 33846907537 scopus 로고    scopus 로고
    • The influence of stochastic organ conductivity in 2-D ECG forward modeling: A stochastic finite element study
    • Shanghai, China, Sep. 1-4
    • S. E. Geneser, S. Choe, R. M. Kirby, and R. S. Macleod, "The influence of stochastic organ conductivity in 2-D ECG forward modeling: A stochastic finite element study," in Proc. 27th Annu. IEEE Eng. Medicine Biology Conf., Shanghai, China, Sep. 1-4, 2005, pp. 5528-5531.
    • (2005) Proc. 27th Annu. IEEE Eng. Medicine Biology Conf , pp. 5528-5531
    • Geneser, S.E.1    Choe, S.2    Kirby, R.M.3    Macleod, R.S.4
  • 15
    • 37349120636 scopus 로고    scopus 로고
    • 2D stochastic finite element study of the influence of organ conductivity in ECG
    • S. E. Geneser, S.-K. Choe, R. M. Kirby, and R. S. Macleod, "2D stochastic finite element study of the influence of organ conductivity in ECG," Int. J. Bioelectromagn., vol. 7, no. 1, pp. 321-324, 2005.
    • (2005) Int. J. Bioelectromagn , vol.7 , Issue.1 , pp. 321-324
    • Geneser, S.E.1    Choe, S.-K.2    Kirby, R.M.3    Macleod, R.S.4
  • 16
    • 33947434079 scopus 로고    scopus 로고
    • Stochastic Markovian modeling of electrophysiology of ion channels: Reconstruction of standard deviations in macroscopic currents
    • S. E. Geneser, D. B. Xiu, R. M. Kirby, and F. B. Sachse, "Stochastic Markovian modeling of electrophysiology of ion channels: Reconstruction of standard deviations in macroscopic currents," J. Theor. Biol., vol. 245, no. 4, pp. 627-637, 2007.
    • (2007) J. Theor. Biol , vol.245 , Issue.4 , pp. 627-637
    • Geneser, S.E.1    Xiu, D.B.2    Kirby, R.M.3    Sachse, F.B.4
  • 20
    • 0035441628 scopus 로고    scopus 로고
    • Computational stochastic mechanics - Recent advances
    • G. I. Schueller, "Computational stochastic mechanics - Recent advances," Comput. Struct., vol. 79, pp. 2225-2234, 2001.
    • (2001) Comput. Struct , vol.79 , pp. 2225-2234
    • Schueller, G.I.1
  • 21
    • 0000932182 scopus 로고
    • Monte Carlo sensitivity analysis of input-output models
    • C. W. Bullard and A. V. Sebald, "Monte Carlo sensitivity analysis of input-output models," Rev. Econ. Statistics, vol. 70, pp. 708-712, 1988.
    • (1988) Rev. Econ. Statistics , vol.70 , pp. 708-712
    • Bullard, C.W.1    Sebald, A.V.2
  • 23
    • 84900847494 scopus 로고
    • Small sample sensitivity analysis techniques for computer models, with an application to risk assessment
    • R. L. Iman and W. J. Conover, "Small sample sensitivity analysis techniques for computer models, with an application to risk assessment," Commun. Stat. Theory Meth., vol. A9, no. 17, pp. 1749-1842, 1980.
    • (1980) Commun. Stat. Theory Meth , vol.A9 , Issue.17 , pp. 1749-1842
    • Iman, R.L.1    Conover, W.J.2
  • 24
    • 15844394066 scopus 로고    scopus 로고
    • A comparison of uncertainty and sensitivity analysis results obtained with random and Latin hypercube sampling
    • J. C. Helton, F. J. Davis, and J. D. Johnson, "A comparison of uncertainty and sensitivity analysis results obtained with random and Latin hypercube sampling," Reliab. Eng. Syst. Saf., vol. 89, pp. 305-330, 2005.
