Numerical algorithms for stiff stochastic differential equations are developed using linear approximations of the fast diffusion processes, under the assumption of decoupling between fast and slow processes. Three numerical schemes are proposed, all of which are based on the linearized formulation albeit with different degrees of approximation. The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems. Convergence analysis is conducted for one of the schemes, that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1. Approximations arriving at the other two schemes are discussed. Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.
Nonlocal models provide a much-needed predictive capability for important Sandia mission applications, ranging from fracture mechanics for nuclear components to subsurface flow for nuclear waste disposal, where traditional partial differential equations (PDEs) models fail to capture effects due to long-range forces at the microscale and mesoscale. However, utilization of this capability is seriously compromised by the lack of a rigorous nonlocal interface theory, required for both application and efficient solution of nonlocal models. To unlock the full potential of nonlocal modeling we developed a mathematically rigorous and physically consistent interface theory and demonstrate its scope in mission-relevant exemplar problems.
This article illustrates the use of unsupervised probabilistic learning techniques for the analysis of planetary reentry trajectories. A three-degree-of-freedom model was employed to generate optimal trajectories that comprise the training datasets. The algorithm first extracts the intrinsic structure in the data via a diffusion map approach. We find that data resides on manifolds of much lower dimensionality compared to the high-dimensional state space that describes each trajectory. Using the diffusion coordinates on the graph of training samples, the probabilistic framework subsequently augments the original data with samples that are statistically consistent with the original set. The augmented samples are then used to construct conditional statistics that are ultimately assembled in a path planning algorithm. In this framework, the controls are determined stage by stage during the flight to adapt to changing mission objectives in real-time.
A Bayesian inference strategy has been used to estimate uncertain inputs to global impurity transport code (GITR) modeling predictions of tungsten erosion and migration in the linear plasma device, PISCES-A. This allows quantification of GITR output uncertainty based on the uncertainties in measured PISCES-A plasma electron density and temperature profiles (ne, Te) used as inputs to GITR. The technique has been applied for comparison to dedicated experiments performed for high (4 × 1022 m–2 s–1) and low (5 × 1021 m–2 s–1) flux 250 eV He–plasma exposed tungsten (W) targets designed to assess the net and gross erosion of tungsten, and corresponding W impurity transport. The W target design and orientation, impurity collector, and diagnostics, have been designed to eliminate complexities associated with tokamak divertor plasma exposures (inclined target, mixed plasma species, re-erosion, etc) to benchmark results against the trace impurity transport model simulated by GITR. The simulated results of the erosion, migration, and re-deposition of W during the experiment from the GITR code coupled to materials response models are presented. Specifically, the modeled and experimental W I emission spectroscopy data for a 429.4 nm line and net erosion through the target and collector mass difference measurements are compared. Furthermore, the methodology provides predictions of observable quantities of interest with quantified uncertainty, allowing estimation of moments, together with the sensitivities to plasma temperature and density.
The TChem open-source software is a toolkit for computing thermodynamic properties, source term, and source term’s Jacobian matrix for chemical kinetic models that involve gas and surface reactions.
CSPlib is an open source software library for analyzing general ordinary differential equation (ODE) systems and detailed chemical kinetic ODE/DAE systems. It relies on the computational singular perturbation (CSP) method for the analysis of these systems.
We present a new geodesic-based method for geometry optimization in a basis set of redundant internal coordinates. Our method updates the molecular geometry by following the geodesic generated by a displacement vector on the internal coordinate manifold, which dramatically reduces the number of steps required to converge to a minimum. Our method can be implemented in any existing optimization code, requiring only implementation of derivatives of the Wilson B-matrix and the ability to numerically solve an ordinary differential equation.
