Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)
Recent Submissions

Simulation of soot size distribution in an ethylene counterflow flame(20140106)Soot, an aggregate of carbonaceous particles produced during the rich combustion of fossil fuels, is an undesirable pollutant and health hazard. Soot evolution involves various dynamic processes: nucleation soot formation from polycyclic aromatic hydrocarbons (PAHs) condensation PAHs condensing on soot particle surface surface processes hydrogenabstractionC2H2addition, oxidation coagulation two soot particles coagulating to form a bigger particle This simulation work investigates soot size distribution and morphology in an ethylene counterflow flame, using i). Chemkin with a method of moments to deal with the coupling between vapor consumption and soot formation; ii). Monte Carlo simulation of soot dynamics.

Fast Estimation of Expected Information Gain for Bayesian Experimental Design Based on Laplace Approximation(20140106)Shannontype expected information gain is an important utility in evaluating the usefulness of a proposed experiment that involves uncertainty. Its estimation, however, cannot rely solely on Monte Carlo sampling methods, that are generally too computationally expensive for realistic physical models, especially for those involving the solution of stochastic partial differential equations. In this work we present a new methodology, based on the Laplace approximation of the posterior probability density function, to accelerate the estimation of expected information gain in the model parameters and predictive quantities of interest. Furthermore, in order to deal with the issue of dimensionality in a complex problem, we use sparse quadratures for the integration over the prior. We show the accuracy and efficiency of the proposed method via several nonlinear numerical examples, including a single parameter design of one dimensional cubic polynomial function and the current pattern for impedance tomography.

Time dependent meanfield games(20140106)We consider time dependent meanfield games (MFG) with a local powerlike dependence on the measure and Hamiltonians satisfying both sub and superquadratic growth conditions. We establish existence of smooth solutions under a certain set of conditions depending both on the growth of the Hamiltonian as well as on the dimension. In the subquadratic case this is done by combining a GagliardoNirenberg type of argument with a new class of polynomial estimates for solutions of the FokkerPlanck equation in terms of LrLp norms of DpH. These techniques do not apply to the superquadratic case. In this setting we recur to a delicate argument that combines the nonlinear adjoint method with polynomial estimates for solutions of the FokkerPlanck equation in terms of L1L1norms of DpH. Concerning the subquadratic case, we substantially improve and extend the results previously obtained. Furthermore, to the best of our knowledge, the superquadratic case has not been addressed in the literature yet. In fact, it is likely that our estimates may also add to the current understanding of HamiltonJacobi equations with superquadratic Hamiltonians.

Multivariate MaxStable Spatial Processes(20140106)Analysis of spatial extremes is currently based on univariate processes. Maxstable processes allow the spatial dependence of extremes to be modelled and explicitly quantified, they are therefore widely adopted in applications. For a better understanding of extreme events of real processes, such as environmental phenomena, it may be useful to study several spatial variables simultaneously. To this end, we extend some theoretical results and applications of maxstable processes to the multivariate setting to analyze extreme events of several variables observed across space. In particular, we study the maxima of independent replicates of multivariate processes, both in the Gaussian and Studentt cases. Then, we define a Poisson process construction in the multivariate setting and introduce multivariate versions of the Smith Gaussian extremevalue, the Schlather extremalGaussian and extremalt, and the BrownResnick models. Inferential aspects of those models based on composite likelihoods are developed. We present results of various Monte Carlo simulations and of an application to a dataset of summer daily temperature maxima and minima in Oklahoma, U.S.A., highlighting the utility of working with multivariate models in contrast to the univariate case. Based on joint work with Simone Padoan and Huiyan Sang.

Optimal Design of Shock Tube Experiments for Parameter Inference(20140106)We develop a Bayesian framework for the optimal experimental design of the shock tube experiments which are being carried out at the KAUST Clean Combustion Research Center. The unknown parameters are the preexponential parameters and the activation energies in the reaction rate expressions. The control parameters are the initial mixture composition and the temperature. The approach is based on first building a polynomial based surrogate model for the observables relevant to the shock tube experiments. Based on these surrogates, a novel MAP based approach is used to estimate the expected information gain in the proposed experiments, and to select the best experimental setups yielding the optimal expected information gains. The validity of the approach is tested using synthetic data generated by sampling the PC surrogate. We finally outline a methodology for validation using actual laboratory experiments, and extending experimental design methodology to the cases where the control parameters are noisy.

Inverse Problems and Uncertainty Quantification(20140106)In a Bayesian setting, inverse problems and uncertainty quantification (UQ)  the propagation of uncertainty through a computational (forward) modelare strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time consuming and slowly convergent Monte Carlo sampling. The developed sampling free nonlinear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisa tion to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.

