Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
3
5
2013
FRAMEWORK FOR CONVERGENCE AND VALIDATION OF STOCHASTIC UNCERTAINTY QUANTIFICATION AND RELATIONSHIP TO DETERMINISTIC VERIFICATION AND VALIDATION
371-395
S. Maysam
Mousaviraad
IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa 52242, USA
Wei
He
Visiting scholar from NAOCE, Shanghai Jiao Tong University, Shanghai, China
Matteo
Diez
Visiting scholar from CNR-INSEAN, Via di Vallerano 139, 00128 Rome, Italy
Frederick
Stern
IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa 52242, USA
A framework is described for convergence and validation of nonintrusive uncertainty quantification (UQ) methods; the relationship between deterministic verification and validation (V&V) and stochastic UQ is studied, and an example is provided for a unit problem. Convergence procedures are developed for Monte Carlo (MC) without and with metamodels, showing that in addition to the usual user-defined acceptable confidence intervals, convergence studies with systematic refinement ratio are required. A UQ validation procedure is developed using the benchmark UQ results and defining the comparison error and its uncertainty to evaluate validation. A stochastic influence factor is defined to evaluate the effects of input variability on the performance expectation and four possibilities are identified in making design decisions. The unit problem studies a two-dimensional airfoil with variable Re and normal distribution using high-fidelity Reynolds-averaged Navier-Stokes (RANS) simulations. Deterministic V&V studies achieve monotonic grid convergence and validation at the validation uncertainty interval of 2.2% D, averaged between lift and drag, with an average error of 0.25% D. For MC with Latin hypercube sampling the converged results are obtained with 400 computational fluid dynamics (CFD) simulations and are used as validation benchmark in the absence of experimental UQ. The stochastic influence factor is small such that the output expected value is not distinguishable from the deterministic solution. The output uncertainty is one order of magnitude smaller for lift than drag, implying that lift is only weakly dependent on Re. Several metamodels are used with MC, reducing the number of CFD simulations to a minimum of 4. The results are converged and validated at the average intervals of 0.1% for expected value (EV) and 10.7% for standard deviation (SD). The Gauss quadrature and the polynomial chaos (PC) method are validated using 8 and 7 CFD simulations, respectively, at the average intervals of 0.08% for EV and 7.5% for SD. The error values are smallest for the metamodels, followed by the PC method and then the Gauss quadratures.
PARAMETER SENSITIVITY OF AN EDDY VISCOSITY MODEL: ANALYSIS, COMPUTATION AND ITS APPLICATION TO QUANTIFYING MODEL RELIABILITY
397-419
Faranak
Pahlevani
Division of Science and Engineering, Penn State Abington, Abington, Pennsylvania 19001, USA
Lisa
Davis
Department of Mathematical Sciences, Montana State University, Bozeman, Montana 59717, USA
An eddy viscosity model can be used as a computationally tractable alternative to that of the Navier-Stokes equations. Model errors immediately become a concern when considering such an approach, and quantifying this error is essential to understanding and using model predictions within an engineering design process. In this paper, sensitivity analysis is presented for a subgrid eddy viscosity model with respect to variations of the eddy viscosity parameter. We demonstrate the analysis utilizing the sensitivity equation method. Approximating the sensitivity requires the solution of the eddy viscosity model. Therefore, the eddy viscosity model and sensitivity equation are coupled in our analysis and computations. An implicit-explicit time-stepping method is developed and analyzed for this set of equations. Our numerical assessments present the role of the sensitivity in quantifying the modeling error arising from the choice of various values of the eddy viscosity parameter. The sensitivity computation allows one to identify an interval of reliability for the eddy viscosity parameter. This gives the user a range of parameter values for which the eddy viscosity model can be considered to be a reliable approximation to the Navier-Stokes equations. A two-dimensional cavity problem is used to illustrate the ideas. In addition, for the standard model problem of two-dimensional flow around a cylinder, the sensitivity computations are shown to be very useful in improving the flow functional approximations that may be used within an optimal design algorithm.
PROPAGATION OF UNCERTAINTY BY SAMPLING ON CONFIDENCE BOUNDARIES
421-444
Jan Peter
Hessling
SP Technical Research Institute of Sweden, Measurement Technology, Box 857, SE-50115 Boras, Sweden
Thomas
Svensson
SP Technical Research Institute of Sweden, Building Technology and Mechanics, Box 857, SE-50115 Boras, Sweden
A new class of methods for propagation of uncertainty through complex models nonlinear-in-parameters is proposed. It is derived from a recent idea of propagating covariance within the unscented Kalman filter. The nonlinearity could be due to a pole-zero parametrization of a dynamic model in the Laplace domain, finite element model (FEM) or other large computer models, models of mechanical fatigue etc. Two approximate methods of this class are evaluated against Monte Carlo simulations and compared to the application of the Gauss approximation formula. Three elementary static models illustrate pros and cons of the methods, while one dynamic model provides a realistic simple example of its use.
ASYMPTOTICALLY INDEPENDENT MARKOV SAMPLING: A NEW MARKOV CHAIN MONTE CARLO SCHEME FOR BAYESIAN INFERENCE
445-474
James L.
Beck
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
Konstantin M.
Zuev
Department of Computing and Mathematical Sciences, Division of Engineering and Applied Science, 1200 E California Blvd., California Institute of Technology, Pasadena, California 91125, USA
In Bayesian inference, many problems can he expressed as the evaluation of the expectation of an uncertain quantity of interest with respect to the posterior distribution based on relevant data. Standard Monte Carlo method is often not applicable because the encountered posterior distributions cannot be sampled directly. In this case, the most popular strategies are the importance sampling method, Markov chain Monte Carlo, and annealing. In this paper, we introduce a new scheme for Bayesian inference, called asymptotically independent Markov sampling (AIMS), which is based on the above methods. We derive important ergodic properties of AIMS. In particular, it is shown that, under certain conditions, the AIMS algorithm produces a uniformly ergodic Markov chain. The choice of the free parameters of the algorithm is discussed and recommendations are provided for this choice, both theoretically and heuristically based. The efficiency of AIMS is demonstrated with three numerical examples, which include both multimodal and higher-dimensional target posterior distributions.