    • (2005) Reliab. Eng. Syst. Saf , vol.89 , pp. 305-330
    • Helton, J.C.1    Davis, F.J.2    Johnson, J.D.3
  • 26
    • 0028465265 scopus 로고
    • Reliability evaluation of reinforced concrete beam
    • R. Lu, Y. Luo, and J. P. Conte, "Reliability evaluation of reinforced concrete beam," Struct. Saf., vol. 14, pp. 277-298, 1994.
    • (1994) Struct. Saf , vol.14 , pp. 277-298
    • Lu, R.1    Luo, Y.2    Conte, J.P.3
  • 27
    • 0026841760 scopus 로고
    • Sensitivity estimation within first and second order reliability methods
    • A. Karamchandani and C. A. Cornel, "Sensitivity estimation within first and second order reliability methods," Struct. Saf., vol. 11, pp. 95-107, 1992.
    • (1992) Struct. Saf , vol.11 , pp. 95-107
    • Karamchandani, A.1    Cornel, C.A.2
  • 28
    • 0019926821 scopus 로고
    • Global sensitivity analysis - A computational implementation of the Fourier amplitude sensitivity test (FAST)
    • G. J. McRae, J. W. Tilden, and J. H. Seinfeld, "Global sensitivity analysis - A computational implementation of the Fourier amplitude sensitivity test (FAST)," Comput. Chem. Eng., vol. 6, pp. 15-25, 1982.
    • (1982) Comput. Chem. Eng , vol.6 , pp. 15-25
    • McRae, G.J.1    Tilden, J.W.2    Seinfeld, J.H.3
  • 29
    • 37349129153 scopus 로고
    • Sensitivity of complex urban air quality model to input data
    • S. Seigneur, T. W. Teche, R. M. Roth, and L. E. Reid, "Sensitivity of complex urban air quality model to input data," J. Appl. Meteorol., vol. 20, pp. 157-177, 1981.
    • (1981) J. Appl. Meteorol , vol.20 , pp. 157-177
    • Seigneur, S.1    Teche, T.W.2    Roth, R.M.3    Reid, L.E.4
  • 30
    • 0028183888 scopus 로고
    • Sensitivity analysis of the requirements of rate coefficients for the operational models of photochemical oxidants formation in the troposphere
    • J. C. Dechaux, V. Zimmerman, and V. Nollet, "Sensitivity analysis of the requirements of rate coefficients for the operational models of photochemical oxidants formation in the troposphere," Atmosph. Environ., vol. 28, pp. 195-211, 1994.
    • (1994) Atmosph. Environ , vol.28 , pp. 195-211
    • Dechaux, J.C.1    Zimmerman, V.2    Nollet, V.3
  • 31
    • 0020401378 scopus 로고
    • Direct measurements of ozone and nitrogen dioxide photolysis rates in the troposphere
    • R. R. Dickerson, D. H. Stedman, and A. C. Delany, "Direct measurements of ozone and nitrogen dioxide photolysis rates in the troposphere," J. Geophys. Res., vol. 87, pp. 4933-4946, 1982.
    • (1982) J. Geophys. Res , vol.87 , pp. 4933-4946
    • Dickerson, R.R.1    Stedman, D.H.2    Delany, A.C.3
  • 32
    • 84986765250 scopus 로고
    • A computational algorithm for the Green's function method of sensitivity analysis in chemical kinetics
    • E. P. Doughtery and H. Rabitz, "A computational algorithm for the Green's function method of sensitivity analysis in chemical kinetics," Int. J. Chem. Kinet., vol. 11, pp. 1237-1249, 1979.
    • (1979) Int. J. Chem. Kinet , vol.11 , pp. 1237-1249
    • Doughtery, E.P.1    Rabitz, H.2
  • 33
    • 36749106262 scopus 로고
    • Further developments and applications of the Green's function method of sensitivity analysis in chemical kinetics
    • E. P. Doughtery, J. T. Hwang, and H. Rabitz, "Further developments and applications of the Green's function method of sensitivity analysis in chemical kinetics," Int. J. Chem. Kin., vol. 71, pp. 1794-1808, 1979.
