Ernesto E. Prudencio

Bayesian Uncertainty Analysis with Applications to Turbulence Modeling

Abstract In this paper, we apply Bayesian uncertainty quantification techniques to the processes of calibrating complex mathematical models and predicting quantities of interest (QoIs) with such models. These techniques also enable the systematic comparison of competing model classes. The processes of calibration and comparison constitute the building blocks of a larger validation process, the goal of which is to accept or reject a given mathematical model for the prediction of a particular QoI for a particular scenario. In this work, we take the first step in this process by applying the methodology to the analysis of the Spalart–Allmaras turbulence model in the context of incompressible, boundary layer flows. Three competing model classes based on the Spalart–Allmaras model are formulated, calibrated against experimental data, and used to issue a prediction with quantified uncertainty. The model classes are compared in terms of their posterior probabilities and their prediction of QoIs. The model posterior probability represents the relative plausibility of a model class given the data. Thus, it incorporates the models ability to fit experimental observations. Alternatively, comparing models using the predicted QoI connects the process to the needs of decision makers that use the results of the model. We show that by using both the model plausibility and predicted QoI, one has the opportunity to reject some model classes after calibration, before subjecting the remaining classes to additional validation challenges.

SELECTION AND ASSESSMENT OF PHENOMENOLOGICAL MODELS OF TUMOR GROWTH

We address general approaches to the rational selection and validation of mathematical and computational models of tumor growth using methods of Bayesian inference. The model classes are derived from a general diffuse-interface, continuum mixture theory and focus on mass conservation of mixtures with up to four species. Synthetic data are generated using higher-order base models. We discuss general approaches to model calibration, validation, plausibility, and selection based on Bayesian-based methods, information theory, and maximum information entropy. We also address computational issues and provide numerical experiments based on Markov chain Monte Carlo algorithms and high performance computing implementations.

The parallel c++ statistical library 'QUESO': quantification of uncertainty for estimation, simulation and optimization

QUESO is a collection of statistical algorithms and programming constructs supporting research into the uncertainty quantification (UQ) of models and their predictions. It has been designed with three objectives: it should (a) be sufficiently abstract in order to handle a large spectrum of models, (b) be algorithmically extensible, allowing an easy insertion of new and improved algorithms, and (c) take advantage of parallel computing, in order to handle realistic models. Such objectives demand a combination of an object-oriented design with robust software engineering practices. QUESO is written in C++, uses MPI, and leverages libraries already available to the scientific community. We describe some UQ concepts, present QUESO, and list planned enhancements.

Parallel Full Space SQP Lagrange--Newton--Krylov--Schwarz Algorithms for PDE-Constrained Optimization Problems

Optimization problems constrained by nonlinear partial differential equations have been the focus of intense research in scientific computing lately. Current methods for the parallel numerical solution of such problems involve sequential quadratic programming (SQP), with either reduced or full space approaches. In this paper we propose and investigate a class of parallel full space SQP Lagrange--Newton--Krylov--Schwarz (LNKSz) algorithms. In LNKSz, a Lagrangian functional is formed and differentiated to obtain a Karush--Kuhn--Tucker (KKT) system of nonlinear equations. An inexact Newton method with line search is then applied. At each Newton iteration the linearized KKT system is solved with a Schwarz preconditioned Krylov subspace method. We apply LNKSz to the parallel numerical solution of some boundary control problems of two-dimensional incompressible Navier--Stokes equations. Numerical results are reported for different combinations of Reynolds number, mesh size, and number of parallel processors. We also compare the application of the LNKSz method to flow control problems against the application of the Newton--Krylov--Schwarz (NKSz) method to flow simulation problems.

Estimation of the nitrogen ionization reaction rate using electric arc shock tube data and Bayesian model analysis

In this paper, we apply a Bayesian analysis to calibrate the parameters of a model for atomic nitrogen ionization using experimental data from the electric arc shock tube (EAST) wind-tunnel at NASA. We use a one-dimensional plasma flow solver coupled with a radiation solver for the simulation of the radiative signature emitted in the shock-heated air plasma as well as a Park’s two-temperature model for the thermal and chemical non-equilibrium effects. We simultaneously quantify model parameter uncertainties and physical model inadequacies when solving the statistical inverse problem. Prior to the solution of such a problem, we perform a sensitivity analysis of the radiative heat flux in order to identify important sources of uncertainty. This analysis clearly shows the importance of the direct ionization of atomic nitrogen as it mostly influences the radiative heating. We then solve the statistical inverse problem and compare the calibrated reaction rates against values available in the literature. Our ca...

