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Articles Published in Journal of Computational Physics

  1. Asokan, B.V. and Zabaras, N.. "Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations" Journal of Computational Physics. vol. 202. 2005. pp. 94--133.


  2. 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.


  3. Yu, Y., Zhao, M., Lee, T., Pestieau , N., Bo, W., Glimm, J., and Grove, J.W.. "Uncertainty quantification for chaotic computational fluid dynamics" Journal of Computational Physics. 217 (1). SEP 2006. pp. 200--216.

    We seek error models for simulations that model chaotic flow. Stable statistics for the solution and for the error are obtained after suitable averaging procedures.

    We seek error models for simulations that model chaotic flow. Stable statistics for the solution and for the error are obtained after suitable averaging procedures. (c) 2006 Elsevier Inc. All rights reserved.


  4. 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.


  5. Asokan, B.V. and Zabaras, N.. "A stochastic variational multiscale method for diffusion in heterogeneous random media" Journal of Computational Physics. 218 (2). 2006. pp. 654--676.

    A stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media. The exact solution of the governing equations is split into two components: a coarse-scale solution that can be captured on a coarse mesh and a subgrid solution. A localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse-scale solution. The coarse-scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model. The generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models. Finally, various numerical examples are considered for evaluating the method.


  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. Tartakovsky, D.M. and Xiu, D.. "Stochastic analysis of transport in tubes with rough walls" Journal of Computational Physics. 217 (1). 2006. pp. 248--259.

    Flow and transport in tubes with rough surfaces play an important role in a variety of applications. Often the topology of such surfaces cannot be accurately described in all of its relevant details due to either insufficient data or measurement errors or both. In such cases, this topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we use a computational framework, which is based on stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a generalized polynomial chaos and Monte Carlo simulations to solve the transformed stochastic problem. We use our approach to describe transport of a passive scalar in Stokes' flow and to quantify the corresponding predictive uncertainty.


  9. Zabaras, N. and Ganapathysubramanian, B.. "A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach" Journal of Computational Physics. 227 (9). 2008. pp. 4697--4735.

    Experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions. Recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties. These advances raise the possibility of solving the corresponding stochastic inverse problem: the problem of designing/estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information. A scalable, parallel methodology for stochastic inverse/design problems is developed in this article. The representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology. A novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems. The stochastic inverse/design problem is transformed to a deterministic optimization problem in a larger-dimensional space that is subsequently solved using deterministic optimization algorithms. The design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems, thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes. Various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework.


  10. Chen, Q.-Y., Gottlieb, D., and Hesthaven, J.S.. "Uncertainty analysis for the steady-state flows in a dual throat nozzle" Journal of Computational Physics. 204 (1). 2005. pp. 378--398.

    It is well known that the steady state of an isentropic flow in a dual-throat nozzle with equal throat areas is not unique. In particular there is a possibility that the flow contains a shock wave, whose location is determined solely by the initial condition. In this paper, we consider cases with uncertainty in this initial condition and use generalized polynomial chaos methods to study the steady-state solutions for stochastic initial conditions. Special interest is given to the statistics of the shock location. The polynomial chaos (PC) expansion modes are shown to be smooth functions of the spatial variable x, although each solution realization is discontinuous in the spatial variable x. When the variance of the initial condition is small, the probability density function of the shock location is computed with high accuracy. Otherwise, many terms are needed in the PC expansion to produce reasonable results due to the slow convergence of the PC expansion, caused by non-smoothness in random space.


  11. Lin, G., Su, C.-H., and Karniadakis, G.E.. "Predicting shock dynamics in the presence of uncertainties" Journal of Computational Physics. 217 (1). 2006. pp. 260--276.

    We revisit the classical aerodynamics problem of supersonic flow past a wedge but subject to random inflow fluctuations or random wedge oscillations around its apex. We first obtain analytical solutions for the inviscid flow, and subsequently we perform stochastic simulations treating randomness both as a steady as well as a time-dependent process. We use a multi-element generalized polynomial chaos (ME-gPC) method to solve the two-dimensional stochastic Euler equations. A Galerkin projection is employed in the random space while WENO discretization is used in physical space. A key issue is the characteristic flux decomposition in the stochastic framework for which we propose different approaches. The results we present show that the variance of the location of perturbed shock grows quadratically with the distance from the wedge apex for steady randomness. However, for a time-dependent random process the dependence is quadratic only close to the apex and linear for larger distances. The multi-element version of polynomial chaos seems to be more effective and more efficient in stochastic simulations of supersonic flows compared to the global polynomial chaos method.