A Highly Stable MarchingoninTime Volume Integral Equation Solver for Analyzing Transient Wave Interactions on HighContrast Scatterers(20140106)Time domain integral equation (TDIE) solvers represent an attractive alternative to finite difference (FDTD) and finite element (FEM) schemes for analyzing transient electromagnetic interactions on composite scatterers. Current induced on a scatterer, in response to a transient incident field, generates a scattered field. First, the scattered field is expressed as a spatiotemporal convolution of the current and the Green function of the background medium. Then, a TDIE is obtained by enforcing boundary conditions and/or fundamental field relations. TDIEs are often solved for the unknown current using marching onintime (MOT) schemes. MOTTDIE solvers expand the current using local spatiotemporal basis functions. Inserting this expansion into the TDIE and testing the resulting equation in space and time yields a lower triangular system of equations (termed MOT system), which can be solved by marching in time for the coefficients of the current expansion. Stability of the MOT scheme often depends on how accurately the spatiotemporal convolution of the current and the Green function is discretized. In this work, bandlimited prolatebased interpolation functions are used as temporal bases in expanding the current and discretizing the spatiotemporal convolution. Unfortunately, these functions are two sided, i.e., they require ”future” current samples for interpolation, resulting in a noncausal MOT system. To alleviate the effect of noncausality and restore the ability to march in time, an extrapolation scheme can be used to estimate the future values of the currents from their past values. Here, an accurate, stable and bandlimited extrapolation scheme is developed for this purpose. This extrapolation scheme uses complex exponents, rather than commonly used harmonics, so that propagating and decaying mode fields inside the dielectric scatterers are accurately modeled. The resulting MOT scheme is applied to solving the time domain volume integral equation (VIE). Numerical results demonstrate that this new MOTVIE solver maintains its stability and accuracy even when used in analyzing transient wave interactions on highcontrast scatterers.

Spectrum Scarcity and Free Space Optical Communications(20140106)Exact and asymptotic studies of the average error probability of wireless communication systems over generalized fading channels have been extensively pursued over the last two decades. In contrast, studies and results dealing with the channel capacity in these environments have been more scarce. In the first part of this talk, we present a generic moment generating functionbased approach for the exact computation of the channel capacity in such kind of environments. The resulting formulas are applicable to systems having channel state information (CSI) at the receiver and employing maximalratio combining or equalgain combining multichannel reception. The analysis covers the case where the combined paths are not necessarily independent or identically distributed. In all cases, the proposed approach leads to an expression of the ergodic capacity involving a single finiterange integral, which can be easily computed numerically. In the second part of the talk, we focus on the asymptotic analysis of the capacity in the high and low signaltonoise ratio (SNR) regimes. More specifically, we offer new simple closedform formulas that give an intuitive understanding of the capacity behavior at these two extreme regimes. Our characterization covers not only the case where the CSI is available only at the receiver but also the case where the CSI is available at both the transmitter and receiver.

Higherorder Solution of Stochastic Diffusion equation with Nonlinear Losses Using WHEP technique(20140106)Using WienerHermite expansion with perturbation (WHEP) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The WienerHermite expansion is the only known expansion that handles the white/colored noise exactly. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic WienerHermite coefficients. In this poster, the WHEP technique is used to solve the 2D diffusion equation with nonlinear losses and excited with white noise. The solution will be obtained numerically and will be validated and compared with the analytical solution that can be obtained from any symbolic mathematics package such as Mathematica.

Solution of Stochastic Nonlinear PDEs Using Automated WienerHermite Expansion(20140106)The solution of the stochastic differential equations (SDEs) using WienerHermite expansion (WHE) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic WienerHermite coefficients. In WHE approach, there is no randomness directly involved in the computations. One does not have to rely on pseudo random number generators, and there is no need to solve the SDEs repeatedly for many realizations. Instead, the deterministic system is solved only once. For previous research efforts see [2, 4].

DataDriven Model Order Reduction for Bayesian Inverse Problems(20140106)One of the major challenges in using MCMC for the solution of inverse problems is the repeated evaluation of computationally expensive numerical models. We develop a datadriven projection based model order reduction technique to reduce the computational cost of numerical PDE evaluations in this context.

On the Predictability of Computer simulations: Advances in Verification and Validation(20140106)We will present recent advances on the topics of Verification and Validation in order to assess the reliability and predictability of computer simulations. The first part of the talk will focus on goaloriented error estimation for nonlinear boundaryvalue problems and nonlinear quantities of interest, in which case the error representation consists of two contributions: 1) a first contribution, involving the residual and the solution of the linearized adjoint problem, which quantifies the discretization or modeling error; and 2) a second contribution, combining higherorder terms that describe the linearization error. The linearization error contribution is in general neglected with respect to the discretization or modeling error. However, when nonlinear effects are significant, it is unclear whether ignoring linearization effects may produce poor convergence of the adaptive process. The objective will be to show how both contributions can be estimated and employed in an adaptive scheme that simultaneously controls the two errors in a balanced manner. In the second part of the talk, we will present novel approach for calibration of model parameters. The proposed inverse problem not only involves the minimization of the misfit between experimental observables and their theoretical estimates, but also an objective function that takes into account some design goals on specific design scenarios. The method can be viewed as a regularization approach of the inverse problem, one, however, that best respects some design goals for which mathematical models are intended. The inverse problem is solved by a Bayesian method to account for uncertainties in the data. We will show that it shares the same structure as the deterministic problem that one would obtain by multiobjective optimization theory. The method is illustrated on an example of heat transfer in a twodimensional fin. The proposed approach has the main benefit that it increases the confidence in predictive capabilities of mathematical models.