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Fractional and Stochastic PDEs/Uncertainty Quantification

Fractional and Stochastic PDEs/Uncertainty Quantification

    1. F. Zeng, Z. Zhang, G. E. Karniadakis, "Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions." Comput. Methods Appl. Mech. Engrg. 327, 478-502.  (2017), http://dx.doi.org/10.1016/j.cma.2017.08.029.

    2. F. Song, C. Xu, G. E. Karniadakis, "Computing fractional laplacians on complex-geometry domains: algorithms and simulations." SIAM J. Sci. Comput. 39(4),A1320-A1344.

    3. G. Pang, P. Perdikaris, W, Cai, G.E. Karniadakis, "Discovering variable fractional orders of advection-dispersion equations from field data using multi-fidelity Bayesian optimization." J. Comput. Phys. (2017), http://dx.doi.org/10.1016/j.jcp.2017.07.052

    4. E. Kharazmi, M. Zayernouri, G. E. Karniadakis, "A Petrov-Galerkin spectral element method for fractional elliptic problems." Comput. Methods Appl. Mech. Engrg. 324, 512-536, 2017

    5. F. Song, G. E. Karniadakis, "Fractional spectral vanishing viscosity method: Application to the quasi-geostrophic equation." Chaos Solitons and Fractals, 102; 327-352, 2017

    6. L. Bonfiglio, P. Perdikaris, S. Brizzolara, and G. E. Karniadakis. "A multi-fidelity framework for investigating the performance of super-cavitating hydrofoils under uncertain flow conditions." AIAA Non-Deterministic Approaches, 2017.

    7. F. Zeng, Z. Mao, G.E. Karniadakis, "A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities." SIAM J. Sci. Comput. 2017, 39(1);A360-A383.

    8. F. Sang, F. Zeng, W. Cai, W. Chen, G. E. Karniadakis, "Efficient two-dimensional simulations of the fractional Szabo equation with different time-stepping schemes." Computers and Mathematics with Applications, 2017, 73; 1286-1297.

    9. X. Chen, F. Zeng, G.E. Karniadakis, "A tunable finite difference method for fractional differential equations with non-smooth solutions." Comput. Methods Appl. Mech. Engrg. 2017, 318;193-214.

    10. Z. Mao, G.E. Karniadakis, "Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods." J. Comput. Phys. 2017, 336;143-163.

    11. L. Parussini, D. Venturi, P. Perdikaris, G.E. Karniadakis, "Multi-fidelity Gaussian process regression for prediction of random fields". J. Comput. Phys. 2017, 336; 36-50. 

    12. Z. Mao, G.E. Karniadakis, "Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin Methods". J. Comput. Phys. 2017, http://dx.doi.org/10.1016/j.jcp.2017.01.048

    13. F. Song, F. Zeng, W. Cai, W. Chen and G.E. Karniadakis,"Efficient two-dimensional simulations of the fractional Szabo equation with different time-stepping schemes",  Computers and Mathematics with Applications.  http://dx.doi.org/10.1016/j.camwa.2016.11.018

    14. C. Kim, O.Borodin and G.E. Karniadakis, "Quantification of sampling uncertainty for molecular dynamics simulation: Time-dependent diffusion coefficient in simple fluids", J. Comp. Phys., 302,485-508,2016

    15. J. Suzuki, M. Zayernouri and G.E. Karniadakis, "Fractional-order uniaxial visco-elasto-plastic models for structural analysis", Computer Methods in Applied Mechanics and Engineering, 308, 443-467, 2016

    16. Y. Yu, P.Perdikaris and G.E. Karniadakis, "Fractional Modeling of Viscoelasticity in 3D Cerebral Arteries and Aneurysms", J. Comput. Phys.,  In Press, 2016, DOI:10.1016/J.JCP.2016.06.038

    17. Z. Zhang, B. Rozovskii, G.E. Karniadakis, " Strong and weak convergence order of finite element methods for stochastic PDEs with spatial white noise,"  Numer.Math, 134, 61-89 2016

    18. F. Song, C. Xu, G.E.Karniadakis, "A fractional phase-field model for two-phase flows with tunable sharpness:Algorithms and simulations," Comput. Methods Appl. Mech. Engrg. 305, 376-404 2016.

