In a recent paper, the authors proposed a general methodology for probabilistic learning on manifolds. The method was used to generate numerical samples that are statistically consistent with an existing dataset construed as a realization from a non-Gaussian random vector. The manifold structure is learned using diffusion manifolds and the statistical sample generation is accomplished using a projected Itô stochastic differential equation. This probabilistic learning approach has been extended to polynomial chaos representation of databases on manifolds and to probabilistic nonconvex constrained optimization with a fixed budget of function evaluations. The methodology introduces an isotropic-diffusion kernel with hyperparameter ε. Currently, ε is more or less arbitrarily chosen. In this paper, we propose a selection criterion for identifying an optimal value of ε, based on a maximum entropy argument. The result is a comprehensive, closed, probabilistic model for characterizing data sets with hidden constraints. This entropy argument ensures that out of all possible models, this is the one that is the most uncertain beyond any specified constraints, which is selected. Applications are presented for several databases.
A stable explicit time-scale splitting algorithm for stiff chemical Langevin equations (CLEs) is developed, based on the concept of computational singular perturbation. The drift term of the CLE is projected onto basis vectors that span the fast and slow subdomains. The corresponding fast modes exhaust quickly, in the mean sense, and the system state then evolves, with a mean drift controlled by slow modes, on a random manifold. The drift-driven time evolution of the state due to fast exhausted modes is modeled algebraically as an exponential decay process, while that due to slow drift modes and diffusional processes is integrated explicitly. This allows time integration step sizes much larger than those required by typical explicit numerical methods for stiff stochastic differential equations. The algorithm is motivated and discussed, and extensive numerical experiments are conducted to illustrate its accuracy and stability with a number of model systems.
We describe the load flow formulation and the solution algorithms available in the Electric power Grid Simulator (EGSim) software toolkit. EGSim contains tools aimed at simulating static load flow solutions for electric power grids. It parses power grid models described in IEEE Common Data Format, and generates solutions for the bus voltages and voltage angles, and real and reactive power values through the transmission lines. The software, written in C++, implements both Gauss-Seidel and Newton solution methods. Example results for the 118 bus models and 300 bus models are also presented.
Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.
Model error estimation remains one of the key challenges in uncertainty quantification and predictive science. For computational models of complex physical systems, model error, also known as structural error or model inadequacy, is often the largest contributor to the overall predictive uncertainty. This work builds on a recently developed framework of embedded, internal model correction, in order to represent and quantify structural errors, together with model parameters,within a Bayesian inference context. We focus specifically on a Polynomial Chaos representation with additive modification of existing model parameters, enabling a non-intrusive procedure for efficient approximate likelihood construction, model error estimation, and disambiguation of model and data errors’ contributions to predictive uncertainty. The framework is demonstrated on several synthetic examples, as well as on a chemical ignition problem.
Model error estimation remains one of the key challenges in uncertainty quantification and predictive science. For computational models of complex physical systems, model error, also known as structural error or model inadequacy, is often the largest contributor to the overall predictive uncertainty. This work builds on a recently developed frame- work of embedded, internal model correction, in order to represent and quantify structural errors, together with model parameters, within a Bayesian inference context.We focus specifically on a polynomial chaos representation with addi- tive modification of existing model parameters, enabling a nonintrusive procedure for efficient approximate likelihood construction, model error estimation, and disambiguation of model and data errors’ contributions to predictive uncer- tainty. The framework is demonstrated on several synthetic examples, as well as on a chemical ignition problem.
A procedure for determining the joint uncertainty of Arrhenius parameters across multiple combustion reactions of interest is demonstrated. This approach is capable of constructing the joint distribution of the Arrhenius parameters arising from the uncertain measurements performed in specific target experiments without having direct access to the underlying experimental data. The method involves constructing an ensemble of hypothetical data sets with summary statistics consistent with the available information reported by the experimentalists, followed by a fitting procedure that learns the structure of the joint parameter density across reactions using this consistent hypothetical data as evidence. The procedure is formalized in a Bayesian statistical framework, employing maximum-entropy and approximate Bayesian computation methods and utilizing efficient Markov chain Monte Carlo techniques to explore data and parameter spaces in a nested algorithm. We demonstrate the application of the method in the context of experiments designed to measure the rates of selected chain reactions in the H2-O2 system and highlight the utility of this approach for revealing the critical correlations between the parameters within a single reaction and across reactions, as well as for maximizing consistency when utilizing rate parameter information in predictive combustion modeling of systems of interest.
We present an overview of optimization under uncertainty efforts under the DARPA Enabling Quantification of Uncertainty in Physical Systems (EQUiPS) ScramjetUQ project. We introduce the mathematical frameworks and computational tools employed for performing this task. In particular, we provide details in the optimization and multilevel uncertainty quantification algorithms, which are available through the SNOWPAC and DAKOTA software packages. The overall workflow is first demonstrated on a simplified model design problem with non-reacting inviscid supersonic flows. Preliminary results and updates are then reported for a in-progress scramjet design optimization case using large-eddy simulations of supersonic reactive flows inside the HIFiRE Direct Connect Rig.