This report investigates the use of unsupervised probabilistic learning techniques for the analysis of hypersonic trajectories. The algorithm first extracts the intrinsic structure in the data via a diffusion map approach. Using the diffusion coordinates on the graph of training samples, the probabilistic framework augments the original data with samples that are statistically consistent with the original set. The augmented samples are then used to construct conditional statistics that are ultimately assembled in a path-planing algorithm. In this framework the controls are determined stage by stage during the flight to adapt to changing mission objectives in real-time. A 3DOF model was employed to generate optimal hypersonic trajectories that comprise the training datasets. The diffusion map algorithm identfied that data resides on manifolds of much lower dimensionality compared to the high-dimensional state space that describes each trajectory. In addition to the path-planing worflow we also propose an algorithm that utilizes the diffusion map coordinates along the manifold to label and possibly remove outlier samples from the training data. This algorithm can be used to both identify edge cases for further analysis as well as to remove them from the training set to create a more robust set of samples to be used for the path-planing process.
This report outlines the mathematical formulation for the atomic environment vector (AEV) construction used in the aevmod software package. The AEV provides a summary of the geometry of a molecule or atomic configuration. We also present the formulation for the analytical Jacobian of the AEV with respect to the atomic Cartesian coordinates. The software provides functionality for both the AEV and AEV-Jacobian, as well as the AEV-Hessian which is available via reliance on the third party library Sacado.
There currently exist two methods for analysing an explosive mode introduced by chemical kinetics in a reacting process: the Computational Singular Perturbation (CSP) algorithm and the Chemical Explosive Mode Analysis (CEMA). CSP was introduced in 1989 and addressed both dissipative and explosive modes encountered in the multi-scale dynamics that characterize the process, while CEMA was introduced in 2009 and addressed only the explosive modes. It is shown that (i) the algorithmic tools incorporated in CEMA were developed previously on the basis of CSP and (ii) the examination of explosive modes has been the subject of CSP-based works, reported before the introduction of CEMA.
We have extended the computational singular perturbation (CSP) method to differential algebraic equation (DAE) systems and demonstrated its application in a heterogeneous-catalysis problem. The extended method obtains the CSP basis vectors for DAEs from a reduced Jacobian matrix that takes the algebraic constraints into account. We use a canonical problem in heterogeneous catalysis, the transient continuous stirred tank reactor (T-CSTR), for illustration. The T-CSTR problem is modelled fundamentally as an ordinary differential equation (ODE) system, but it can be transformed to a DAE system if one approximates typically fast surface processes using algebraic constraints for the surface species. We demonstrate the application of CSP analysis for both ODE and DAE constructions of a T-CSTR problem, illustrating the dynamical response of the system in each case. We also highlight the utility of the analysis in commenting on the quality of any particular DAE approximation built using the quasi-steady state approximation (QSSA), relative to the ODE reference case.
CSPlib is an open source software library for analyzing general ordinary differential equation (ODE) systems and detailed chemical kinetic ODE systems. It relies on the computational singular perturbation (CSP) method for the analysis of these systems. The software provides support for: General ODE models (gODE model class) for computing source terms and Jacobians for a generic ODE system; TChem model (ChemElemODETChem model class) for computing source term, Jacobian, other necessary chemical reaction data, as well as the rates of progress for a homogenous batch reactor using an elementary step detailed chemical kinetic reaction mechanism. This class relies on the TChem [2] library; A set of functions to compute essential elements of CSP analysis (Kernel class). This includes computations of the eigensolution of the Jacobian matrix, CSP basis vectors and co-vectors, time scales (reciprocals of the magnitudes of the Jacobian eigenvalues), mode amplitudes, CSP pointers, and the number of exhausted modes. This class relies on the Tines library; A set of functions to compute the eigensolution of the Jacobian matrix using Tines library GPU eigensolver; A set of functions to compute CSP indices (Index Class). This includes participation indices and both slow and fast importance indices.
Fundamental results and an efficient algorithm for constructing eigenvectors corresponding to non-zero eigenvalues of matrices with zero rows and/or columns are developed. The formulation is based on the relation between eigenvectors of such matrices and the eigenvectors of their submatrices after removing all zero rows and columns. While being easily implemented, the algorithm decreases the computation time needed for numerical eigenanalysis, and resolves potential numerical eigensolver instabilities.
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.
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.
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.
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.
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.