Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems

Najm, Habib N.; Valorani, Mauro

doi:10.1016/j.jcp.2014.03.061

Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems

Journal Article · Sat Apr 12 04:00:00 EDT 2014 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2014.03.061· OSTI ID:1106517

Najm, Habib N. ^[1]; Valorani, Mauro ^[2]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sapienza Univ. of Rome (Italy)

We explore the relation between the development of a non-negligible probability of negative states and the instability of numerical integration of the intrusive Galerkin ordinary differential equation system describing uncertain chemical ignition. To prevent this instability without resorting to either multi-element local polynomial chaos (PC) methods or increasing the order of the PC representation in time, we propose a procedure aimed at modifying the amplitude of the PC modes to bring the probability of negative state values below a user-defined threshold. This modification can be effectively described as a filtering procedure of the spectral PC coefficients, which is applied on-the-fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold. We demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor. This is carried out by comparing different observables and error measures as obtained by non-intrusive Monte Carlo and Gauss-quadrature integration and the filtered intrusive procedure. Lastly, the filtering procedure has been shown to effectively stabilize divergent intrusive solutions, and also to improve the accuracy of stable intrusive solutions which are close to the stability limits.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-CA), Livermore, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1106517

Alternate ID(s):: OSTI ID: 22314884

Report Number(s):: SAND2013--6540J; PII: S0021999114002484

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 270; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (86)

