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Articles written by Najm, H.N.

  1. Le Maitre, O.P., Knio, O.M., Debusschere, B.J., Najm, H.N., and Ghanem, R.G.. "A multigrid solver for two-dimensional stochastic diffusion equations" Methods in Applied Mechanics and Engineering. vol. 192. 2003. pp. 4723--4744.


  2. Le Maitre, O.P., Knio, O.M., Najm, H.N., and Ghanem, R.G.. "Uncertainty propagation using Wiener-Haar expansions" J. Comput. Phys.. 197 (1). 2004. pp. 28--57.

    An uncertainty quantification scheme is constructed based on generalized Polynomial Chaos (PC) representations. Two such representations are considered, based on the orthogonal projection of uncertain data and solution variables using either a Haar or a Legendre basis. Governing equations for the unknown coefficients in the resulting representations are derived using a Galerkin procedure and then integrated in order to determine the behavior of the stochastic process. The schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of Rayleigh-Bénard instability. For situations involving random parameters close to a critical point, the computational implementations show that the Wiener-Haar (WHa) representation provides more robust predictions that those based on a Wiener-Legendre (WLe) decomposition. However, when the solution depends smoothly on the random data, the WLe scheme exhibits superior convergence. Suggestions regarding future extensions are finally drawn based on these experiences.


  3. Le Maitre, O.P., Najm, H.N., Ghanem, R.G., and Knio, O.M.. "Multi-resolution analysis of wiener-type uncertainty propagation schemes" Journal of Computational Physics. 197 (2). 2004. pp. 502--531.

    A multi-resolution analysis (MRA) is applied to an uncertainty propagation scheme based on a generalized polynomial chaos (PC) representation. The MRA relies on an orthogonal projection of uncertain data and solution variables onto a multi-wavelet basis, consisting of compact piecewise-smooth polynomial functions. The coefficients of the expansion are computed through a Galerkin procedure. The MRA scheme is applied to the simulation of the Lorenz system having a single random parameter. The convergence of the solution with respect to the resolution level and expansion order is investigated. In particular, results are compared to two Monte-Carlo sampling strategies, demonstrating the superiority of the MRA. For more complex problems, however, the MRA approach may require excessive CPU times. Adaptive methods are consequently developed in order to overcome this drawback. Two approaches are explored: the first is based on adaptive refinement of the multi-wavelet basis, while the second is based on adaptive block-partitioning of the space of random variables. Computational tests indicate that the latter approach is better suited for large problems, leading to a more efficient, flexible and parallelizable scheme.


  4. Debusschere, B.J., Najm, H.N., Pebay, P.P., Knio, O.M., Ghanem, R.G., and Le Maitre, O.P.. "Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes" SIAM J. Sci. Comput.. 26 (2). 2005. pp. 698--719.

    This paper gives an overview of the use of polynomial chaos (PC) expansions to represent stochastic processes in numerical simulations. Several methods are presented for performing arithmetic on, as well as for evaluating polynomial and nonpolynomial functions of variables represented by PC expansions. These methods include Taylor series, a newly developed integration method, as well as a sampling-based spectral projection method for nonpolynomial function evaluations. A detailed analysis of the accuracy of the PC representations, and of the different methods for nonpolynomial function evaluations, is performed. It is found that the integration method offers a robust and accurate approach for evaluating nonpolynomial functions, even when very high-order information is present in the PC expansions.


  5. Marzouk, Y.M., Najm, H.N., and Rahn, L.A.. "Stochastic spectral methods for efficient Bayesian solution of inverse problems" Journal of Computational Physics. 224 (2). 2007. pp. 560--586.

    We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation, the choice of PC basis, and the decomposition of the support of the prior. The computational cost of the new scheme shows significant gains over direct sampling.


  6. Le Maitre, O.P., Knio, O.M., Najm, H.N., and Ghanem, R.G.. "A stochastic projection method for fluid flow. I: basic formulation" Journal of Computational Physics. 173 (2). 2001. pp. 481--511.


  7. Le Maitre, O.P., Reagan, M.T., Najm, H.N., Ghanem, R.G., and Knio, O.M.. "A stochastic projection method for fluid flow II.: random process" Journal of Computational Physics. 181 (1). 2002. pp. 9--44.

    An uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes. The scheme relies on spectral representation of uncertainty using the polynomial chaos (PC) system. The solver combines a Galerkin procedure for the determination of PC coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations. Implementation of the numerical scheme is illustrated through simulations of natural convection in a 2D square cavity with stochastic temperature distribution at the cold wall. The properties of the uncertainty representation scheme are analyzed, and the predictions are contrasted with results obtained using a Monte Carlo approach.


  8. Le Maitre, O.P., Reagan, M.T., Debusschere, B.J., Najm, H.N., Ghanem, R.G., and Knio, O.M.. "Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions" SIAM Journal on Scientific Computing. 26 (2). 2004. pp. 375--394.

