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.