The development of scramjet engines is crucial for attaining efficient and stable propulsion under hypersonic flight conditions. Design for well-performing scramjet engines requires accurate flow simulations in conjunction with uncertainty quantification (UQ). We advance computational methods in bringing together UQ and large-eddy simulations for scramjet computations, with a focus on the HIFiRE Direct Connect Rig combustor. In particular, we perform uncertainty propagation for spatially dependent field quantities of interest (QoIs) by treating them as random fields, and numerically compute low-dimensional Karhunen-Loève expansions (KLEs) using a finite number of simulations on non-uniform grids. We also describe a formulation and procedure to extract conditional KLEs that characterize the stochasticity induced by uncertain parameters at given designs. This is achieved by first building a single KLE for each QoI via samples drawn jointly from the parameter and design spaces, and then leverage polynomial chaos expansions to insert input dependencies into the KLE. The ability to access conditional KLEs will be immensely useful for subsequent efforts in design optimization under uncertainty as well as model calibration with field variable measurements.
The computational burden of a large-eddy simulation for reactive flows is exacerbated in the presence of uncertainty in flow conditions or kinetic variables. A comprehensive statistical analysis, with a sufficiently large number of samples, remains elusive. Statistical learning is an approach that allows for extracting more information using fewer samples. Such procedures, if successful, will greatly enhance the predictability of models in the sense of improving exploration and characterization of uncertainty due to model error and input dependencies, all while being constrained by the size of the associated statistical samples. In this paper, it is shown how a recently developed procedure for probabilistic learning on manifolds can serve to improve the predictability in a probabilistic framework of a scramjet simulation. The estimates of the probability density functions of the quantities of interest are improved together with estimates of the statistics of their maxima. It is also demonstrated how the improved statistical model adds critical insight to the performance of the model.
In this work, we provide a method for enhancing stochastic Galerkin moment calculations to the linear elliptic equation with random diffusivity using an ensemble of Monte Carlo solutions. This hybrid approach combines the accuracy of low-order stochastic Galerkin and the computational efficiency of Monte Carlo methods to provide statistical moment estimates which are significantly more accurate than performing each method individually. The hybrid approach involves computing a low-order stochastic Galerkin solution, after which Monte Carlo techniques are used to estimate the residual. We show that the combined stochastic Galerkin solution and residual is superior in both time and accuracy for a one-dimensional test problem and a more computational intensive two-dimensional linear elliptic problem for both the mean and variance quantities.
This study explores a Bayesian calibration framework for the RAMPAGE alloy potential model for Cu-Ni and Cu-Zr systems, respectively. In RAMPAGE potentials, it is proposed that once calibrated potentials for individual elements are available, the inter-species interactions can be described by fitting a Morse potential for pair interactions with three parameters, while densities for the embedding function can be scaled by two parameters from the elemental densities. Global sensitivity analysis tools were employed to understand the impact each parameter has on the MD simulation results. A transitional Markov Chain Monte Carlo algorithm was used to generate samples from the multimodal posterior distribution consistent with the discrepancy between MD simulation results and DFT data. For the Cu-Ni system the posterior predictive tests indicate that the fitted interatomic potential model agrees well with the DFT data, justifying the basic RAMPAGE assumptions. For the Cu-Zr system, where the phase diagram suggests more complicated atomic interactions than in the case of Cu-Ni, the RAMPAGE potential captured only a subset of the DFT data. The resulting posterior distribution for the 5 model parameters exhibited several modes, with each mode corresponding to specific simulation data and a suboptimal agreement with the DFT results.
This investigation tackles the probabilistic parameter estimation problem involving the Arrhenius parameters for the rate coefficient of the chain branching reaction H + O2 → OH + O. This is achieved in a Bayesian inference framework that uses indirect data from the literature in the form of summary statistics by approximating the maximum entropy solution with the aid of approximate bayesian computation. The summary statistics include nominal values and uncertainty factors of the rate coefficient, obtained from shock-tube experiments performed at various initial temperatures. The Bayesian framework allows for the incorporation of uncertainty in the rate coefficient of a secondary reaction, namely OH + H2 → H2O + H, resulting in a consistent joint probability density on Arrhenius parameters for the two rate coefficients. It also allows for uncertainty quantification in numerical ignition predictions while conforming with the published summary statistics. The method relies on probabilistic reconstruction of the unreported data, OH concentration profiles from shock-tube experiments, along with the unknown Arrhenius parameters. The data inference is performed using a Markov chain Monte Carlo sampling procedure that relies on an efficient adaptive quadrature in estimating relevant integrals needed for data likelihood evaluations. For further efficiency gains, local Padé–Legendre approximants are used as surrogates for the time histories of OH concentration, alleviating the need for 0-D auto-ignition simulations. The reconstructed realisations of the missing data are used to provide a consensus joint posterior probability density on the unknown Arrhenius parameters via probabilistic pooling. Uncertainty quantification analysis is performed for stoichiometric hydrogen–air auto-ignition computations to explore the impact of uncertain parameter correlations on a range of quantities of interest.