  12. Lin, G., Grinberg, L., and Karniadakis, G.E.. "Numerical studies of the stochastic Korteweg-de Vries equation" Journal of Computational Physics. 213 (2). 2006. pp. 676--703.

    We present numerical solutions of the stochastic Korteweg-de Vries equation for three cases corresponding to additive time-dependent noise, multiplicative space-dependent noise and a combination of the two. We employ polynomial chaos for discretization in random space, and discontinuous Galerkin and finite difference for discretization in physical space. The accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and Monte Carlo simulation results. Of particular interest is the interplay of spatial discretization error with the stochastic approximation error, which is examined for different orders of spatial and stochastic approximation.


  13. Asokan, B.V. and Zabaras, N.. "Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations" Journal of Computational Physics. 202 (1). 2005. pp. 94--133.

    An extension of the deterministic variational multiscale (VMS) approach with algebraic subgrid scale (SGS) modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic Navier-Stokes equations. The stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method (SSFEM). Generalized polynomial chaos and Karhunen-Loève expansion techniques are used for representation of uncertain quantities. The proposed stabilized method is then applied to various standard advection-diffusion and fluid-flow examples with uncertainty in essential boundary conditions. Comparisons are drawn between the numerical solutions and Monte-Carlo/analytical solutions wherever possible.


  14. Asokan, B.V. and Zabaras, N.. "Using stochastic analysis to capture unstable equilibrium in natural convection" Journal of Computational Physics. 208 (1). 2005. pp. 134--153.

    A stabilized stochastic finite element implementation for the natural convection system of equations under Boussinesq assumptions with uncertainty in inputs is considered. The stabilized formulations are derived using the variational multiscale framework assuming a one-step trapezoidal time integration rule. The stabilization parameters are shown to be functions of the time-step size. Provision is made for explicit tracking of the subgrid-scale solution through time. A support-space/stochastic Galerkin approach and the generalized polynomial chaos expansion (GPCE) approach are considered for input-output uncertainty representation. Stochastic versions of standard Rayleigh-Benard convection problems are used to evaluate the approach. It is shown that for simulations around critical points, the GPCE approach fails to capture the highly non-linear input uncertainty propagation whereas the support-space approach gives fairly accurate results. A summary of the results and findings is provided.


  15. Zhang, D. and Lu, Z.. "An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Lo\`eve and polynomial expansions" Journal of Computational Physics. 194 (2). 2004. pp. 773--794.

    In this study, we attempt to obtain higher-order solutions of the means and (co)variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of Karhunen-Loève decomposition, polynomial expansion, and perturbation methods. We first decompose the log hydraulic conductivity Y = ln Ks as an infinite series on the basis of a set of orthogonal Gaussian standard random variables ξi. The coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity. We then write head as an infinite series whose terms h(n) represent head of nth order in terms of σY, the standard deviation of Y, and derive a set of recursive equations for h(n). We then decompose h(n) with polynomial expansions in terms of the products of n Gaussian random variables. The coefficients in these series are determined by substituting decompositions of Y and h(n) into those recursive equations. We solve the mean head up to fourth-order in σY and the head variances up to third-order in σY2. We conduct Monte Carlo (MC) simulation and compare MC results against approximations of different orders from the moment-equation approach based on Karhunen-Loève decomposition (KLME). We also explore the validity of the KLME approach for different degrees of medium variability and various correlation scales. It is evident that the KLME approach with higher-order corrections is superior to the conventional first-order approximations and is computationally more efficient than the Monte Carlo simulation.


  16. Asokan, B.V. and Zabaras, N.. "Using stochastic analysis to capture unstable equilibrium in natural convection" Journal of Computational Physics. 208 (1). 2005. pp. 134--153.

