Fractional and Stochastic PDEs/Uncertainty Quantification

Fractional and Stochastic PDEs/Uncertainty Quantification

  1. I. C. Tsantili, M. H. Cho, W. Cai, G. E. Karniadakis, "A computational stochastic methodology for the design of random meta-materials under geometric constraints." SIAM J. Sci. Comput. 40(2), B353-B378, 2018.
  2. A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, G. E. Karniadakis, "What is the fractional Laplacian?"  https://arxiv.org/pdf/1801.09767.pdf, 2018.
  3. Z. Zhang, W. Deng, G.E. Karniadakis, "A Riesz basis Galerkin method for the tempered fractional Laplacian." SIAM Journal on Numerical Analysis, to appear, 2018.
  4. Z. Mao, G. E. Karniadakis, "A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative." SIAM J. Numer. Anal. 56(1), 24-49, 2018.
  5. 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. 
  6. 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, 2017.
  7. 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. 348, 694-714, 2017. 
  8. 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.
  9. A. Lischke, M. Zayermouri, G. E. Karniadakis, "A Petrov-Galerkin spectral method of linear complexity for fractional multi-term ODEs on the half line." SIAM J. Sci. Comput. 39(3), A922-946, 2017.
  10. F. Song, G. E. Karniadakis, "Fractional spectral vanishing viscosity method: Application to the quasi-geostrophic equation." Chaos Solitons and Fractals, 102; 327-352, 2017.
  11. L. Bonfiglio, P. Perdikaris, S. Brizzolara, G. E. Karniadakis. "A multi-fidelity framework for investigating the performance of super-cavitating hydrofoils under uncertain flow conditions." AIAA Non-Deterministic Approaches, 2017.
  12. 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. 39(1), A360-A383, 2017.
  13. 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, 73, 1286-1297, 2017.
  14. 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. 318, 193-214, 2017.
  15. Z. Mao, G. E. Karniadakis, "Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods." J. Comput. Phys. 336, 143-163, 2017.
  16. L. Parussini, D. Venturi, P. Perdikaris, G. E. Karniadakis, "Multi-fidelity Gaussian process regression for prediction of random fields." J. Comput. Phys. 336; 36-50, 2017. 
  17. C. Kim, O. Borodin, 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.
  18. W. Cao, Z. Zhang, G. E. Karniadakis, "Implicit-explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions." SIAM J. Sci. Comput. 38(5), A3070-3093, 2016.
  19. J. Suzuki, M. Zayernouri, G. E. Karniadakis, "Fractional-order uniaxial visco-elasto-plastic models for structural analysis." Comp. Methods Applied Mech. and Engin. 308, 443-467, 2016.
  20. Y. Yu, P. Perdikaris, G. E. Karniadakis, "Fractional Modeling of Viscoelasticity in 3D Cerebral Arteries and Aneurysms." J. Comput. Phys. 323, 219-242, 2016. 
  21. 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
  22. 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.
  23. F. Zeng, Z. Zhang, G. E. Karniadakis, "Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations." J. Comput. Phys. 307, 15-33, 2016.
  24. H. Cho, D. Venturi and G.E. Karniadakis, “Numerical methods for high-dimensional probability density function equations.” J. Comput. Phys. 305, 817-837, 2016.
  25. P. Perdikaris, D. Venturi, J. Royset, G. E. Karniadakis, "Multi-fidelity modelling via recursive co-kriging and Gaussian-Markov random fields." P. R. Soc. A. 471, 2017.
  26. M. Zayernouri, M. Ainsworth, G. E. Karniadakis,  "Tempered Fractional Sturm-Liouville eigen-problems." SIAM J. Sci. Comput. 37(4), A1777-A1800, 2015.
  27. H. Cho, Y. Yang, D. Venturi, G. E. Karniadakis, "Algorithms for propagating uncertainty across heterogeneous domains." SIAM J. Sci. Comput. 37(6), A3030-A3064, 2015.
  28. X. Zhao, Z.Z. Sun, and G.E. Karniadakis, “Second-order approximations for variable order fractional derivatives: Algorithms and applications.” J. Comput. Physics, 293, 184-200, 2015.
  29. F. Zeng, Z. Zhang and G.E. Karniadakis, “A generalized spectral collocation method with turnable accuracy for variable-order fractional differential equations.” SIAM J. Sci Comput. 37(6), A2710-A2732, 2015.
  30. Z. Zhang, F. Zeng and G.E. Karniadakis “Optimal error estimates of spectral Petrov-Galerkin and collocation methods for initial value problems of fractional differential equations.” SIAM J. Numer. Anal. 53(4) 2074-2096, 2015.
