Fractional and Stochastic PDEs/Uncertainty Quantification

F. Zeng, Z. Zhang, G. E. Karniadakis, "Secondorder numerical methods for multiterm fractional differential equations: Smooth and nonsmooth solutions." Comput. Methods Appl. Mech. Engrg. (2017), http://dx.doi.org/10.1016/j.cma.2017.08.029.

F. Song, C. Xu, G. E. Karniadakis, "Computing fractional laplacians on complexgeometry domains: algorithms and simulations." SIAM J. Sci. Comput. 39(4),A1320A1344.

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

E. Kharazmi, M. Zayernouri, G. E. Karniadakis, "A PetrovGalerkin spectral element method for fractional elliptic problems." Comput. Methods Appl. Mech. Engrg. 324, 512536, 2017

S. Lee, I. G. Kevrekidis, G. E. Karniadakis, "A general CFD framework for faultresilient simulations based on multiresolution information fusion." J. Comp. Physics, 347; 290304, 2017.

F. Song, G. E. Karniadakis, "Fractional spectral vanishing viscosity method: Application to the quasigeostrophic equation." Chaos Solitons and Fractals, 102; 327352, 2017

M. Raissi, P. Perdikaris, and G. E. Karniadakis, "Numerical Gaussian processes for timedependent and nonlinear partial differential equations." SIAM J. Sci. Comput (under Review), 2017.

M. Raissi, P. Perdikaris, and G. E. Karniadakis, "Machine learning of linear differential equations using Gaussian processes." P.R. Soc. A, (Under review), 2017.

P. Perdikaris, M. Raissi, A. Damianou, N. D. Lawrence, and G. E. Karniadakis, "Nonlinear information fusion algorithms for dataefficient multifidelity modeling." P. R. Soc. A. 473(2198), 2017.

L. Bonfiglio, P. Perdikaris, S. Brizzolara, and G. E. Karniadakis. "A multifidelity framework for investigating the performance of supercavitating hydrofoils under uncertain flow conditions." AIAA NonDeterministic Approaches, 2017.

S. Lee, I.G. Kevrekidis, G. E. Karniadakis, "A resilient and efficient CFD framework: Statistical learning tools for multifidelity and heterogeneous information fusion." J. Comput. Phys. 2017, http://dx.doi.org/10.1016/j.jcp.2017.05.021

F. Zeng, Z. Mao, G.E. Karniadakis, "A generalized spectral collocation method with tunable accuracy for fractional differential equations with endpoint singularities." SIAM J. Sci. Comput. 2017, 39(1);A360A383.

F. Sang, F. Zeng, W. Cai, W. Chen, G. E. Karniadakis, "Efficient twodimensional simulations of the fractional Szabo equation with different timestepping schemes." Computers and Mathematics with Applications, 2017, 73; 12861297.

X. Chen, F. Zeng, G.E. Karniadakis, "A tunable finite difference method for fractional differential equations with nonsmooth solutions." Comput. Methods Appl. Mech. Engrg. 2017, 318;193214.

Z. Mao, G.E. Karniadakis, "Fractional Burgers equation with nonlinear nonlocality: Spectral vanishing viscosity and local discontinuous Galerkin methods." J. Comput. Phys. 2017, 336;143163.

L. Parussini, D. Venturi, P. Perdikaris, G.E. Karniadakis, "Multifidelity Gaussian process regression for prediction of random fields". J. Comput. Phys. 2017, 336; 3650.

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

M. Raissi, P. Perdikaris, G.E. Karniadakis, "Inferring solution of differential equations using noisy multifidelity data". J. Comp. Physics, 2017. http://dx.doi.org/10.1016/j.jcp.2017.01.060

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

C. Kim, O.Borodin and G.E. Karniadakis, "Quantification of sampling uncertainty for molecular dynamics simulation: Timedependent diffusion coefficient in simple fluids", J. Comp. Phys., 302,485508,2016

P. Perdikaris, D. Venturi and G.E. Karniadakis, "Multifidelity information fusion algorithms for highdimensional systems and massive data sets", SIAM J. Sci. Comput., 38(4), B521538, 2016

P.Perdikaris and G.E. Karniadakis, "Model inversion via multifidelity Bayesian optimization: a new paradigm for parameter estimation in haemodynamics, and beyond", Journal of The Royal Society Interface, 13.118, 2016, 20151107

J. Suzuki, M. Zayernouri and G.E. Karniadakis, "Fractionalorder uniaxial viscoelastoplastic models for structural analysis", Computer Methods in Applied Mechanics and Engineering, 308, 443467, 2016

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

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, 6189 2016

F. Song, C. Xu, G.E.Karniadakis, "A fractional phasefield model for twophase flows with tunable sharpness:Algorithms and simulations," Comput. Methods Appl. Mech. Engrg. 305, 376404 2016.