    19. F. Zeng, Z. Zhang and G.E. Karniadakis, "Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations", J. Comput. Phys, 307, 15-33, 2016

    20. P. Perdikaris, D. Venturi, J. Royset, and G. E. Karniadakis, "Multi-fidelity modelling via recursive co-kriging and Gaussian-Markov random fields." P. R. Soc. A., vol 471, 2017.

    21. M. Zayernouri, M. Ainsworth and G.E. Karniadakis, "Tempered Fractional Sturm-Liouville eigen-problems", SIAM J. Sci. Comput., 37(4) A1777-A1800, 2015

    22. H. Cho, Y.Yang, D.Venturi and G.E. Karniadakis, "Algorithms for propagating uncertainty across heterogeneous domains", SIAM J. Sci. Comput. 37(6), A3030-A3064, 2015

    23. L. Xuejin, L. Zhen, X. Bian, M. Deng, C. Kim, Y. Tang, A. Yazdani, G.E. Karniadakis, "Dissipative Particle Dynamics, Overview," Encyclopedia of Nanotechnology, doi 10.1007/978-94-007-6178-0_100954-1.

    24. G.E. Karniadakis, J.S. Hesthaven, I. Podlubny, "Special Issue on "Fractional PDEs: Theory, Numerics, and Applications," J. Comput. Phys. 293, 1-3 2015.
       
    25. M. Zayernouri and G.E. Karnidakis, "Fractional spectral collocation methods for linear and nonlinar variable order FPDEs," J. Comput. Phys. 293, 312-338, 2015.
       
    26. W. Cao, Z. Zhang and G.E. Karniadakis, "Numerical methods for stochastic delay differential equations via the Wong-Zakai approximation," SIAM J. Sci. Comput, 37(1), A295-A318, 2015.
       
    27. B. Yildirim and G.E. Karniadakis, "Stochastic simulations of ocean waves: An uncertainty quantification study," Ocean Modelling, 86, 15-35, 2015.
       
    28. M. Zayernouri, M. Ainsworth and G.E. Karnidakis, "A unified Petrov-Galerkin spectral method for fractiona PDEs," Computer Methods in Applied Mechanics and Engineering,  283, 1545-1569,  DOI/10.1016/j.cma.2014.10.051, 2015.
       
    29. Z. Zhang, M.V. Tretyakov, B. Rozovskii and G.E. Karniadakis, "A recursive sparse grid collocation method for differential equations with white noise," SIAM J. Sci. Comput. 36(4), A1652-A1677, 2014.
       
    30. M. Zayernouri and G.E. Karniadais, "Discontinuous spectral element methods for time-and space-fractional advection equations," SIAM J. Sci. Comput. 36(4), B684-B707, 2014.
       
    31. D. Venturi and G.E. Karniadakis, "Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems," Proc. Royal Soc. A, 470, 20130754, 2014.
       
    32. M. Zayernouri and G.E. Karnidakis, "Exponentially accurate spectral and spectral element methods for fractional ODEs," J. Comput. Phys., 247, Part A, 460-480, 2014.
       
    33. M. Zayernouri and G.E. Karniadakis, "Fractional spectral collocation methods," SIAM J. Sci. Comput. 36(1), A40-A62, 2014.
       
    34. M.V. Tretyakov and  Z. Zhang, "A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications," SIAM J. Numer. Anal., 51(6), 3135-3162, 2013.
       
    35. M. Zayernouri and G.E. Karniadakis, "Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation," J. Comput. Phys., 252, 495-517, 2013.
       
    36. H. Cho, D. Venturi and G.E. Karniadakis, "Karhunen-Loeve expansion for multi-correlated stochastic processes," Probabilistic Engineering Mechanics, 34, 157-167, 2013.
       
    37. H. Cho, D. Venturi and G.E. Karniadakis, "Adaptive discontinuous Galerkin method for response-excitation PDF equations," SIAM J. Sci. Comput. 35(4), B890-B911, 2013.
       
    38. W. Cao and Z. Zhang, "On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations," J. Comput. Appl. Math., 244, 182-193, 2013.
       