  31. 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.
  32. G. E. Karniadakis, J. S. Hesthaven, I. Podlubny, "Special Issue on "Fractional PDEs: Theory, Numerics, and Applications." J. Comput. Phys. 293, 1-3, 2015. 
  33. M. Zayernouri, G. E. Karniadakis, "Fractional spectral collocation methods for linear and nonlinar variable order FPDEs." J. Comput. Phys. 293, 312-338, 2015. 
  34. W. Cao, Z. Zhang and G.E. Karniadakis “Time-splitting schemes for fractional differential equations I: Smooth solu­tions.” SIAM J. Sci. Comput. 37(4), A1752-A1776, 2015.
  35. Zheng M.* and G.E. Karniadakis. “Numerical methods for SPDEs with tempered stable processes.” SIAM J. Sci. Comp. 37(3): A1197-A1217, 2015.
  36. Zheng, M., Rozovsky, B., & Karniadakis, G. E. “Adaptive Wick–Malliavin Approximation to Nonlinear SPDEs with Discrete Random Variables.” SIAM J. Sci. Comput. 37(4), A1872-A1890, 2015.
  37. W. Cao, Z. Zhang, G. E. Karniadakis, "Numerical methods for stochastic delay differential equations via the Wong-Zakai approximation." SIAM J. Sci. Comput. 37(1), A295-A318, 2015. 
  38. B. Yildirim, G. E. Karniadakis, "Stochastic simulations of ocean waves: An uncertainty quantification study." Ocean Modelling. 86, 15-35, 2015. 
  39. M. Zayernouri, M. Ainsworth, G. E. Karniadakis, "A unified Petrov-Galerkin spectral method for fractiona PDEs." Comput. Methods Appl. Mech. Enging. 283, 1545-1569, 2015. 
  40. Zheng M.*, X. Wan*, and GE Karniadakis. “Adaptive-multi-element polynomial chaos with discrete measure: Algo­rithms and application to SPDEs.” Appl. Numer. Math. 90, 91-110, 2015.
  41. H. Cho, D. Venturi and G.E. Karniadakis, “Statistical analysis and simulation of random shocks in Burgers turbulence.” P. R. Soc. A, 470, 20140080, 2014.
  42. Zhang, Z., X. Yang*, G. Marucci, P. Maffezzoni, I. Abe M. Elfadel, G.E. Karniadakis, and L. Daniel. “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques.” In Custom Integrated Circuits Conference (CICC), 2014 IEEE Proceedings of the, pp. 1-8. IEEE, 2014.
  43. Choi M.*, T. Sapsis, and G.E. Karniadakis, “On the equivalence of dynamically orthogonal and dynamically bi­orthogonal methods: Theory and numerical simulations.” J. Comput. Phys. 270, 1-20, 2014.
  44. M. Zayernouri, W. Cao, Z. Zhang, G. E. Karniadakis, “Spectral and Discontinuous Spectral Element Methods for Fractional Delay Equations.” SIAM J. Sci. Comput., 36 (6), B904-B929, 2014.
  45. Z. Zhang, M. V. Tretyakov, B. Rozovskii, G. E. Karniadakis, "A recursive sparse grid collocation method for differential equations with white noise." SIAM J. Sci. Comput. 36(4), A1652-A1677, 2014. 
  46. M. Zayernouri, G. E. Karniadakis, "Discontinuous spectral element methods for time-and space-fractional advection equations." SIAM J. Sci. Comput. 36(4), B684-B707, 2014. 
  47. D. Venturi, G. E. Karniadakis, "Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems." Proc. R. Soc. A. 470, 20130754, 2014. 
  48. D. Venturi, X. Wan, R. Mikulevicius, B. Rozovskii and G.E. Karniadakis, “Wick-Malliavin approximation to nonlinear stochastic PDEs: Analysis and Simulation.” P. R. Soc. A. 34:157-167, 2014.
  49. M. Zayernouri, G. E. Karniadakis, "Exponentially accurate spectral and spectral element methods for fractional ODEs." J. Comput. Phys. 247, 460-480, 2014. 
  50. M. Zayernouri, G. E. Karniadakis, "Fractional spectral collocation methods." SIAM J. Sci. Comput. 36(1), A40-A62, 2014. 
  51. M. V. Tretyakov, 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. 
  52. M. Zayernouri, G. E. Karniadakis, "Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation." J. Comput. Phys. 252, 495-517, 2013. 
  53. H. Cho, D. Venturi, G. E. Karniadakis, "Karhunen-Loeve expansion for multi-correlated stochastic processes." Probab. Enging. Mech. 34, 157-167, 2013. 