F. Zeng, Z. Zhang and G.E. Karniadakis, "Fast difference schemes for solving highdimensional timefractional subdiffusion equations", J. Comput. Phys, 307, 1533, 2016

P. Perdikaris, D. Venturi, J. Royset, and G. E. Karniadakis, "Multifidelity modelling via recursive cokriging and GaussianMarkov random fields." P. R. Soc. A., vol 471, 2017.

M. Zayernouri, M. Ainsworth and G.E. Karniadakis, "Tempered Fractional SturmLiouville eigenproblems", SIAM J. Sci. Comput., 37(4) A1777A1800, 2015

H. Cho, Y.Yang, D.Venturi and G.E. Karniadakis, "Algorithms for propagating uncertainty across heterogeneous domains", SIAM J. Sci. Comput. 37(6), A3030A3064, 2015

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 10.1007/9789400761780_1009541.
 G.E. Karniadakis, J.S. Hesthaven, I. Podlubny, "Special Issue on "Fractional PDEs: Theory, Numerics, and Applications," J. Comput. Phys. 293, 13 2015.
 M. Zayernouri and G.E. Karnidakis, "Fractional spectral collocation methods for linear and nonlinar variable order FPDEs," J. Comput. Phys. 293, 312338, 2015.
 W. Cao, Z. Zhang and G.E. Karniadakis, "Numerical methods for stochastic delay differential equations via the WongZakai approximation," SIAM J. Sci. Comput, 37(1), A295A318, 2015.
 B. Yildirim and G.E. Karniadakis, "Stochastic simulations of ocean waves: An uncertainty quantification study," Ocean Modelling, 86, 1535, 2015.
 M. Zayernouri, M. Ainsworth and G.E. Karnidakis, "A unified PetrovGalerkin spectral method for fractiona PDEs," Computer Methods in Applied Mechanics and Engineering, 283, 15451569, DOI/10.1016/j.cma.2014.10.051, 2015.
 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), A1652A1677, 2014.
 M. Zayernouri and G.E. Karniadais, "Discontinuous spectral element methods for timeand spacefractional advection equations," SIAM J. Sci. Comput. 36(4), B684B707, 2014.
 D. Venturi and G.E. Karniadakis, "Convolutionless NakajimaZwanzig equations for stochastic analysis in nonlinear dynamical systems," Proc. Royal Soc. A, 470, 20130754, 2014.
 M. Zayernouri and G.E. Karnidakis, "Exponentially accurate spectral and spectral element methods for fractional ODEs," J. Comput. Phys., 247, Part A, 460480, 2014.
 M. Zayernouri and G.E. Karniadakis, "Fractional spectral collocation methods," SIAM J. Sci. Comput. 36(1), A40A62, 2014.
 M.V. Tretyakov and Z. Zhang, "A fundamental meansquare convergence theorem for SDEs with locally Lipschitz coefficients and its applications," SIAM J. Numer. Anal., 51(6), 31353162, 2013.
 M. Zayernouri and G.E. Karniadakis, "Fractional SturmLiouville eigenproblems: Theory and numerical approximation," J. Comput. Phys., 252, 495517, 2013.
 H. Cho, D. Venturi and G.E. Karniadakis, "KarhunenLoeve expansion for multicorrelated stochastic processes," Probabilistic Engineering Mechanics, 34, 157167, 2013.
 H. Cho, D. Venturi and G.E. Karniadakis, "Adaptive discontinuous Galerkin method for responseexcitation PDF equations," SIAM J. Sci. Comput. 35(4), B890B911, 2013.
 W. Cao and Z. Zhang, "On exponential meansquare stability of twostep Maruyama methods for stochastic delay differential equations," J. Comput. Appl. Math., 244, 182193, 2013.
 C. Kim and G.E. Karniadakis, "Microscopic theory of Brownian motion revised: The Rayleigh model," Phy. Review E, 87(3), 032129, 2013.
 X. Yang and G.E. Karniadakis,"Reweighted l_{1 }minimization method for stochastic elliptic differential equations," J. Comp. Phys., 248, 87108, 2013.
 Venturi, D.M. Tartakovsky, A.M. Tartakovsky and G.E. Karniadakis, "Exact PDF equations and closure approximations for advectivereactive transport," J. Comp. Phys., 243, 323343, 2013.
 M. Zayernouri, SW. 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, 99112, 2013.