    39. C. Kim and G.E. Karniadakis, "Microscopic theory of Brownian motion revised: The Rayleigh model," Phy. Review E, 87(3), 032129, 2013.
       
    40. X. Yang and G.E. Karniadakis,"Reweighted lminimization method for stochastic elliptic differential equations," J. Comp. Phys., 248, 87-108, 2013.
       
    41. Venturi, D.M. Tartakovsky, A.M. Tartakovsky and G.E. Karniadakis, "Exact PDF equations and closure approximations for advective-reactive transport," J. Comp. Phys., 243, 323-343, 2013.|
       
    42. M. Zayernouri, S-W. Park, D.M. Tartakovsky and G.E. Karniadakis, "Stochastic smoothed profile method for modeling random roughness in flow problems," Comput. Methods Appl. Mech. Engrg. 263, 99-112, 2013.
    43. M. Choi, T. Sapsis and G.E. Karniadakis, "A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes," J. Comp. Phys., 245, 281-301, 2013.
       
    44. Z. Zhang, X. Yang, G. Lin and G.E. Karniadakis, "Numerical solution of the Stratonovich- and Ito-Euler equations: Application to the stochastic piston problem," J. Comp. Phys.  236, 15-27, 2013.
       
    45. W. Cao and Z. Zhang, "Simulations of two-step Maruyama methods for nonlinear stochastic delay differential equations," Adv. Appl. Math. Mech., 4(6), 821-832, 2012.|
       
    46. X. Yang, M. Choi, G. Lin and G.E. Karniadakis, "Adaptive ANOV decomposition of stochastic incompressible and compressible flows," J. Comp. Phys., 231, 1587-1614, 2012.
       
    47. Z. Zhang, B. Rosvoskii, M.V. Tretyakov and G.E. Karniadakis, "A multi-stage Wiener chaos expansion method for stochastic advection-diffusion-reaction equations," SIAM J. Sci. Comput., 34(2), A914-A936, 2012.
       
    48. Z. Zhang, M. Choi and G.E. Karniadakis,"Error estimates for the ANOVA method with polynomial chaos interpolations: Tensor product functions," SIAM J. Sci. Comp., 34(2), A1165-A1186, 2012.
       
    49. D. Venturi, M. Choi and G.E. Karniadakis, Supercritical quasi-continuum states in stochastic Rayleigh-Bernard convection," Int. J. Heat & Mass Transfer, 55, 3732-3743, 2012.
       
    50. D. Venturi and G.E. Karniadakis, "Differential constrains for the probability density function of stochastic solutions to the wave equations," Int. J. for Uncertainty Quantification, 2(3), 195-213, 2012.
       
    51. D. Venturi, T. Sapsis and G.E. Karniadakis, "A computable evolution equation for the probability density function of stochastic dynamical systems," Proc. Roy. Soc. A, 468(2139), 759-783, 2012.
       
    52. D. Venturi and G.E. Karniadakis, "New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs," J. Comp. Phys., 231, 7450-7474, 2012.
       
    53. Z. Zhang, B. Rosovskii, M.V. Tretyakov and G.E. Karniadakis, "A multi-stage Wiener chaos expansion method for stochastic advection-diffusion-reaction equations," SIAM J. Sci. Comput. 34(2), A914-A936, 2012.
       
    54. J. Foo and G.E. Karniadakis, "Multi-element probabilistic collocation method in high dimensions," J. Comp. Phys., 229(5), 1536-1577, 2010.
       
    55. D. Venturi, X. Wan and G.E. Karniadakis "Stochastic bifurcation analysis of Rayleigh-Benard Convection," J. Fluid Mech., 650, 391-413, 2010.
       
    56. M. Gerritsma, J-B. van der Steen, P. Vos and G.E. Karniadakis, "Time-dependent generalized polynomial chaos," J. Comput. Phys., 229, 8333-8363, 2010.
       
    57. P. Prempraneerach, F.S. Hover, M.S. Triantafyllou and G.E. Karniadakis, "Uncertainty quantification in simulations of power systems: Multi-element polynomial chaos methods," Reliability Engineering and System Safety", 95, 632-646, 2010.
       