    • (1979) Int. J. Chem. Kin , vol.71 , pp. 1794-1808
    • Doughtery, E.P.1    Hwang, J.T.2    Rabitz, H.3
  • 34
    • 37349074645 scopus 로고    scopus 로고
    • G. Adomian, Applied Stochastic Processes. New York: Stochastic System Analysis, 1980, pp. 1-17.
    • G. Adomian, Applied Stochastic Processes. New York: Stochastic System Analysis, 1980, pp. 1-17.
  • 35
    • 0026833191 scopus 로고
    • Aerodynamic design optimization using sensitivity analysis and computational fluid dynamics
    • O. Baysal and M. E. Eleshaky, "Aerodynamic design optimization using sensitivity analysis and computational fluid dynamics," Amer. Inst. Aeron. Astron., vol. 30, no. 3, pp. 718-725, 1992.
    • (1992) Amer. Inst. Aeron. Astron , vol.30 , Issue.3 , pp. 718-725
    • Baysal, O.1    Eleshaky, M.E.2
  • 36
    • 0028501607 scopus 로고
    • A review of techniques for parameter sensitivity analysis of environmental models
    • D. M. Hamby, "A review of techniques for parameter sensitivity analysis of environmental models," Environ. Monitor. Assess., vol. 32, no. 2, pp. 135-154, 1994.
    • (1994) Environ. Monitor. Assess , vol.32 , Issue.2 , pp. 135-154
    • Hamby, D.M.1
  • 37
    • 0024888725 scopus 로고
    • On using perturbation analysis to do sensitivity analysis: Derivatives versus differences
    • Tampa, FL, Dec. 13-15
    • J. M. Holtzman, "On using perturbation analysis to do sensitivity analysis: Derivatives versus differences," in Proc. 28th IEEE Conf. Decision Control, Tampa, FL, Dec. 13-15, 1989, pp. 2018-2023.
    • (1989) Proc. 28th IEEE Conf. Decision Control , pp. 2018-2023
    • Holtzman, J.M.1
  • 38
    • 0027879248 scopus 로고
    • An uncertainty analysis of the spectral correction factor
    • Louisville, KY, May 10-14
    • H. Feild and K. Emery, "An uncertainty analysis of the spectral correction factor," in Conf. Rec. 23rd IEEE Photovoltaic Specialists Conf., Louisville, KY, May 10-14, 1993, pp. 1180-1187.
    • (1993) Conf. Rec. 23rd IEEE Photovoltaic Specialists Conf , pp. 1180-1187
    • Feild, H.1    Emery, K.2
  • 39
    • 0030189059 scopus 로고    scopus 로고
    • Application of the first-order perturbation method to optimal structural design
    • B. W. Lee and O. K. Lim, "Application of the first-order perturbation method to optimal structural design," Struct. Eng. Mechan., vol. 4, pp. 425-436, 1996.
    • (1996) Struct. Eng. Mechan , vol.4 , pp. 425-436
    • Lee, B.W.1    Lim, O.K.2
  • 40
    • 0000786435 scopus 로고
    • The homogeneous chaos
    • N. Wiener, "The homogeneous chaos," Amer. J. Math., vol. 60, no. 4, pp. 897-936, 1938.
    • (1938) Amer. J. Math , vol.60 , Issue.4 , pp. 897-936
    • Wiener, N.1
  • 41
    • 0000476520 scopus 로고
    • Use of Wwiener-Hermite expansion for nearly normal turbulence
    • W. C. Meecham and D. T. Jeng, "Use of Wwiener-Hermite expansion for nearly normal turbulence," J. Fluid Mech., no. 32, pp. 225-249, 1968.