  54. H. Cho, D. Venturi, G. E. Karniadakis, "Adaptive discontinuous Galerkin method for response-excitation PDF equations." SIAM J. Sci. Comp. 35(4), B890-B911, 2013. 
  55. W. Cao, 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. 
  56. C. Kim, G. E. Karniadakis, "Microscopic theory of Brownian motion revised: The Rayleigh model." Phy. Review E. 87(3), 032129, 2013. 
  57. X. Yang, G. E. Karniadakis,"Reweighted lminimization method for stochastic elliptic differential equations." J. Comp. Phys. 248, 87-108, 2013. 
  58. D. Venturi, D. M. Tartakovsky, A. M. Tartakovsky, G. E. Karniadakis, "Exact PDF equations and closure approximations for advective-reactive transport." J. Comp. Phys. 243, 323-343, 2013. 
  59. M. Zayernouri, S. W. Park, D. M. Tartakovsky, G. E. Karniadakis, "Stochastic smoothed profile method for modeling random roughness in flow problems." Comput. Methods Appl. Mech. Engrg. 263, 99-112, 2013. 
  60. M. Choi, T. Sapsis, G. E. Karniadakis, "A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes." J. Comp. Phys. 245, 281-301, 2013. 
  61. Z. Zhang, X. Yang, G. Lin, 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. 
  62. W. Cao, Z. Zhang, "Simulations of two-step Maruyama methods for nonlinear stochastic delay differential equations." Adv. Appl. Math. Mech. 4(6), 821-832, 2012. 
  63. X. Yang, M. Choi, G. Lin, G. E. Karniadakis, "Adaptive ANOV decomposition of stochastic incompressible and compressible flows." J. Comp. Phys. 231, 1587-1614, 2012. 
  64. Z. Zhang, B. Rosvoskii, M. V. Tretyakov, 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. 
  65. Z. Zhang, M. Choi, 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. 
  66. D. Venturi, M. Choi, G. E. Karniadakis, Supercritical quasi-continuum states in stochastic Rayleigh-Bernard convection." Int. J. Heat & Mass Transfer. 55, 3732-3743, 2012. 
  67. D. Venturi, 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. 
  68. D. Venturi, T. Sapsis, G. E. Karniadakis, "A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems." Proc. R. Soc. A. 468(2139), 759-783, 2012. 
  69. D. Venturi, 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. 
  70. J. Foo, G. E. Karniadakis, "Multi-element probabilistic collocation method in high dimensions." J. Comp. Phys. 229(5), 1536-1577, 2010. 
  71. D. Venturi, X. Wan, G. E. Karniadakis "Stochastic bifurcation analysis of Rayleigh-Benard Convection." J. Fluid Mech. 650, 391-413, 2010. 
  72. M. Gerritsma, J. B. van der Steen, P. Vos, G. E. Karniadakis, "Time-dependent generalized polynomial chaos." J. Comput. Phys. 229, 8333-8363, 2010. 
  73. P. Prempraneerach, F. S. Hover, M. S. Triantafyllou, G. E. Karniadakis, "Uncertainty quantification in simulations of power systems: Multi-element polynomial chaos methods." Reliability Enging System Safety. 95, 632-646, 2010. 
  74. Z. Zhang, M. Choi, G. E. Karniadakis, "Anchor points matter in ANOVA decomposition," ICOSAHOM09
  75. G. Lin, G. E. Karniadakis, "Sensitivity analysis and stochastic simulations of non-equilibrium plasma flow," Int. J. Numer. Meth. Engng. 80, 738-766, 2009.  
  76. J. Foo, S. Sindi, G. E. Karniadakis, "Multi-element probabilistic collocation for sensitivity analysis in cellular signaling networks." IET Systems Biology. 3(4), 239-254, 2009. 
  77. X. Wan. B. Rozovski, G. E. Karniadakis, "A stochastic modeling method based on weighted Wiener chaos and Malliavan calculus." PNAS, 106, 14189-14194, 2009. 
  78. X. Wan, G. E. Karniadakis, "Solving elliptic problems with non-Gaussian spatially-dependent random coefficients." Comput. Methods Appl. Mech. Engrg. 198, 1985-1995, 2009. 
  79. X. Wan, G. E. Karniadakis, "Error control in multi-element generalized polynomial chaos methods for elliptic problems with random coefficient." Commun. Comput. Phys. 5, 793-820, 2009. 