    58. Z. Zhang, M. Choi and G.E. Karniadakis, "Anchor points matter in ANOVA decomposition," ICOSAHOM09.
       
    59. G. Lin and G.E. Karniadakis, "Sensitivity analysis and stochastic simulations of non-equilibrium plasma flow," Int. J. Numer. Meth. Engng., 80, 738-766, published online DOI.10.1002/nme.2582, 2009. 
       
    60. J. Foo, S. Sindi and G.E. Karniadakis, "Multi-element probabilistic collocation for sensitivity analysis in cellular signaling networks," IET Systems Biology, 3(4), 239-254, DOI.10.1049/iet-sub.2009.0126, 2009.
       
    61. X. Wan. B. Rozovskii and G.E. Karniadakis, "A stochastic modeling method based on weighted Wiener chaos and Malliavan calculus," PNAS, 106, 14189-14194, DOI: 10.1073/pnas.0902348106, 2009.
       
    62. X. Wan and G.E. Karniadakis, "Solving elliptic problems with non-Gaussian spatially-dependent random coefficients." Comput. Methods Appl. Mech. Engrg., 198, 1985-1995, 2009.
       
    63. X. Wan and G.E. Karniadakis, "Error control in multi-element generalized polynomial chaos methods for elliptic problems with random coefficient," Communication in Computational Physics, 4, 793-820, 2009.
       
    64. G. Lin, C-H. Su and G.E. Karniadakis, "Stochastic modeling of random roughness in shock scattering problems: Theory and simulations," Comput. Methods Appl. Mech. Engrg. 197, 3420-3334, 2008.
       
    65. J. Foo, X. Wan and G.E. Karniadakis, "The multi-element probabilistic collocation methods (ME-PCM): Error analysis and applications," J. Comp. Phys. 227(22), 9572-9595, 2008.
       
    66. P. Prempraneerach, C. Chryssostomidis, M.S. Triantafyllou, and G.E. Karniadakis, "Stochastic modeling of integrated power system coupled to hydrodynamics in the all-electric ship," International Symposium on Power Electronics, Electrical Drives, Automation and Motion, June 2008.
       
    67. P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, C. Chryssostomidis and G.E. Karniadakis, "Sensitivity analysis and low-dimensional stochastic modeling of shipboard integrated power systems," 39th Power Electronics Specialists Conference, June 2008.
       
    68. X. Wan and G.E. Karniadakis, "Recent advances in polynomial chaos methods and extensions," NATO/OTAN, Paper Reference Number: RTO-MP-IST-999, Computational Uncertainty in Military Vehicle Design Meeting Proceedings 2008.
       
    69. P. Prempraneerach, J. Foo, M.S. Triantafyllou, C. Chryssostomidis and G.E. Karniadakis, "Gradient-free stochastic sensitivity analysis of the shipboard power system, GCMS'08, Edinburgh, Scotland, June 16-19 2008.
       
    70. D. Venturi, X. Wan and G.E. Karniadakis ''Stochastic low-dimensional modeling of random laminar wake past a circular cylinder,'' J. Fluid Mech. J. Fluid Mech. 606, 339-367, 2008.
       
    71. P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, T.J. McCoy, C. Chryssostomidis and G.E. Karniadakis, "Stochastic sensitivity methods and application to the shipboard power system," submitted to IEEE Transaction on Power system Engineering, April 2007 with 1st revision in November 2007.
       
    72. P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, T.J. McCoy, C. Chryssostomidis and G.E. Karniadakis, "Sensitivity analysis of the shipboard integrated power system," Naval Engineering Journal, 119(2), 2007.
       
    73. G. Lin, C.-H. Su and G.E. Karniadakis ''Random roughness enhances lift in supersonic flow,'' Phy. Rev. Letts. 99(10), 104501-1 - 104501-4, 2007. 
       
    74. J. Foo, Z. Yosibash and G.E. Karniadakis, "Stochastic simulation of riser-sections with uncertain measured pressure loads and/or uncertain material properties," Comput. Methods Appl. Mech. Engr., 196, 4250-4271, 2007.
       
    75. G. Lin, X. Wan, C.-H. Su and G.E. Karniadakis ''Stochastic computational fluid mechanics,'' IEEE Computing in Sci. Engin. 9(2) 21-29, March-April 2007.
       