Letter to the Editor: Probabilistic models and uncertainty quantification for the ionization reaction rate of atomic Nitrogen

The objective in this paper is to analyze some stochastic models for estimating the ionization reaction rate constant of atomic Nitrogen (N+e^-->N^++2e^-). Parameters of the models are identified by means of Bayesian inference using spatially resolved absolute radiance data obtained from the Electric Arc Shock Tube (EAST) wind-tunnel. The proposed methodology accounts for uncertainties in the model parameters as well as physical model inadequacies, providing estimates of the rate constant that reflect both types of uncertainties. We present four different probabilistic models by varying the error structure (either additive or multiplicative) and by choosing different descriptions of the statistical correlation among data points. In order to assess the validity of our methodology, we first present some calibration results obtained with manufactured data and then proceed by using experimental data collected at EAST experimental facility. In order to simulate the radiative signature emitted in the shock-heated air plasma, we use a one-dimensional flow solver with Parks two-temperature model that simulates non-equilibrium effects. We also discuss the implications of the choice of the stochastic model on the estimation of the reaction rate and its uncertainties. Our analysis shows that the stochastic models based on correlated multiplicative errors are the most plausible models among the four models proposed in this study. The rate of the atomic Nitrogen ionization is found to be (6.2+/-3.3)x10^1^1cm^3mol^-^1s^-^1 at 10,000K.

Parallel Multilevel Restricted Schwarz Preconditioners with Pollution Removing for PDE-Constrained Optimization

We develop a class of V-cycle-type multilevel restricted additive Schwarz (RAS) methods and study the numerical and parallel performance of the new fully coupled methods for solving large sparse Jacobian systems arising from the discretization of some optimization problems constrained by nonlinear partial differential equations. Straightforward extensions of the one-level RAS to multilevel do not work due to the pollution effects of the coarse interpolation. We then introduce, in this paper, a pollution removing coarse-to-fine interpolation scheme for one of the components of the multicomponent linear system and show numerically that the combination of the new interpolation scheme with the RAS smoothed multigrid method provides an effective family of techniques for solving rather difficult PDE-constrained optimization problems. Numerical examples involving the boundary control of incompressible Navier-Stokes flows are presented in detail.

On the (in)validation of a thermochemical model with EAST shock tube radiation measurements

Kenji Miki∗, Marco Panesi∗, Ernesto E. Prudencio†, Andre Maurente∗, Sai Hung Cheung∗, Jeremy Jagodzinski∗, David Goldstein‡, Serge Prudhomme§, Karl Schulz¶, Chris Simmons‖, James Strand∗∗ , and Philip Varghese†† Center for Predictive Engineering and Computational Sciences (PECOS), Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 1 University Station C0200, Austin, Texas 78712, USA

Calibration of rates parameters for multi-temperature models using Bayesian formulation

In this paper, we apply a Bayesian analysis to calibrate the parameters of a model for atomic Nitrogen ionization using experimental data from the Electric Arc Shock Tube (EAST, from NASA) wind-tunnel. We use a one-dimensional ow solver coupled with a radiation solver for the simulation of the radiative signature emitted in the shock-heated air plasma, as well as a Park’s two-temperature model for the thermal and chemical nonequilibrium eects. We simultaneously quantify model parameter uncertainties and physical model inadequacies when solving the statistical inverse problem. Prior to the solution of such a problem, we perform a sensitivity analysis of the radiative heat ux in order to identify important sources of uncertainty. This analysis clearly shows the importance of the direct ionization of atomic Nitrogen as it mostly inuences the radiative heating. We then solve the statistical inverse problem and compare the calibrated reaction rates against values available in the literature. Our calculations estimate the reaction rate of the atomic Nitrogen ionization to be (3:7 1:5) 10 11 cm 3 mol 1 s 1 at 10,000 K, a range consistent with Park’s estimation. Finally, in order to assess the validity of the estimated parameters, we propagate their uncertainties through a statistical forward problem dened on a prediction scenario dierent from the calibration scenarios and compare the model predictions against other experimental data.

On the use of Bayesian Approach, Experimental Data, and Articially Designed Data, for the Identication of Missing Reactions

We propose and analyze the use of a Bayesian approach for the investigation of missing reactions, considering both modeling and experimental uncertainties. The main idea is the use of two calibration data sets: one is experimental and the other is articially designed