  M. Choi, T. Sapsis and G.E. Karniadakis, "A convergence study for SPDEs using combined Polynomial Chaos and DynamicallyOrthogonal schemes," J. Comp. Phys., 245, 281301, 2013.
 Z. Zhang, X. Yang, G. Lin and G.E. Karniadakis, "Numerical solution of the Stratonovich and ItoEuler equations: Application to the stochastic piston problem," J. Comp. Phys. 236, 1527, 2013.
 W. Cao and Z. Zhang, "Simulations of twostep Maruyama methods for nonlinear stochastic delay differential equations," Adv. Appl. Math. Mech., 4(6), 821832, 2012.
 X. Yang, M. Choi, G. Lin and G.E. Karniadakis, "Adaptive ANOV decomposition of stochastic incompressible and compressible flows," J. Comp. Phys., 231, 15871614, 2012.
 Z. Zhang, B. Rosvoskii, M.V. Tretyakov and G.E. Karniadakis, "A multistage Wiener chaos expansion method for stochastic advectiondiffusionreaction equations," SIAM J. Sci. Comput., 34(2), A914A936, 2012.
 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), A1165A1186, 2012.
 D. Venturi, M. Choi and G.E. Karniadakis, Supercritical quasicontinuum states in stochastic RayleighBernard convection," Int. J. Heat & Mass Transfer, 55, 37323743, 2012.
 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), 195213, 2012.
 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), 759783, 2012.
 D. Venturi and G.E. Karniadakis, "New evolution equations for the joint responseexcitation probability density function of stochastic solutions to firstorder nonlinear PDEs," J. Comp. Phys., 231, 74507474, 2012.
 Z. Zhang, B. Rosovskii, M.V. Tretyakov and G.E. Karniadakis, "A multistage Wiener chaos expansion method for stochastic advectiondiffusionreaction equations," SIAM J. Sci. Comput. 34(2), A914A936, 2012.
 J. Foo and G.E. Karniadakis, "Multielement probabilistic collocation method in high dimensions," J. Comp. Phys., 229(5), 15361577, 2010.
 D. Venturi, X. Wan and G.E. Karniadakis "Stochastic bifurcation analysis of RayleighBenard Convection," J. Fluid Mech., 650, 391413, 2010.
 M. Gerritsma, JB. van der Steen, P. Vos and G.E. Karniadakis, "Timedependent generalized polynomial chaos," J. Comput. Phys., 229, 83338363, 2010.
 P. Prempraneerach, F.S. Hover, M.S. Triantafyllou and G.E. Karniadakis, "Uncertainty quantification in simulations of power systems: Multielement polynomial chaos methods," Reliability Engineering and System Safety", 95, 632646, 2010.
 Z. Zhang, M. Choi and G.E. Karniadakis, "Anchor points matter in ANOVA decomposition," ICOSAHOM09.
 G. Lin and G.E. Karniadakis, "Sensitivity analysis and stochastic simulations of nonequilibrium plasma flow," Int. J. Numer. Meth. Engng., 80, 738766, published online DOI.10.1002/nme.2582, 2009.
 J. Foo, S. Sindi and G.E. Karniadakis, "Multielement probabilistic collocation for sensitivity analysis in cellular signaling networks," IET Systems Biology, 3(4), 239254, DOI.10.1049/ietsub.2009.0126, 2009.
 X. Wan. B. Rozovskii and G.E. Karniadakis, "A stochastic modeling method based on weighted Wiener chaos and Malliavan calculus," PNAS, 106, 1418914194, DOI: 10.1073/pnas.0902348106, 2009.
 X. Wan and G.E. Karniadakis, "Solving elliptic problems with nonGaussian spatiallydependent random coefficients." Comput. Methods Appl. Mech. Engrg., 198, 19851995, 2009.
 X. Wan and G.E. Karniadakis, "Error control in multielement generalized polynomial chaos methods for elliptic problems with random coefficient," Communication in Computational Physics, 4, 793820, 2009.
 G. Lin, CH. Su and G.E. Karniadakis, "Stochastic modeling of random roughness in shock scattering problems: Theory and simulations," Comput. Methods Appl. Mech. Engrg. 197, 34203334, 2008.