    76. P. Prempraneerach, F.S. Hover, M.S. Triantafyllou and G.E. Karniadakis, "Uncertainty quantification in simulations of power systems: Multi-element polynomial chaos methods," submitted to International Journal of Electrical Power and Energy Systems, November 2007.
       
    77. P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, T.J. McCoy, C. Chryssostomidis and G.E. Karniadakis, "Sensitivity analysis of the shipboard integrated power system," ASNE Proceedings on Fuel Tank to Target, Building the Electric Fighting Ship, June 2007.
       
    78. P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, C. Chryssostomidis and G.E. Karniadakis, "Effects of Uncertain Propulsion Loads on the Electric Ship Power Bus," Electric Ship Research and Development consortium (ESRDC), MIT, May 2007.
       
    79. G. Lin, L. Grinberg and G.E. Karniadakis ''Numerical studies of the stochastic Korteweg de Vries equation,'' J. Comp. Phys., 213(2), 676-703, 2006.
       
    80. X. Wan and G.E. Karniadakis "A sharp error estimate for the fast Gauss transform," J. Comp. Phys. 219(1), 7-12, 2006.
       
    81. H. Gunes, S. Sirsisup and G.E. Karniadakis, "Gappy data: To Krig or not to Krig?," J. Comp. Phys., 212(1), 358-382, 2006.
       
    82. G. Lin, C.-H. Su and G.E. Karniadakis, ''Modeling uncertainties in supersonic flow past a wedge,'' 44th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 9-12, 2006.
       
    83. G. Lin, C.-H. Su and G.E. Karniadakis, ''Predicting shock dynamics in the presence of uncertainties,'' J. Comp. Phys. 217(1), 1-4, 2006.
       
    84. X. Wan and G.E. Karniadakis, ''Stochastic heat transfer enhancement in a grooved channel,'' J. Fluid Mech., 565, 255-278, 2006.
       
    85. X. Wan and G.E. Karniadakis, ''Beyond Wiener-Askey expansions: Handling arbitrary PDFs,'' J. Sci. Comp., 27(1-3), 455-464, 2006.
       
    86. X. Wan and G.E. Karniadakis, ''Multi-element generalized polynomial chaos for arbitrary probability measures,'' SIAM J. Sci. Comp., 28(3), 901-928, 2006.
       
    87. X. Wan and G.E. Karniadakis, ''Long-term behavior of polynomial chaos in stochastic flow simulations,'' Comput. Methods Appl. Mech. Engrg., 195(41-43), 5582-5596, 2006.
       
    88. X. Wan and G. Karniadakis, ''Simulation of heat transfer with uncertainty,'' Proceedings of HT2005, 2005 ASME Summer Heat Transfer Conference, San Francisco, CA, July 17-22 2005.
       
    89. X. Wan and G.E. Karniadakis, "An adaptive multi-element generalized polynomial chaos methods for stochastic differential equations," J. Comp. Phys. 209(2), 617-642, 2005.
       
    90. G. Lin, C.-H. Su and G.E. Karniadakis, ''Stochastic solvers for the Euler equations,'' 43rd Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 10-13, 2005.
       
    91. G. Lin, C.-H. Su and G.E. Karniadakis, ''The stochastic piston problem,'' Proc. National Academy of Sciences, 101, pp. 15840-15845, 2004.
       
    92. D. Lucor, C.-H. Su and G.E. Karniadakis, ''Generalized polynomial chaos and random oscillators,'' Int. J. Numer. Meth. Eng., 60(3), 571-596, 2004.
       
    93. D. Lucor, C.-H. Su and G.E. Karniadakis, ''Karhunen-Loeve representation of periodic second-order autoregressive processes,'' International Conference on Computational Science, June 6-9 2004, Krakow, Poland.
       
    94. D. Lucor and G.E. Karniadakis, ''Adaptive generalized polynomial chaos for nonlinear random oscillators,'' SIAM J. Sci. Comp., 26(2), 720-735, 2004.
       
    95. X. Wan, D. Xiu and G.E. Karniadakis, ''Modeling uncertainty in three-dimensional heat transfer problems,'' Heat Transfer 2004, March 24-26 2004, Lisbon, Portugal.
       
