Begell House Inc. International Journal for Uncertainty Quantification IJUQ 2152-5080 3 1 2013
POLYNOMIAL CHAOS FOR SEMIEXPLICIT DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX 1 1-23 Roland Pulch Department of Mathematics and Computer Science University of Greifswald Domstraße 11, 17489 Greifswald, Germany Mathematical modeling of technical applications often yields systems of differential algebraic equations. Uncertainties of physical parameters can be considered by the introduction of random variables. A corresponding uncertainty quantification requires one to solve the stochastic model. We focus on semiexplicit systems of nonlinear differential algebraic equations with index 1. The stochastic model is solved using the expansion of the generalised polynomial chaos. We investigate both the stochastic collocation technique and the stochastic Galerkin method to determine the unknown coefficient functions. In particular, we analyze the index of the larger coupled systems, which result from the stochastic Galerkin method. Numerical simulations of test examples are presented, where the two approaches are compared with respect to their efficiency.
VISUALIZATION OF COVARIANCE AND CROSS-COVARIANCE FIELDS 25-38 Chao Yang School of Computing, University of Utah, Salt Lake City, UT 84112, USA Dongbin Xiu Ohio State University Mike Kirby Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, Utah, 84112, USA We present a numerical technique to visualize covariance and cross-covariance fields of a stochastic simulation. The method is local in the sense that it demonstrates the covariance structure of the solution at a point with its neighboring locations. When coupled with an efficient stochastic simulation solver, our framework allows one to effectively concurrently visualize both the mean and (cross-)covariance information for two-dimensional (spatial) simulation results. Most importantly, the visualization provides the scientist a means to identify the interesting correlation structure of the solution field. The mathematical setup is discussed, along with several examples to demonstrate the efficacy of this approach.
COMPUTING GREEN'S FUNCTIONS FOR FLOW IN HETEROGENEOUS COMPOSITE MEDIA 39-46 David A. Barajas-Solano Department of Mechanical and Aerospace Engineering, University of California, San Diego, USA Daniel M. Tartakovsky Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA Green's functions lie at the foundation of many uncertainty quantification and uncertainty reduction techniques (e.g., the moment differential equation approach, parameter and/or source identification, and data assimilation). We discuss an accurate and numerically efficient approach to compute Green's functions for transport processes in heterogeneous composite media. We focus on elliptic partial differential equations with (random) discontinuous coefficients. The approach relies on a regularization technique to obtain an associated regular problem, which can be solved using standard finite element methods. We perform numerical experiments to assess the performance of the regularization approach and to evaluate the effects of strong coefficient discontinuities on the Green's function behavior.
PHYSICS-BASED COVARIANCE MODELS FOR GAUSSIAN PROCESSES WITH MULTIPLE OUTPUTS 47-71 Emil M. Constantinescu Mathematics and Computer Science Division, Argonne National Laboratory, USA Mihai Anitescu Mathematics and Computer Science Division, Argonne National Laboratory, USA Gaussian process analysis of processes with multiple outputs is limited by the fact that far fewer good classes of covariance functions exist compared with the scalar (single-output) case. To address this difficulty, we turn to covariance function models that take a form consistent in some sense with physical laws that govern the underlying simulated process. Models that incorporate such information are suitable when performing uncertainty quantification or inferences on multidimensional processes with partially known relationships among different variables, also known as cokriging. One example is in atmospheric dynamics where pressure and wind speed are driven by geostrophic assumptions (wind &prop; &#8706;/&#8706;x pressure). In this study we develop both analytical and numerical auto-covariance and cross-covariance models that are consistent with physical constraints or can incorporate automatically sensible assumptions about the process that generated the data. We also determine high-order closures, which are required for nonlinear dependencies among the observables. We use these models to study Gaussian process regression for processes with multiple outputs and latent processes (i.e., processes that are not directly observed and predicted but inter-relate the output quantities). Our results demonstrate the effectiveness of the approach on both synthetic and real data sets.
VALIDATION OF A PROBABILISTIC MODEL FOR MESOSCALE ELASTICITY TENSOR OF RANDOM POLYCRYSTALS 73-100 Arash Noshadravan University of Southern California, Department of Civil and Environmental Engineering, USA Roger Ghanem Department of Aerospace and Mechanical Engineering, University of Southern California, 210 KAP Hall, Los Angeles, California 90089, USA Johann Guilleminot Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME, Marne la Vallée, France Ikshwaku Atodaria Arizona State University, School for Engineering of Matter, Transport and Energy, Tempe, Arizona 85287-6106, USA Pedro Peralta Arizona State University, School for Engineering of Matter, Transport and Energy, Tempe, Arizona 85287-6106, USA In this paper, we present validation of a probabilistic model for mesoscale elastic behavior of materials with microstructure. The linear elastic constitutive matrix of this model is described mathematically as a bounded random matrix. The bounds reflect theoretical constraints consistent with the theory of elasticity. We first introduce a statistical characterization of an experimental database on morphology and crystallography of polycrystalline microstructures. The resulting statistical model is used as a surrogate to further experimental data, required for calibration and validation. We then recall the construction of a probabilistic model for the random matrix characterizing the apparent elasticity tensor of a heterogeneous random medium. The calibration of this coarse scale probabilistic model using an experimental database of microstructural measurements and utilizing the developed microstructural simulation tool is briefly discussed. Before using the model as a predictive tool in a system level simulation for the purpose of detection and prognosis, the credibility of the model must be established through evaluating the degree of agreement between the predictions of the model and the observations. As such, a procedure is presented to validate the probabilistic model from simulated data resulting from subscale simulations. Suitable quantities of interest are introduced and predictive accuracy of the model is studied by comparing probability density functions of response quantities of interest. The validation task is exercised under both static and dynamic loading condition. The results indicate that the probabilistic model of mesoscale elasticity tensor is adequate to predict the response quantity of interest in the elastostatic regime. The scatter in the model predictions is found to be consistent with the fine scale response. In the case of elastodynamic, the model predicts the mean behavior for lower frequency for which we have a quasistatic regime.