    In recent years, there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal flow problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. The sparse grid collocation method based on the Smolyak algorithm offers a viable alternate method for solving high-dimensional stochastic partial differential equations. An extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation. We show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions. Equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured. Comparison of the present method with the generalized polynomial chaos expansion and Monte-Carlo methods are mad


  17. Schwab, C. and Todor, R.A.. "Karhunen-Lo\`eve approximation of random fields by generalized fast multipole methods" Journal of Computational Physics. 217 (1). 2006. pp. 100--122.


  18. Hou, T.Y., Luo , W., Rozovskii, B., and Zhou, H.-M.. "Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics" Journal of Computational Physics. 216 (2). 2006. pp. 687--706.

    In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier-Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. The separation principle effectively reduces a stochastic equation to its associated propagator, a system of deterministic equations for the coefficients of the Wiener Chaos expansion. Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. We demonstrate that for short time solutions the numerical methods based on the Wiener Chaos expansion are more efficient and accurate than those based on the Monte Carlo simulations.


  19. Ganapathysubramanian, B. and Zabaras, N.. "Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method" Journal of Computational Physics. 226 (1). SEP 10 2007. pp. 326--353.

    In recent years, there has been intense interest in understanding various physical phenomena in random heterogeneous media. Any accurate description/simulation of a process in such media has to satisfactorily account for the twin issues of randomness as well as the multilength scale variations in the material properties. An accurate model of the material property variation in the system is an important prerequisite towards complete characterization of the system response. We propose a general methodology to construct a data-driven, reduced-order model to describe property variations in realistic heterogeneous media. This reduced-order model then serves as the input to the stochastic partial differential equation describing thermal diffusion through random heterogeneous media. A decoupled scheme is used to tackle the problems of stochasticity and multilength scale variations in properties. A sparse-grid collocation strategy is utilized to reduce the solution of the stochastic partial differential equation to a set of deterministic problems. A variational multiscale method with explicit subgrid modeling is used to solve these deterministic problems. An illustrative example using experimental data is provided to showcase the effectiveness of the proposed methodology. (C) 2007 Elsevier Inc. All rights reserved.


  20. Paffrath, M. and Wever, U.. "Adapted polynomial chaos expansion for failure detection" Journal of Computational Physics. 226 (1). SEP 10 2007. pp. 263--281.

    In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem. (C) 2007 Elsevier Inc. All rights reserved.


  21. Rupert, C.P. and Miller, C.T.. "An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems" Journal of Computational Physics. 226 (2). OCT 2007. pp. 2175--2205.

    We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method. (c) 2007 Elsevier Inc. All rights reserved.


  22. Vasilyev, O.V. and Paolucci, S.. "A fast adaptive wavlet collocation algorithm for multidimensional PDEs" Journal of Computational Physics. vol. 138. 1997. pp. 16--56.


  23. Xiu, D. and Sherwin, S.J.. "Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network" Journal of Computational Physics. 226 (2). 2007. pp. 1385--1407.

    Reduced models of human arterial networks are an efficient approach to analyze quantitative macroscopic features of human arterial flows. The justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in the networks. Although these types of models have been employed extensively and many issues associated with their implementations have been widely researched, the issue of data uncertainty has received comparatively little attention. Similar to many biological systems, a large amount of uncertainty exists in the value of the parameters associated with the models. Clearly reliable assessment of the system behaviour cannot be made unless the effect of such data uncertainty is quantified. In this paper we present a study of parametric data uncertainty in reduced modelling of human arterial networks which is governed by a hyperbolic system. The uncertain parameters are modelled as random variables and the governing equations for the arterial network therefore become stochastic. This type stochastic hyperbolic systems have not been previously systematically studied due to the difficulties introduced by the uncertainty such as a potential change in the mathematical character of the system and imposing boundary conditions. We demonstrate how the application of a high-order stochastic collocation method based on the generalized polynomial chaos expansion, combined with a discontinuous Galerkin spectral/hp element discretization in physical space, can successfully simulate this type of hyperbolic system subject to uncertain inputs with bounds. Building upon a numerical study of propagation of uncertainty and sensitivity in a simplified model with a single bifurcation, a systematical parameter sensitivity analysis is conducted on the wave dynamics in a multiple bifurcating human arterial network. Using the physical understanding of the dynamics of pulse waves in these types of networks we are able to provide an insight into the results of the stochastic simulations, thereby demonstrating the effects of uncertainty in physiologically accurate human arterial networks.