    96. D. Xiu and G.E. Karniadakis, ''Generalized Polynomial Chaos: Performance Evaluation and Applications,'' DDDAS/Kluwer 2004.
       
    97. D. Lucor and G.E. Karniadakis, ''Noisy inflows cause shedding-mode switching in flow past an oscillating cylinder,'' Physical Review Letters, 92(15), 154501-1 - 154501-4 2004.
       
    98. D. Lucor and G.E. Karniadakis, ''Predictability and uncertainty in flow-structure interactions,'' European Journal of Mechanics/B Fluids, Vol. 23 pp. 41-49 (2004) Special Issue: Proc. Conference on Bluff Body Wakes and Vortex-Induced Vibrations (BBVIV3; invited paper), 17-20 December 2002, Queensland, Australia, Edited by T. Leweke, K. Hourigan and C.H.K. Williamson.
       
    99. D. Xiu and G.E. Karniadakis, ''Supersensitivity due to uncertain boundary conditions,'' Int. J. Num. Meth. Eng., 61, 2114-2138, 2004.
       
    100. D. Venturi and G.E. Karniadakis, "Gappy data and reconstruction procedures for flow past a cylinder," J. Fluid Mech., 519, 315-336.
       
    101. X. Wan, D. Xiu and G.E. Karniadakis, ''Stochastic solutions for the two-dimensional advection-diffusion equation,'' SIAM J. Sci. Comp., 26(2),  578-590, 2004.
       
    102. D. Xiu and G.E. Karniadakis, ''Uncertainty modeling of Burgers' Equation by generalized polynomial chaos,'' Computational Stochastic Mechanics, Proceedings of the 4th International Conference on Computational Stochastic Mechanics, Corfu, Greece, June 2002. Edited by P.D. Spanos and G. Deodatis, pp. 655-661, Millpress Rotterdam, 2003.
       
    103. D. Lucor, D. Xiu, C.-H. Su and G.E. Karniadakis, ''Predictability and Uncertainty in CFD,'' Int. J. Num. Meth. Fluids, 43(5), 485-505, 2003.
       
    104. D. Xiu and G.E. Karniadakis, "A new stochastic approach to transient heat conduction modeling with uncertainty," Int. J. Heat & Mass Transfer, 46, 4681-4693, 2003.
       
    105. D. Xiu and G.E. Karniadakis, ''Modeling uncertainty in flow simulations via generalized polynomial chaos,'' J. Comp. Phys., 187, 137-167, 2003.
       
    106. D. Xiu and G.E. Karniadakis, ''Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, ''Computer Methods in Applied Mechanics and Engineering, 191, 4927-4948, 2002.
       
    107. G.E. Karniadakis, ''Quantifying Uncertainty in CFD,'' Editor of special issue of J. Fluids Engineering, March 2002.
       
    108. D. Xiu and G.E. Karniadakis, ''The Wiener-Askey polynomial chaos for stochastic differential equations,'' SIAM J. Sci. Comput., 24(2), 619-644, 2002.
       
    109. D. Xiu, D. Lucor, C.-H. Su and G.E. Karniadakis, ''Stochastic modeling of flow-structure interactions using generalized polynomial chaos,'' J. Fluids Engin., 124, 51-59, 2002.
       
    110. D. Xiu and G.E. Karniadakis, ''Modeling uncertainty of elliptic partial differential equations via generalized polynomial chaos,'' Proceedings of the 5th ASCE Engineering Mechanics Division Conference, Columbia University, New York City, June 2002.
       
    111. M. Jardak, C.-H. Su and G.E. Karniadakis, ''Spectral polynomial chaos solutions of the stochastic advection equation,'' J. Sci. Comp., 17(1-4), 319-338, 2002.
       
    112. D. Xiu, D. Lucor and G.E. Karniadakis, ''Modeling uncertainty in flow-structure interactions,'' Computational Fluid and Solid Mechanics, Proc. First MIT Conference, Cambridge, Massachusetts, USA, June 12-15, 2001, edited K.J. Bathe, 2, 1420-1423, 2001.
       
    113. G.E. Karniadakis, ''Towards a numerical error bar in CFD,'' Editorial Article, J. Fluids Engineering, March 1995.