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.
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.
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.
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.
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.
Here, compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several compressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers l1_ls, SpaRSA, CGIST, FPC_AS, and ADMM, we develop techniques to mitigate overfitting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendations on parameter settings for these techniques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-crossflow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy, and computational trade-offs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.
Increasing penetration levels of renewables have transformed how power systems are operated. High levels of uncertainty in production make it increasingly difficulty to guarantee operational feasibility; instead, constraints may only be satisfied with high probability. We present a chance-constrained economic dispatch model that efficiently integrates energy storage and high renewable penetration to satisfy renewable portfolio requirements. Specifically, we require that wind energy contribute at least a prespecified proportion of the total demand and that the scheduled wind energy is deliverable with high probability. We develop an approximate partial sample average approximation (PSAA) framework to enable efficient solution of large-scale chance-constrained economic dispatch problems. Computational experiments on the IEEE-24 bus system show that the proposed PSAA approach is more accurate, closer to the prescribed satisfaction tolerance, and approximately 100 times faster than standard sample average approximation. Finally, the improved efficiency of our PSAA approach enables solution of a larger WECC-240 test system in minutes.