Quantifying uncertainty in chemical systems modeling Reagan, M. T.; Najm, H. N.; Pébay, P. P. International Journal of Chemical Kinetics, Vol. 37, Issue 6 https://doi.org/10.1002/kin.20081	journal	January 2005
Uncertainty quantification in chemical systems Najm, H. N.; Debusschere, B. J.; Marzouk, Y. M. International Journal for Numerical Methods in Engineering, Vol. 80, Issue 6â7 https://doi.org/10.1002/nme.2551	journal	November 2009
A Stochastic Projection Method for Fluid Flow Le Maı̂ tre, Olivier P.; Knio, Omar M.; Najm, Habib N. Journal of Computational Physics, Vol. 173, Issue 2 https://doi.org/10.1006/jcph.2001.6889	journal	November 2001
A Stochastic Projection Method for Fluid Flow Le Maı̂tre, Olivier P.; Reagan, Matthew T.; Najm, Habib N. Journal of Computational Physics, Vol. 181, Issue 1 https://doi.org/10.1006/jcph.2002.7104	journal	September 2002
Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics Le Maître, O. P.; Knio, Omar M. Scientific Computation https://doi.org/10.1007/978-90-481-3520-2	book	January 2010
Stochastic Galerkin techniques for random ordinary differential equations Augustin, F.; Rentrop, P. Numerische Mathematik, Vol. 122, Issue 3 https://doi.org/10.1007/s00211-012-0466-8	journal	April 2012
Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection Alexanderian, Alen; Le Maître, Oliver P.; Najm, Habib N. Journal of Scientific Computing, Vol. 50, Issue 2 https://doi.org/10.1007/s10915-011-9486-2	journal	April 2011
Stochastic Approaches to Uncertainty Quantification in CFD Simulations Mathelin, Lionel; Hussaini, M. Yousuff; Zang, Thomas A. Numerical Algorithms, Vol. 38, Issue 1 https://doi.org/10.1007/s11075-004-2866-z	journal	March 2005
Finite elements for elliptic problems with stochastic coefficients Frauenfelder, Philipp; Schwab, Christoph; Todor, Radu Alexandru Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 2-5 https://doi.org/10.1016/j.cma.2004.04.008	journal	February 2005
Long-term behavior of polynomial chaos in stochastic flow simulations Wan, Xiaoliang; Karniadakis, George Em Computer Methods in Applied Mechanics and Engineering, Vol. 195, Issue 41-43 https://doi.org/10.1016/j.cma.2005.10.016	journal	August 2006
A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations Nouy, Anthony Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 45-48 https://doi.org/10.1016/j.cma.2007.05.016	journal	September 2007
Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms Nouy, Anthony Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 51-52 https://doi.org/10.1016/j.cma.2008.06.012	journal	October 2008
An alternative unsteady adaptive stochastic finite elements formulation based on interpolation at constant phase Witteveen, Jeroen A. S.; Bijl, Hester Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 3-4 https://doi.org/10.1016/j.cma.2008.09.005	journal	December 2008
Modeling physical uncertainties in dynamic stall induced fluid–structure interaction of turbine blades using arbitrary polynomial chaos Witteveen, Jeroen A. S.; Sarkar, Sunetra; Bijl, Hester Computers & Structures, Vol. 85, Issue 11-14 https://doi.org/10.1016/j.compstruc.2007.01.004	journal	June 2007
Uncertainty propagation using Wiener–Haar expansions Le Maı̂tre, O. P.; Knio, O. M.; Najm, H. N. Journal of Computational Physics, Vol. 197, Issue 1 https://doi.org/10.1016/j.jcp.2003.11.033	journal	June 2004
Multi-resolution analysis of Wiener-type uncertainty propagation schemes Le Maı̂tre, O. P.; Najm, H. N.; Ghanem, R. G. Journal of Computational Physics, Vol. 197, Issue 2 https://doi.org/10.1016/j.jcp.2003.12.020	journal	July 2004
Variational multiscale stabilized FEM formulations for transport equations: stochastic advection–diffusion and incompressible stochastic Navier–Stokes equations Badri Narayanan, Velamur Asokan; Zabaras, Nicholas Journal of Computational Physics, Vol. 202, Issue 1 https://doi.org/10.1016/j.jcp.2004.06.019	journal	January 2005
Using stochastic analysis to capture unstable equilibrium in natural convection Asokan, Badrinarayanan Velamur; Zabaras, Nicholas Journal of Computational Physics, Vol. 208, Issue 1 https://doi.org/10.1016/j.jcp.2005.02.007	journal	September 2005
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations Wan, Xiaoliang; Karniadakis, George Em Journal of Computational Physics, Vol. 209, Issue 2 https://doi.org/10.1016/j.jcp.2005.03.023	journal	November 2005
Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics Hou, Thomas Y.; Luo, Wuan; Rozovskii, Boris Journal of Computational Physics, Vol. 216, Issue 2 https://doi.org/10.1016/j.jcp.2006.01.008	journal	August 2006
A stochastic variational multiscale method for diffusion in heterogeneous random media Velamur Asokan, Badrinarayanan; Zabaras, Nicholas Journal of Computational Physics, Vol. 218, Issue 2 https://doi.org/10.1016/j.jcp.2006.02.026	journal	November 2006
