Begell House
International Journal for Uncertainty Quantification
International Journal for Uncertainty Quantification
2152-5080
4
6
2014
ANALYSIS OF VARIANCE-BASED MIXED MULTISCALE FINITE ELEMENT METHOD AND APPLICATIONS IN STOCHASTIC TWO-PHASE FLOWS
The stochastic partial differential systems have been widely used to model physical processes, where the inputs involve large uncertainties. Flows in random and heterogeneous porous media is one of the cases where the random inputs (e.g., permeability) are often modeled as a stochastic field with high-dimensional random parameters. To treat the high dimensionality and heterogeneity efficiently, model reduction is employed in both stochastic space and physical space. An analysis of variance (ANOVA)-based mixed multiscale finite element method (MsFEM) is developed to decompose the high-dimensional stochastic problem into a set of lower-dimensional stochastic subproblems, which require much less computational complexity and significantly reduce the computational cost in stochastic space, and the mixed MsFEM can capture the heterogeneities on a coarse grid to greatly reduce the computational cost in the spatial domain. In addition, to enhance the efficiency of the traditional ANOVA method, an adaptive ANOVA method based on a new adaptive criterion is developed, where the most active dimensions can be selected to greatly reduce the computational cost before conducting ANOVA decomposition. This novel adaptive criterion is based on variance-decomposition method coupled with sparse-grid probabilistic collocation method or multilevel Monte Carlo method. The advantage of this adaptive criterion lies in its much lower computational overhead for identifying the active dimensions and interactions. A number of numerical examples in two-phase stochastic flows are presented and demonstrate the accuracy and performance of the adaptive ANOVA-based mixed MsFEM.
Jia
Wei
Department of Mathematics, Texas A&M University, College Station, Texas 77840; Computational Science & Mathematics Division, Pacific Northwest National Laboratory, Richland, Washington 99352, USA
Guang
Lin
Computational Science & Mathematics Division, Pacific Northwest National Laboratory, Richland, Washington 99352; Department of Mathematics, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, USA
Lijian
Jiang
College of Mathematics and Econometrics, Hunan University, China
Yalchin
Efendiev
Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia; Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843-3368, USA
455-477
GRADIENT-BASED STOCHASTIC OPTIMIZATION METHODS IN BAYESIAN EXPERIMENTAL DESIGN
Optimal experimental design (OED) seeks experiments expected to yield the most useful data for some purpose. In
practical circumstances where experiments are time-consuming or resource-intensive, OED can yield enormous savings. We pursue OED for nonlinear systems from a Bayesian perspective, with the goal of choosing experiments that are optimal for parameter inference. Our objective in this context is the expected information gain in model parameters, which in general can only be estimated using Monte Carlo methods. Maximizing this objective thus becomes a stochastic optimization problem. This paper develops gradient-based stochastic optimization methods for the design of experiments on a continuous parameter space. Given a Monte Carlo estimator of expected information gain, we use infinitesimal perturbation analysis to derive gradients of this estimator.We are then able to formulate two gradient-based stochastic optimization approaches: (i) Robbins-Monro stochastic approximation, and (ii) sample average approximation combined with a deterministic quasi-Newton method. A polynomial chaos approximation of the forward model
accelerates objective and gradient evaluations in both cases.We discuss the implementation of these optimization methods, then conduct an empirical comparison of their performance. To demonstrate design in a nonlinear setting with partial differential equation forward models, we use the problem of sensor placement for source inversion. Numerical results yield useful guidelines on the choice of algorithm and sample sizes, assess the impact of estimator bias, and quantify tradeoffs of computational cost versus solution quality and robustness.
Xun
Huan
Massachusetts Institute of Technology
Youssef
Marzouk
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Room 33-305 Cambridge, MA 02139 USA
479-510
INHERENT AND EPISTEMIC UNCERTAINTY ANALYSIS FOR COMPUTATIONAL FLUID DYNAMICS SIMULATIONS OF SYNTHETIC JET ACTUATORS
A mixed uncertainty quantification method was applied to computational fluid dynamics (CFD) modeling of a synthetic
jet actuator. A test case, flow over a hump model with synthetic jet actuators, was selected from the CFDVAL2004
workshop to apply the second-order probability framework implemented with a stochastic response surface obtained from quadrature-based nonintrusive polynomial chaos. Three uncertainty sources were considered: (1) epistemic uncertainty in turbulence model, (2) inherent uncertainty in free stream velocity, and (3) inherent uncertainty
in actuation frequency. Uncertainties in both long-time averaged and phase averaged quantities were quantified using a fourth-order polynomial chaos expansion. A global sensitivity analysis with Sobol indices was utilized to rank the importance of each uncertainty source to the overall output uncertainty. The results indicated that for the long-time averaged separation bubble size, the uncertainty in turbulence model had a dominant contribution, which was also observed in the long-time averaged skin-friction coefficients at three selected locations. The mixed uncertainty results for phase-averaged x-velocity distributions at three selected locations showed that the 95% confidence interval could
generally envelop the experimental data. The Sobol indices showed that near the wall, the uncertainty in turbulence
model had a main influence on the x-velocity. While approaching the main stream, the uncertainty in free stream velocity became a larger contributor. The mixed uncertainty quantification approach demonstrated in this study can also be applied to other CFD problems with inherent and epistemic uncertainties.
Daoru
Han
Laboratory for Astronautical Plasma Dynamics, Department of Astronautical Engineering, 233 Robert Glenn Rapp Engineering Research Building, University of Southern California, Los Angeles, California 90089-1192, USA
Serhat
Hosder
Aerospace Simulations Laboratory, Department of Mechanical and Aerospace Engineering, 290B Toomey Hall, Missouri University of Science and Technology, Rolla, Missouri 65409-0500, USA
511-533
EFFECTIVE SAMPLING SCHEMES FOR BEHAVIOR DISCRIMINATION IN NONLINEAR SYSTEMS
Behavior discrimination is the problem of identifying sets of parameters for which the system does (or does not) reach
a given set of states. While there are a variety of methods to address this problem for linear systems, few successful techniques have been developed for nonlinear models. Existing methods often rely on numerical simulations without rigorous bounds on the numerical errors and usually require a large number of model evaluations, rendering those methods impractical for studies of high-dimensional and expensive systems. In this work, we describe a probabilistic framework to estimate the boundary that separates contrasting behaviors and to quantify the uncertainty in this estimation. In our approach, we directly parameterize the, yet unknown, boundary by the zero level-set of a polynomial function, then use statistical inference on available data to identify the coefficients of the polynomial. Building upon this framework, we consider the problem of choosing effective data sampling schemes for behavior discrimination of nonlinear systems in two different settings: the low-discrepancy sampling scheme, and the uncertainty-based sequential sampling scheme. In both cases, we successfully derive theoretical results about the convergence of the expected boundary to the true boundary of interest. We then demonstrate the efficacy of the method in several application contexts with a focus on biological models. Our method outperforms previous approaches to this problem in several ways and proves to be effective to study high-dimensional and expensive systems.
Vu
Dinh
Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907, USA
Ann E.
Rundell
Weldon School of Biomedical Engineering, Purdue University, 206 S. Martin Jischke Drive, West Lafayette, Indiana 47907, USA
Gregery T.
Buzzard
Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907, USA
535-554
INDEX - Volume 4
555-559