    • (1968) J. Fluid Mech , Issue.32 , pp. 225-249
    • Meecham, W.C.1    Jeng, D.T.2
  • 42
    • 0001203226 scopus 로고
    • Hermite expansions in Monte Carlo computation
    • A. J. Chorin, "Hermite expansions in Monte Carlo computation," J. Comput. Phys., vol. 8, pp. 471-482, 1971.
    • (1971) J. Comput. Phys , vol.8 , pp. 471-482
    • Chorin, A.J.1
  • 43
    • 0347659106 scopus 로고
    • Gaussian fields and random flow
    • A. J. Chorin, "Gaussian fields and random flow," J. Fluid Mech., vol. 63, pp. 21-32, 1974.
    • (1974) J. Fluid Mech , vol.63 , pp. 21-32
    • Chorin, A.J.1
  • 44
    • 0000531673 scopus 로고
    • Variance reduction in Monte Carlo computations using multi-dimensional Hermite polynomials
    • F. H. Maltz and D. L. Hitzl, "Variance reduction in Monte Carlo computations using multi-dimensional Hermite polynomials," J. Comput. Phys., vol. 32, pp. 345-376, 1979.
    • (1979) J. Comput. Phys , vol.32 , pp. 345-376
    • Maltz, F.H.1    Hitzl, D.L.2
  • 45
    • 0035964634 scopus 로고    scopus 로고
    • M. K. Deb, I. M. Babu?ska, and J. T. Oden, Solutions of stochastic partial differential equations using Galerkin finite element techniques, Comput. Methods Appl. Mech. Eng., 190, pp. 6359-6372, 2001.
    • M. K. Deb, I. M. Babu?ska, and J. T. Oden, "Solutions of stochastic partial differential equations using Galerkin finite element techniques," Comput. Methods Appl. Mech. Eng., vol. 190, pp. 6359-6372, 2001.
  • 46
    • 0036742814 scopus 로고    scopus 로고
    • A stochastic projection method for fluid flow II: Random process
    • O. P. Le Maître, M. Reagan, H. N. Najm, R. G. Ghanem, and O. M. Knio, "A stochastic projection method for fluid flow II: Random process," J. Comput. Phys., vol. 181, no. 1, pp. 9-44, 2002.
    • (2002) J. Comput. Phys , vol.181 , Issue.1 , pp. 9-44
    • Le Maître, O.P.1    Reagan, M.2    Najm, H.N.3    Ghanem, R.G.4    Knio, O.M.5
  • 47
    • 0037184252 scopus 로고    scopus 로고
    • Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos
    • D. B. Xiu and G. E. Karniadakis, "Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos," Comput. Methods Appl. Mech. Eng., vol. 191, pp. 4927-4948, 2002.
    • (2002) Comput. Methods Appl. Mech. Eng , vol.191 , pp. 4927-4948
    • Xiu, D.B.1    Karniadakis, G.E.2
  • 48
    • 0037408312 scopus 로고    scopus 로고
    • Modeling uncertainty in flow simulations via generalized polynomial chaos
    • D. B. Xiu and G. E. Karniadakis, "Modeling uncertainty in flow simulations via generalized polynomial chaos," J. Comput. Phys., vol. 187, pp. 137-167, 2003.
    • (2003) J. Comput. Phys , vol.187 , pp. 137-167
    • Xiu, D.B.1    Karniadakis, G.E.2
  • 49
    • 2542421211 scopus 로고    scopus 로고
    • Generalized polynomial chaos and random oscillators
    • D. Lucor, C.-H. Su, and G. E. Karniadakis, "Generalized polynomial chaos and random oscillators," Int. J. Numer. Meth. Eng., vol. 60, pp. 571-596, 2004.