Stochastic analysis of transport in tubes with rough walls Tartakovsky, Daniel M.; Xiu, Dongbin Journal of Computational Physics, Vol. 217, Issue 1 https://doi.org/10.1016/j.jcp.2006.02.029	journal	September 2006
Uncertainty quantification of limit-cycle oscillations Beran, Philip S.; Pettit, Chris L.; Millman, Daniel R. Journal of Computational Physics, Vol. 217, Issue 1 https://doi.org/10.1016/j.jcp.2006.03.038	journal	September 2006
Sparse grid collocation schemes for stochastic natural convection problems Ganapathysubramanian, Baskar; Zabaras, Nicholas Journal of Computational Physics, Vol. 225, Issue 1 https://doi.org/10.1016/j.jcp.2006.12.014	journal	July 2007
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications Foo, Jasmine; Wan, Xiaoliang; Karniadakis, George Em Journal of Computational Physics, Vol. 227, Issue 22 https://doi.org/10.1016/j.jcp.2008.07.009	journal	November 2008
An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations Ma, Xiang; Zabaras, Nicholas Journal of Computational Physics, Vol. 228, Issue 8 https://doi.org/10.1016/j.jcp.2009.01.006	journal	May 2009
A least-squares approximation of partial differential equations with high-dimensional random inputs Doostan, Alireza; Iaccarino, Gianluca Journal of Computational Physics, Vol. 228, Issue 12 https://doi.org/10.1016/j.jcp.2009.03.006	journal	July 2009
Effect of randomness on multi-frequency aeroelastic responses resolved by Unsteady Adaptive Stochastic Finite Elements Witteveen, Jeroen A. S.; Bijl, Hester Journal of Computational Physics, Vol. 228, Issue 18 https://doi.org/10.1016/j.jcp.2009.06.013	journal	October 2009
Multi-element probabilistic collocation method in high dimensions Foo, Jasmine; Karniadakis, George Em Journal of Computational Physics, Vol. 229, Issue 5 https://doi.org/10.1016/j.jcp.2009.10.043	journal	March 2010
Time-dependent generalized polynomial chaos Gerritsma, Marc; van der Steen, Jan-Bart; Vos, Peter Journal of Computational Physics, Vol. 229, Issue 22 https://doi.org/10.1016/j.jcp.2010.07.020	journal	November 2010
A non-adapted sparse approximation of PDEs with stochastic inputs Doostan, Alireza; Owhadi, Houman Journal of Computational Physics, Vol. 230, Issue 8 https://doi.org/10.1016/j.jcp.2011.01.002	journal	April 2011
Spectral and multiresolution Wiener expansions of oscillatory stochastic processes Pettit, C. L.; Beran, P. S. Journal of Sound and Vibration, Vol. 294, Issue 4-5 https://doi.org/10.1016/j.jsv.2005.12.043	journal	July 2006
The role of sensitivity and uncertainty analysis in combustion modelling Tomlin, Alison S. Proceedings of the Combustion Institute, Vol. 34, Issue 1 https://doi.org/10.1016/j.proci.2012.07.043	journal	January 2013
Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection Reagan, Matthew T.; Najm, Habib N.; Ghanem, Roger G. Combustion and Flame, Vol. 132, Issue 3 https://doi.org/10.1016/s0010-2180(02)00503-5	journal	February 2003
Incorporation of parametric uncertainty into complex kinetic mechanisms: Application to hydrogen oxidation in supercritical water Phenix, Brian D.; Dinaro, Joanna L.; Tatang, Menner A. Combustion and Flame, Vol. 112, Issue 1-2 https://doi.org/10.1016/s0010-2180(97)81762-2	journal	January 1998
Making best use of model evaluations to compute sensitivity indices Saltelli, Andrea Computer Physics Communications, Vol. 145, Issue 2 https://doi.org/10.1016/s0010-4655(02)00280-1	journal	May 2002
A new stochastic approach to transient heat conduction modeling with uncertainty Xiu, Dongbin; Karniadakis, George Em International Journal of Heat and Mass Transfer, Vol. 46, Issue 24 https://doi.org/10.1016/s0017-9310(03)00299-0	journal	November 2003
Modeling uncertainty in flow simulations via generalized polynomial chaos Xiu, Dongbin; Karniadakis, George Em Journal of Computational Physics, Vol. 187, Issue 1 https://doi.org/10.1016/s0021-9991(03)00092-5	journal	May 2003
A multigrid solver for two-dimensional stochastic diffusion equations Le Maı̂tre, O. P.; Knio, O. M.; Debusschere, B. J. Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 41-42 https://doi.org/10.1016/s0045-7825(03)00457-2	journal	October 2003
Probabilistic characterization of transport in heterogeneous media Ghanem, Roger Computer Methods in Applied Mechanics and Engineering, Vol. 158, Issue 3-4 https://doi.org/10.1016/s0045-7825(97)00250-8	journal	June 1998
Resolution of gas phase and surface combustion chemistry into elementary reactions Warnatz, Jürgen Symposium (International) on Combustion, Vol. 24, Issue 1 https://doi.org/10.1016/s0082-0784(06)80070-6	journal	January 1992
Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration Li, R.; Ghanem, R. Probabilistic Engineering Mechanics, Vol. 13, Issue 2 https://doi.org/10.1016/s0266-8920(97)00020-9	journal	April 1998
Local and Global Uncertainty Analyses of a Methane Flame Model Zádor, Judit; Zsély, István Gy.; Turányi, Tamás The Journal of Physical Chemistry A, Vol. 109, Issue 43 https://doi.org/10.1021/jp053270i	journal	November 2005