    A stochastic projection method (SPM) is developed for quantitative propagation of uncertainty in compressible zero-Mach-number flows. The formulation is based on a spectral representation of uncertainty using the polynomial chaos (PC) system, and on a Galerkin approach to determining the PC coefficients. Governing equations for the stochastic modes are solved using a mass-conservative projection method. The formulation incorporates a specially tailored stochastic inverse procedure for exactly satisfying the mass-conservation divergence constraints. A brief validation of the zero-Mach-number solver is first performed, based on simulations of natural convection in a closed cavity. The SPM is then applied to analyze the steady-state behavior of the heat transfer and of the velocity and temperature fields under stochastic non-Boussinesq conditions.


  9. Le Maitre, O.P., Najm, H.N., P\'ebay, P.P., Ghanem, R.G., and Knio, O.M.. "Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems" SIAM Journal on Scientific Computing. 29 (2). 2007. pp. 864--889.

    This paper presents a multi-resolution approach for the propagation of parametric uncertainty in chemical systems. It is motivated by previous studies where Galerkin formulations of Wiener-Hermite expansions were found to fail in the presence of steep dependences of the species concentrations with regard to the reaction rates. The multi-resolution scheme is based on representation of the uncertain concentration in terms of compact polynomial multi-wavelets, allowing for the control of the convergence in terms of polynomial order and resolution level. The resulting representation is shown to greatly improve the robustness of the Galerkin procedure in presence of steep dependences. However, this improvement comes with a higher computational cost which drastically increases with the number of uncertain reaction rates. To overcome this drawback an adaptive strategy is proposed to control locally (in the parameter space) and in time the resolution level. The efficiency of the method is demonstrated for an uncertain chemical system having eight random parameters.


  10. Reagan, M.T., Najm, H.N., Debusschere, B.J., Le Maitre, O.P., Knio, O.M., and Ghanem, R.G.. "Spectral stochastic uncertainty quantification in chemical systems" Combustion Theory and Modelling. 8 (3). SEP 2004. pp. 607--632.

    Uncertainty quantification (UQ) in the computational modelling of physical systems is important for scientific investigation, engineering design, and model validation. We have implemented an `intrusive' UQ technique in which (1) model parameters and field variables are modelled as stochastic quantities, and are represented using polynomial chaos (PC) expansions in terms of Hermite polynomial functions of Gaussian random variables, and (2) the deterministic model equations are reformulated using Galerkin projection into a set of equations for the time evolution of the field variable PC mode strengths. The mode strengths relate specific parametric uncertainties to their effects on model outputs. In this work, the intrusive reformulation is applied to homogeneous ignition using a detailed chemistry model through the development of a reformulated pseudospectral chemical source term. We present results analysing the growth of uncertainty during the ignition process. We also discuss numerical issues pertaining to the accurate representation of uncertainty with truncated PC expansions, and ensuing stability of the time integration of the chemical system.


  11. Reagan, M.T., Najm, H.N., Pebay, P.P., Knio, O.M., and Ghanem, R.G.. "Quantifying uncertainty in chemical systems modeling" International Journal of Chemical Kinetics. 37 (6). JUN 2005. pp. 368--382.

    This study compares two techniques for uncertainty quantification in chemistry computations, one based on sensitivity analysis and error propagation, and the other on stochastic analysis using polynomial chaos techniques. The two constructions are studied in the context of H-2-O-2 ignition under supercritical-water conditions. They are compared in terms of their prediction of uncertainty in species concentrations and the sensitivity of selected species concentrations to given parameters. The formulation is extended to one-dimensional reacting-flow simulations. The computations are used to study sensitivities to both reaction rate pre-exponentials and enthalpies, and to examine how this information must be evaluated in light of known, inherent parametric uncertainties in simulation parameters. The results indicate that polynomial chaos methods provide similar first-order information to conventional sensitivity analysis, while preserving higher-order information that is needed for accurate uncertainty quantification and for assigning confidence intervals on sensitivity coefficients. These higher-order effects can be significant, as the analysis reveals substantial uncertainties in the sensitivity coefficients themselves. © 2005 Wiley Periodicals, Inc.


  12. Kim, D., Debusschere, B.J., and Najm, H.N.. "Spectral methods for parametric sensitivity in stochastic dynamical systems" Biophyscial Journal. 92 (2). JAN 2007. pp. 379--393.

    Stochastic dynamical systems governed by the chemical master equation find use in the modeling of biological phenomena in cells, where they provide more accurate representations than their deterministic counterparts, particularly when the levels of molecular population are small. The analysis of parametric sensitivity in such systems requires appropriate methods to capture the sensitivity of the system dynamics with respect to variations of the parameters amid the noise from inherent internal stochastic effects. We use spectral polynomial chaos expansions to represent statistics of the system dynamics as polynomial functions of the model parameters. These expansions capture the nonlinear behavior of the system statistics as a result of finite-sized parametric perturbations. We obtain the normalized sensitivity coefficients by taking the derivative of this functional representation with respect to the parameters. We apply this method in two stochastic dynamical systems exhibiting bimodal behavior, including a biologically relevant viral infection model.