    • (2004) Int. J. Numer. Meth. Eng , vol.60 , pp. 571-596
    • Lucor, D.1    Su, C.-H.2    Karniadakis, G.E.3
  • 50
    • 0032871636 scopus 로고    scopus 로고
    • Higher order sensitivity of heat conduction problems to random data using the spectral stochastic finite element method
    • R. G. Ghanem, "Higher order sensitivity of heat conduction problems to random data using the spectral stochastic finite element method," ASME J. Heat Transfer, vol. 121, pp. 290-299, 1999.
    • (1999) ASME J. Heat Transfer , vol.121 , pp. 290-299
    • Ghanem, R.G.1
  • 51
    • 3142610894 scopus 로고    scopus 로고
    • Stochastic inverse heat conduction using a spectral approach
    • V. A. Narayanan and N. Zabaras, "Stochastic inverse heat conduction using a spectral approach," Int. J. Numer. Meth. Eng., vol. 60, pp. 1569-1593, 2004.
    • (2004) Int. J. Numer. Meth. Eng , vol.60 , pp. 1569-1593
    • Narayanan, V.A.1    Zabaras, N.2
  • 52
    • 0141792397 scopus 로고    scopus 로고
    • A new stochastic approach to transient heat conduction modeling with uncertainty
    • D. B. Xiu and G. E. Karniadakis, "A new stochastic approach to transient heat conduction modeling with uncertainty," Int. J. Heat Mass Transfer, vol. 46, pp. 4681-4693, 2003.
    • (2003) Int. J. Heat Mass Transfer , vol.46 , pp. 4681-4693
    • Xiu, D.B.1    Karniadakis, G.E.2
  • 61
    • 0031214095 scopus 로고    scopus 로고
    • The effects of inhomogeneities and anisotropies on electrocardiographic fields: A three-dimensional finite element study
    • R. N. Klepfer, C. R. Johnson, and R. S. MacLeod, "The effects of inhomogeneities and anisotropies on electrocardiographic fields: A three-dimensional finite element study," IEEE Trans. Biomed. Eng., vol. 44, no. 8, pp. 706-719, 1997.
    • (1997) IEEE Trans. Biomed. Eng , vol.44 , Issue.8 , pp. 706-719
    • Klepfer, R.N.1    Johnson, C.R.2    MacLeod, R.S.3
  • 62
    • 0037249229 scopus 로고    scopus 로고
    • The Wiener-Askey polynomial chaos for stochastic differential equations
    • D. B. Xiu and G. E. Karniadakis, "The Wiener-Askey polynomial chaos for stochastic differential equations," SIAM J. Sci. Comput., vol. 24, pp. 619-644, 2002.
    • (2002) SIAM J. Sci. Comput , vol.24 , pp. 619-644
    • Xiu, D.B.1    Karniadakis, G.E.2
  • 63
    • 0003998877 scopus 로고
    • 4th ed. New York: Springer-Verlag
    • M. Loève, Probability Theory I, 4th ed. New York: Springer-Verlag, 1977.
    • (1977) Probability Theory I
    • Loève, M.1
  • 64
    • 13844319204 scopus 로고    scopus 로고
    • Galerkin finite element approximations of stochastic elliptic differential equations
    • I. M. Babuška, R. Tempone, and G. E. Zouraris, "Galerkin finite element approximations of stochastic elliptic differential equations," SIAM J. Numer. Anal., vol. 42, pp. 800-825, 2004.
    • (2004) SIAM J. Numer. Anal , vol.42 , pp. 800-825
    • Babuška, I.M.1    Tempone, R.2    Zouraris, G.E.3
  • 65
    • 2942555049 scopus 로고    scopus 로고
    • Uncertainty propagation using Wiener-Haar expansions
    • O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, "Uncertainty propagation using Wiener-Haar expansions," J. Comput. Phys., vol. 197, pp. 28-57, 2004.
    • (2004) J. Comput. Phys , vol.197 , pp. 28-57
    • Le Maître, O.P.1    Knio, O.M.2    Najm, H.N.3    Ghanem, R.G.4


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