Slow Manifold Structure in Explosive Kinetics. 1. Bifurcations of Points-at-Infinity in Prototypical Models Creta, F.; Adrover, A.; Cerbelli, S. The Journal of Physical Chemistry A, Vol. 110, Issue 50 https://doi.org/10.1021/jp0636064	journal	November 2006
On the convergence of generalized polynomial chaos expansions Ernst, Oliver G.; Mugler, Antje; Starkloff, Hans-Jörg ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, Issue 2 https://doi.org/10.1051/m2an/2011045	journal	October 2011
Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction Ghiocel, Dan M.; Ghanem, Roger G. Journal of Engineering Mechanics, Vol. 128, Issue 1 https://doi.org/10.1061/(asce)0733-9399(2002)128:1(66)	journal	January 2002
Spectral stochastic uncertainty quantification in chemical systems Reagan, M. T.; Najm 4, H. N.; Debusschere, B. J. Combustion Theory and Modelling, Vol. 8, Issue 3 https://doi.org/10.1088/1364-7830/8/3/010	journal	September 2004
Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients Todor, Radu Alexandru; Schwab, Christoph IMA Journal of Numerical Analysis, Vol. 27, Issue 2 https://doi.org/10.1093/imanum/drl025	journal	April 2007
Noisy Inflows Cause a Shedding-Mode Switching in Flow Past an Oscillating Cylinder Lucor, Didier; Karniadakis, George Em Physical Review Letters, Vol. 92, Issue 15 https://doi.org/10.1103/physrevlett.92.154501	journal	April 2004
Multiparameter Spectral Representation of Noise-Induced Competence in Bacillus Subtilis Sargsyan, Khachik; Safta, Cosmin; Debusschere, Bert IEEE/ACM Transactions on Computational Biology and Bioinformatics, Vol. 9, Issue 6 https://doi.org/10.1109/tcbb.2012.107	journal	November 2012
Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems Isukapalli, S. S.; Roy, A.; Georgopoulos, P. G. Risk Analysis, Vol. 18, Issue 3 https://doi.org/10.1111/j.1539-6924.1998.tb01301.x	journal	June 1998
Hybrid Stochastic Finite Elements and Generalized Monte Carlo Simulation Ghanem, R. Journal of Applied Mechanics, Vol. 65, Issue 4 https://doi.org/10.1115/1.2791894	journal	December 1998
A Two-Scale Nonperturbative Approach to Uncertainty Analysis of Diffusion in Random Composites Xiu, Dongbin; Tartakovsky, Daniel M. Multiscale Modeling & Simulation, Vol. 2, Issue 4 https://doi.org/10.1137/03060268x	journal	January 2004
High-Order Collocation Methods for Differential Equations with Random Inputs Xiu, Dongbin; Hesthaven, Jan S. SIAM Journal on Scientific Computing, Vol. 27, Issue 3 https://doi.org/10.1137/040615201	journal	January 2005
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures Wan, Xiaoliang; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 28, Issue 3 https://doi.org/10.1137/050627630	journal	January 2006
Multi‐Resolution‐Analysis Scheme for Uncertainty Quantification in Chemical Systems Le Maître, O. P.; Najm, H. N.; Pébay, P. P. SIAM Journal on Scientific Computing, Vol. 29, Issue 2 https://doi.org/10.1137/050643118	journal	January 2007
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data Babuška, Ivo; Nobile, Fabio; Tempone, Raúl SIAM Journal on Numerical Analysis, Vol. 45, Issue 3 https://doi.org/10.1137/050645142	journal	January 2007
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data Nobile, F.; Tempone, R.; Webster, C. G. SIAM Journal on Numerical Analysis, Vol. 46, Issue 5 https://doi.org/10.1137/060663660	journal	January 2008
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data Nobile, F.; Tempone, R.; Webster, C. G. SIAM Journal on Numerical Analysis, Vol. 46, Issue 5 https://doi.org/10.1137/070680540	journal	January 2008
Eigenvalues of the Jacobian of a Galerkin-Projected Uncertain ODE System Sonday, Benjamin E.; Berry, Robert D.; Najm, Habib N. SIAM Journal on Scientific Computing, Vol. 33, Issue 3 https://doi.org/10.1137/100785922	journal	January 2011
Uncertainty Quantification given Discontinuous Model Response and a Limited Number of Model Runs Sargsyan, Khachik; Safta, Cosmin; Debusschere, Bert SIAM Journal on Scientific Computing, Vol. 34, Issue 1 https://doi.org/10.1137/100817899	journal	January 2012
A Stochastic Multiscale Coupling Scheme to Account for Sampling Noise in Atomistic-to-Continuum Simulations Salloum, Maher; Sargsyan, Khachik; Jones, Reese Multiscale Modeling & Simulation, Vol. 10, Issue 2 https://doi.org/10.1137/110844404	journal	January 2012
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations Xiu, Dongbin; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 24, Issue 2 https://doi.org/10.1137/s1064827501387826	journal	January 2002
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes Debusschere, Bert J.; Najm, Habib N.; Pébay, Philippe P. SIAM Journal on Scientific Computing, Vol. 26, Issue 2 https://doi.org/10.1137/s1064827503427741	journal	January 2004
Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics Najm, Habib N. Annual Review of Fluid Mechanics, Vol. 41, Issue 1 https://doi.org/10.1146/annurev.fluid.010908.165248	journal	January 2009
The Homogeneous Chaos Wiener, Norbert American Journal of Mathematics, Vol. 60, Issue 4 https://doi.org/10.2307/2371268	journal	October 1938
Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations Matthies, Hermann; Keese, Andreas Institut für Wissenschaftliches Rechnen https://doi.org/10.24355/dbbs.084-200511080100-581	text	January 2003
Polynomial Chaos Expansion Applied to Airfoil Limit Cycle Oscillations Pettit, Chris; Beran, Philip 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference https://doi.org/10.2514/6.2004-1691	conference	April 2004
A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations Hosder, Serhat; Walters, Robert; Perez, Rafael 44th AIAA Aerospace Sciences Meeting and Exhibit https://doi.org/10.2514/6.2006-891	conference	June 2012
Efficient Uncertainty Quantification Applied to the Aeroelastic Analysis of a Transonic Wing Hosder, Serhat; Walters, Robert; Balch, Michael 46th AIAA Aerospace Sciences Meeting and Exhibit https://doi.org/10.2514/6.2008-729	conference	January 2008
Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection Reagan, Matthew T.; Najm, Habib N.; Ghanem, Roger G. Combustion and Flame, Vol. 132, Issue 3 https://doi.org/10.1016/S0010-2180(02)00503-5	journal	February 2003
Incorporation of parametric uncertainty into complex kinetic mechanisms: Application to hydrogen oxidation in supercritical water Phenix, Brian D.; Dinaro, Joanna L.; Tatang, Menner A. Combustion and Flame, Vol. 112, Issue 1-2 https://doi.org/10.1016/S0010-2180(97)81762-2	journal	January 1998
Making best use of model evaluations to compute sensitivity indices Saltelli, Andrea Computer Physics Communications, Vol. 145, Issue 2 https://doi.org/10.1016/S0010-4655(02)00280-1	journal	May 2002
A new stochastic approach to transient heat conduction modeling with uncertainty Xiu, Dongbin; Karniadakis, George Em International Journal of Heat and Mass Transfer, Vol. 46, Issue 24 https://doi.org/10.1016/S0017-9310(03)00299-0	journal	November 2003
Modeling uncertainty in flow simulations via generalized polynomial chaos Xiu, Dongbin; Karniadakis, George Em Journal of Computational Physics, Vol. 187, Issue 1 https://doi.org/10.1016/S0021-9991(03)00092-5	journal	May 2003
A multigrid solver for two-dimensional stochastic diffusion equations Le Maı̂tre, O. P.; Knio, O. M.; Debusschere, B. J. Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 41-42 https://doi.org/10.1016/S0045-7825(03)00457-2	journal	October 2003
Probabilistic characterization of transport in heterogeneous media Ghanem, Roger Computer Methods in Applied Mechanics and Engineering, Vol. 158, Issue 3-4 https://doi.org/10.1016/S0045-7825(97)00250-8	journal	June 1998
Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration Li, R.; Ghanem, R. Probabilistic Engineering Mechanics, Vol. 13, Issue 2 https://doi.org/10.1016/S0266-8920(97)00020-9	journal	April 1998
Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations Matthies, Hermann G.; Keese, Andreas Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 12-16 https://doi.org/10.1016/j.cma.2004.05.027	journal	April 2005
Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media Ghanem, R.; Dham, S. Transport in Porous Media, Vol. 32, Issue 3, p. 239-262 https://doi.org/10.1023/A:1006514109327	journal	January 1998
Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction Ghiocel, Dan M.; Ghanem, Roger G. Journal of Engineering Mechanics, Vol. 128, Issue 1 https://doi.org/10.1061/(ASCE)0733-9399(2002)128:1(66)	journal	January 2002
Noisy Inflows Cause a Shedding-Mode Switching in Flow Past an Oscillating Cylinder Lucor, Didier; Karniadakis, George Em Physical Review Letters, Vol. 92, Issue 15 https://doi.org/10.1103/PhysRevLett.92.154501	journal	April 2004
Multiparameter Spectral Representation of Noise-Induced Competence in Bacillus Subtilis Sargsyan, Khachik; Safta, Cosmin; Debusschere, Bert IEEE/ACM Transactions on Computational Biology and Bioinformatics, Vol. 9, Issue 6 https://doi.org/10.1109/TCBB.2012.107	journal	November 2012
A Two-Scale Nonperturbative Approach to Uncertainty Analysis of Diffusion in Random Composites Xiu, Dongbin; Tartakovsky, Daniel M. Multiscale Modeling & Simulation, Vol. 2, Issue 4 https://doi.org/10.1137/03060268X	journal	January 2004
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations Xiu, Dongbin; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 24, Issue 2 https://doi.org/10.1137/S1064827501387826	journal	January 2002
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes Debusschere, Bert J.; Najm, Habib N.; Pébay, Philippe P. SIAM Journal on Scientific Computing, Vol. 26, Issue 2 https://doi.org/10.1137/S1064827503427741	journal	January 2004

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