  80. G. Lin, C. H. Su, G. E. Karniadakis, "Stochastic modeling of random roughness in shock scattering problems: Theory and simulations." Comput. Methods Appl. Mech. Engrg. 197, 3420-3334, 2008. 
  81. J. Foo, X. Wan, G. E. Karniadakis, "The multi-element probabilistic collocation methods (ME-PCM): Error analysis and applications." J. Comp. Phys. 227(22), 9572-9595, 2008. 
  82. P. Prempraneerach, C. Chryssostomidis, M. S. Triantafyllou, 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. 
  83. P. Prempraneerach, F. S. Hover, M. S. Triantafyllou, C. Chryssostomidis, G. E. Karniadakis, "Sensitivity analysis and low-dimensional stochastic modeling of shipboard integrated power systems." 39th Power Electronics Specialists Conference, June 2008. 
  84. X. Wan, 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. 
  85. P. Prempraneerach, J. Foo, M. S. Triantafyllou, C. Chryssostomidis, G. E. Karniadakis, "Gradient-free stochastic sensitivity analysis of the shipboard power system." GCMS'08, Edinburgh, Scotland, June 16-19 2008. 
  86. D. Venturi, X. Wan, G. E. Karniadakis ''Stochastic low-dimensional modeling of random laminar wake past a circular cylinder.''  J. Fluid Mech. 606, 339-367, 2008. 
  87. P. Prempraneerach, F. S. Hover, M. S. Triantafyllou, T. J. McCoy, C. Chryssostomidis, 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). 
  88. P. Prempraneerach, F. S. Hover, M. S. Triantafyllou, T. J. McCoy, C. Chryssostomidis, G. E. Karniadakis, "Sensitivity analysis of the shipboard integrated power system." Naval Engineering Journal, 119(2), 2007. 
  89. G. Lin, C. H. Su, G. E. Karniadakis ''Random roughness enhances lift in supersonic flow.'' Phy. Rev. Letts. 99(10), 104501-1-104501-4, 2007.  
  90. J. Foo, Z. Yosibash, 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. 
  91. G. Lin, X. Wan, C. H. Su, G. E. Karniadakis ''Stochastic computational fluid mechanics.'' IEEE Computing in Sci. Engin. 9(2) 21-29, Mar-Apr 2007. 
  92. P. Prempraneerach, F. S. Hover, M. S. Triantafyllou, G. E. Karniadakis, "Uncertainty quantification in simulations of power systems: Multi-element polynomial chaos methods." International Journal of Electrical Power and Energy Systems, Nov 2007. 
  93. P. Prempraneerach, F. S. Hover, M. S. Triantafyllou, C. Chryssostomidis, G. E. Karniadakis, "Effects of Uncertain Propulsion Loads on the Electric Ship Power Bus." Electric Ship Research and Development Consortium (ESRDC), MIT, May 2007. 
  94. G. Lin, L. Grinberg, G. E. Karniadakis ''Numerical studies of the stochastic Korteweg de Vries equation.'' J. Comp. Phys. 213(2), 676-703, 2006. 
  95. X. Wan, G. E. Karniadakis "A sharp error estimate for the fast Gauss transform." J. Comp. Phys. 219(1), 7-12, 2006. 
  96. H. Gunes, S. Sirsisup, G. E. Karniadakis, "Gappy data: To Krig or not to Krig?" J. Comp. Phys. 212(1), 358-382, 2006. 
  97. G. Lin, C. H. Su, G. E. Karniadakis, ''Modeling uncertainties in supersonic flow past a wedge.'' 44th Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 9-12, 2006. 
  98. G. Lin, C. H. Su, G. E. Karniadakis, ''Predicting shock dynamics in the presence of uncertainties.'' J. Comp. Phys. 217(1), 1-4, 2006. 
  99. X. Wan, G. E. Karniadakis, ''Stochastic heat transfer enhancement in a grooved channel.'' J. Fluid Mech. 565, 255-278, 2006. 
  100. X. Wan, G. E. Karniadakis, ''Beyond Wiener-Askey expansions: Handling arbitrary PDFs.'' J. Sci. Comp. 27(1-3), 455-464, 2006. 
  101. X. Wan, G. E. Karniadakis, ''Multi-element generalized polynomial chaos for arbitrary probability measures.'' SIAM J. Sci. Comp. 28(3), 901-928, 2006. 
  102. X. Wan, G. E. Karniadakis, ''Long-term behavior of polynomial chaos in stochastic flow simulations.'' Comput. Methods Appl. Mech. Engrg. 195(41-43), 5582-5596, 2006. 