 J. Foo, X. Wan and G.E. Karniadakis, "The multielement probabilistic collocation methods (MEPCM): Error analysis and applications," J. Comp. Phys. 227(22), 95729595, 2008.
 P. Prempraneerach, C. Chryssostomidis, M.S. Triantafyllou, and G.E. Karniadakis, "Stochastic modeling of integrated power system coupled to hydrodynamics in the allelectric ship," International Symposium on Power Electronics, Electrical Drives, Automation and Motion, June 2008.
 P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, C. Chryssostomidis and G.E. Karniadakis, "Sensitivity analysis and lowdimensional stochastic modeling of shipboard integrated power systems," 39th Power Electronics Specialists Conference, June 2008.
 X. Wan and G.E. Karniadakis, "Recent advances in polynomial chaos methods and extensions," NATO/OTAN, Paper Reference Number: RTOMPIST999, Computational Uncertainty in Military Vehicle Design Meeting Proceedings 2008.
 P. Prempraneerach, J. Foo, M.S. Triantafyllou, C. Chryssostomidis and G.E. Karniadakis, "Gradientfree stochastic sensitivity analysis of the shipboard power system, GCMS'08, Edinburgh, Scotland, June 1619 2008.
 D. Venturi, X. Wan and G.E. Karniadakis ''Stochastic lowdimensional modeling of random laminar wake past a circular cylinder,'' J. Fluid Mech. J. Fluid Mech. 606, 339367, 2008.
 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.
 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.
 G. Lin, C.H. Su and G.E. Karniadakis ''Random roughness enhances lift in supersonic flow,'' Phy. Rev. Letts. 99(10), 1045011  1045014, 2007.
 J. Foo, Z. Yosibash and G.E. Karniadakis, "Stochastic simulation of risersections with uncertain measured pressure loads and/or uncertain material properties," Comput. Methods Appl. Mech. Engr., 196, 42504271, 2007.
 G. Lin, X. Wan, C.H. Su and G.E. Karniadakis ''Stochastic computational fluid mechanics,'' IEEE Computing in Sci. Engin. 9(2) 2129, MarchApril 2007.
 P. Prempraneerach, F.S. Hover, M.S. Triantafyllou and G.E. Karniadakis, "Uncertainty quantification in simulations of power systems: Multielement polynomial chaos methods," submitted to International Journal of Electrical Power and Energy Systems, November 2007.
 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.
 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.
 G. Lin, L. Grinberg and G.E. Karniadakis ''Numerical studies of the stochastic Korteweg de Vries equation,'' J. Comp. Phys., 213(2), 676703, 2006.
 X. Wan and G.E. Karniadakis "A sharp error estimate for the fast Gauss transform," J. Comp. Phys. 219(1), 712, 2006.
 H. Gunes, S. Sirsisup and G.E. Karniadakis, "Gappy data: To Krig or not to Krig?," J. Comp. Phys., 212(1), 358382, 2006.
 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 912, 2006.
 G. Lin, C.H. Su and G.E. Karniadakis, ''Predicting shock dynamics in the presence of uncertainties,'' J. Comp. Phys. 217(1), 14, 2006.
 X. Wan and G.E. Karniadakis, ''Stochastic heat transfer enhancement in a grooved channel,'' J. Fluid Mech., 565, 255278, 2006.
 X. Wan and G.E. Karniadakis, ''Beyond WienerAskey expansions: Handling arbitrary PDFs,'' J. Sci. Comp., 27(13), 455464, 2006.
 X. Wan and G.E. Karniadakis, ''Multielement generalized polynomial chaos for arbitrary probability measures,'' SIAM J. Sci. Comp., 28(3), 901928, 2006.
 X. Wan and G.E. Karniadakis, ''Longterm behavior of polynomial chaos in stochastic flow simulations,'' Comput. Methods Appl. Mech. Engrg., 195(4143), 55825596, 2006.
 X. Wan and G. Karniadakis, ''Simulation of heat transfer with uncertainty,'' Proceedings of HT2005, 2005 ASME Summer Heat Transfer Conference, San Francisco, CA, July 1722 2005.
 X. Wan and G.E. Karniadakis, "An adaptive multielement generalized polynomial chaos methods for stochastic differential equations," J. Comp. Phys. 209(2), 617642, 2005.