  103. X. Wan, G. E. Karniadakis, ''Simulation of heat transfer with uncertainty.'' Proceedings of HT2005, 2005 ASME Summer Heat Transfer Conference, San Francisco, CA, July 17-22, 2005. 
  104. X. Wan, G. E. Karniadakis, "An adaptive multi-element generalized polynomial chaos methods for stochastic differential equations." J. Comp. Phys. 209(2), 617-642, 2005. 
  105. G. Lin, C. H. Su, G. E. Karniadakis, ''Stochastic solvers for the Euler equations.'' 43rd Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10-13, 2005. 
  106. G. Lin, C. H. Su, G. E. Karniadakis, ''The stochastic piston problem.'' Proc. National Academy of Sciences, 101, 15840-15845, 2004. 
  107. D. Lucor, C. H. Su, G. E. Karniadakis, ''Generalized polynomial chaos and random oscillators.'' Int. J. Numer. Meth. Eng. 60(3), 571-596, 2004. 
  108. D. Lucor, C. H. Su, G. E. Karniadakis, ''Karhunen-Loeve representation of periodic second-order autoregressive processes.'' International Conference on Computational Science, Krakow, Poland, June 6-9, 2004.
  109. D. Lucor, G. E. Karniadakis, ''Adaptive generalized polynomial chaos for nonlinear random oscillators.'' SIAM J. Sci. Comp. 26(2), 720-735, 2004. 
  110. X. Wan, D. Xiu, G. E. Karniadakis, ''Modeling uncertainty in three-dimensional heat transfer problems.'' Heat Transfer 2004, Lisbon, Portugal, March 24-26, 2004
  111. D. Xiu, G. E. Karniadakis, ''Generalized Polynomial Chaos: Performance Evaluation and Applications.'' DDDAS/Kluwer, 2004. 
  112. D. Lucor, 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. 
  113. D. Lucor, G. E. Karniadakis, ''Predictability and uncertainty in flow-structure interactions.'' European Journal of Mechanics/B Fluids, 23, 41-49, 2004. 
  114. D. Xiu, G. E. Karniadakis, ''Supersensitivity due to uncertain boundary conditions.'' Int. J. Num. Meth. Eng. 61, 2114-2138, 2004. 
  115. D. Venturi, G. E. Karniadakis, "Gappy data and reconstruction procedures for flow past a cylinder." J. Fluid Mech. 519, 315-336. 
  116. X. Wan, D. Xiu, G. E. Karniadakis, ''Stochastic solutions for the two-dimensional advection-diffusion equation.'' SIAM J. Sci. Comp. 26(2), 578-590, 2004. 
  117. D. Xiu, 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.
  118. D. Lucor, D. Xiu, C. H. Su, G. E. Karniadakis, ''Predictability and Uncertainty in CFD.'' Int. J. Num. Meth. Fluids, 43(5), 485-505, 2003. 
  119. D. Xiu, G. E. Karniadakis, "A new stochastic approach to transient heat conduction modeling with uncertainty." Int. J. Heat & Mass Transfer, 46, 4681-4693, 2003. 
  120. D. Xiu, G. E. Karniadakis, ''Modeling uncertainty in flow simulations via generalized polynomial chaos.'' J. Comp. Phys. 187, 137-167, 2003. 
  121. D. Xiu, G. E. Karniadakis, ''Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos.'' Comp. Meth. Appl. Mech. Engrg. 191, 4927-4948, 2002. 
  122. G. E. Karniadakis, ''Quantifying Uncertainty in CFD."  J. Fluids Engrg, 124, 2-3, March 2002. 
  123. D. Xiu, G. E. Karniadakis, ''The Wiener-Askey polynomial chaos for stochastic differential equations.'' SIAM J. Sci. Comput. 24(2), 619-644, 2002. 
  124. D. Xiu, D. Lucor, C. H. Su, G. E. Karniadakis, ''Stochastic modeling of flow-structure interactions using generalized polynomial chaos.'' J. Fluids Engin. 124, 51-59, 2002. 
  125. D. Xiu, 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, NY, June 2002. 
  126. M. Jardak, C. H. Su, G. E. Karniadakis, ''Spectral polynomial chaos solutions of the stochastic advection equation.'' J. Sci. Comp. 17(1-4), 319-338, 2002. 
  127. D. Xiu, D. Lucor, G. E. Karniadakis, ''Modeling uncertainty in flow-structure interactions.'' Computational Fluid and Solid Mechanics, Proc. First MIT Conference, Cambridge, MA, June 12-15, 2001.
  128. G. E. Karniadakis, ''Towards a numerical error bar in CFD.''  J. Fluids Engrg, 117(1), 7-9, March 1995.