 G. Lin, C.H. Su and G.E. Karniadakis, ''Stochastic solvers for the Euler equations,'' 43rd Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 1013, 2005.
 G. Lin, C.H. Su and G.E. Karniadakis, ''The stochastic piston problem,'' Proc. National Academy of Sciences, 101, pp. 1584015845, 2004.
 D. Lucor, C.H. Su and G.E. Karniadakis, ''Generalized polynomial chaos and random oscillators,'' Int. J. Numer. Meth. Eng., 60(3), 571596, 2004.
 D. Lucor, C.H. Su and G.E. Karniadakis, ''KarhunenLoeve representation of periodic secondorder autoregressive processes,'' International Conference on Computational Science, June 69 2004, Krakow, Poland.
 D. Lucor and G.E. Karniadakis, ''Adaptive generalized polynomial chaos for nonlinear random oscillators,'' SIAM J. Sci. Comp., 26(2), 720735, 2004.
 X. Wan, D. Xiu and G.E. Karniadakis, ''Modeling uncertainty in threedimensional heat transfer problems,'' Heat Transfer 2004, March 2426 2004, Lisbon, Portugal.
 D. Xiu and G.E. Karniadakis, ''Generalized Polynomial Chaos: Performance Evaluation and Applications,'' DDDAS/Kluwer 2004.
 D. Lucor and G.E. Karniadakis, ''Noisy inflows cause sheddingmode switching in flow past an oscillating cylinder,'' Physical Review Letters, 92(15), 1545011  1545014 2004.
 D. Lucor and G.E. Karniadakis, ''Predictability and uncertainty in flowstructure interactions,'' European Journal of Mechanics/B Fluids, Vol. 23 pp. 4149 (2004) Special Issue: Proc. Conference on Bluff Body Wakes and VortexInduced Vibrations (BBVIV3; invited paper), 1720 December 2002, Queensland, Australia, Edited by T. Leweke, K. Hourigan and C.H.K. Williamson.
 D. Xiu and G.E. Karniadakis, ''Supersensitivity due to uncertain boundary conditions,'' Int. J. Num. Meth. Eng., 61, 21142138, 2004.
 D. Venturi and G.E. Karniadakis, "Gappy data and reconstruction procedures for flow past a cylinder," J. Fluid Mech., 519, 315336.
 X. Wan, D. Xiu and G.E. Karniadakis, ''Stochastic solutions for the twodimensional advectiondiffusion equation,'' SIAM J. Sci. Comp., 26(2), 578590, 2004.
 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. 655661, Millpress Rotterdam, 2003.
 D. Lucor, D. Xiu, C.H. Su and G.E. Karniadakis, ''Predictability and Uncertainty in CFD,'' Int. J. Num. Meth. Fluids, 43(5), 485505, 2003.
 D. Xiu and G.E. Karniadakis, "A new stochastic approach to transient heat conduction modeling with uncertainty," Int. J. Heat & Mass Transfer, 46, 46814693, 2003.
 D. Xiu and G.E. Karniadakis, ''Modeling uncertainty in flow simulations via generalized polynomial chaos,'' J. Comp. Phys., 187, 137167, 2003.
 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, 49274948, 2002.
 G.E. Karniadakis, ''Quantifying Uncertainty in CFD,'' Editor of special issue of J. Fluids Engineering, March 2002.
 D. Xiu and G.E. Karniadakis, ''The WienerAskey polynomial chaos for stochastic differential equations,'' SIAM J. Sci. Comput., 24(2), 619644, 2002.
 D. Xiu, D. Lucor, C.H. Su and G.E. Karniadakis, ''Stochastic modeling of flowstructure interactions using generalized polynomial chaos,'' J. Fluids Engin., 124, 5159, 2002.
 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.
 M. Jardak, C.H. Su and G.E. Karniadakis, ''Spectral polynomial chaos solutions of the stochastic advection equation,'' J. Sci. Comp., 17(14), 319338, 2002.
 D. Xiu, D. Lucor and G.E. Karniadakis, ''Modeling uncertainty in flowstructure interactions,'' Computational Fluid and Solid Mechanics, Proc. First MIT Conference, Cambridge, Massachusetts, USA, June 1215, 2001, edited K.J. Bathe, 2, 14201423, 2001.
 G.E. Karniadakis, ''Towards a numerical error bar in CFD,'' Editorial Article, J. Fluids Engineering, March 1995.