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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO

In this work, we provide a new approach for the efficient matrix-free application of the transpose of the Jacobian for the spectral element method for the adjoint-based solution of partial differential equation (PDE) constrained optimization. This results in optimizations of nonlinear PDEs using explicit integrators where the integration of the adjoint problem is not more expensive than the forward simulation. Solving PDE constrained optimization problems entails combining expertise from multiple areas, including simulation, computation of derivatives, and optimization. The Portable, Extensible Toolkit for Scientific computation (PETSc) together with its companion package, the Toolkit for Advanced Optimization (TAO), is an integrated numerical software library that contains an algorithmic/software stack for solving linear systems, nonlinear systems, ordinary differential equations, differential algebraic equations, and large-scale optimization problems and, as such, is an ideal tool for performing PDE-constrained optimization. This paper describes an efficient approach in which the software stack provided by PETSc/TAO can be used for large-scale nonlinear time-dependent problems. Time integration can involve a range of high-order methods, both implicit and explicit. The PDE-constrained optimization algorithm used is gradient-based and seamlessly integrated with the simulation of the physical problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Solving differential‐algebraic equations in power system dynamic analysis with quantum computing

Abstract Power system dynamics are generally modeled by high dimensional non‐linear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system non‐linearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.

computational complexity↗
DOE OSTI2024

Two-loop master integrals for leading-color $$ pp\to t\overline{t}H $$ amplitudes with a light-quark loop

Abstract We compute the two-loop master integrals for leading-color QCD scattering amplitudes including a closed light-quark loop in$$ t\overline{t}H $$ t t ¯ H production at hadron colliders. Exploiting numerical evaluations in modular arithmetic, we construct a basis of master integrals satisfying a system of differential equations inϵ-factorized form. We present the analytic form of the differential equations in terms of a minimal set of differential one-forms. We explore properties of the function space of analytic solutions to the differential equations in terms of iterative integrals which can be exploited for studying the analytic form of related scattering amplitudes. Finally, we solve the differential equations using generalized series expansions to numerically evaluate the master integrals in physical phase space. As the first computation of a set of two-loop seven-scale master integrals, our results provide valuable input for analytic studies of scattering amplitudes in processes involving massive particles and a large number of kinematic scales.

Physics↗
DOE OSTI2024

BPZ equations for higher degenerate fields and nonperturbative Dyson-Schwinger equations

In the two-dimensional Liouville conformal field theory, correlation functions involving a degenerate field satisfy partial differential equations due to the decoupling of the null descendant field. On the other hand, the instanton partition function of a four-dimensional N = 2 supersymmetric theory in the Ω -background at a special point of the parameter space also satisfies a partial differential equation resulting from the constraints of the gauge field configurations. This partial differential equation can be proved using the nonperturbative Dyson-Schwinger equations. We show for the next-to-simplest case that the partial differential equations obtained from two different perspectives can be identified, thereby confirming an assertion of the Bogomol’nyi-Prasad-Sommerfield/conformal field theory correspondence. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
DOE OSTI2021

Vibration-Based Sensor Design: A Grey-Box Approach

Knowledge of the internal structure of an object or device under investigation proceeds from the basic idea of constructing its dynamic behavioral relations governed by a set of differential/algebraic equations that characterize its response. These equations can be partial differential equations leading to finite element or finite difference relations requiring a complex numerical solution on a super computer or ordinary differential equations requiring sophisticated numerical integration techniques to obtain the desired solution. Discrete dynamic systems evolving from digitized data acquisition are typically captured by sampled-data (continuous-to-discrete) representations characterized by a set of difference equations specifying the underlying system dynamics. In any case, with a mathematical description in hand, Grey-Box modeling techniques have evolved, concerned with the estimation of model parameters embedded in a prescribed set of equations (the system) governing its behavior, while capturing the underlying physical phenomenology of the problem at hand.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An extended trajectory-mechanics approach for calculating two-phase flow paths

A technique originating in quantum dynamics is used to derive a trajectory-based, semi-analytical solution for two-phase flow. The partial differential equation governing the evolution of the aqueous phase is equivalent to a family of ordinary differential equations defined along a path through the porous medium. The trajectories may be found by solving the differential equations directly or by post-processing the output of a numerical solution to the full set of governing equations. The trajectories, which differ from conventional streamlines, are found to bend downward in response to gravitational forces. The curvature is more pronounced as the dip of the porous layer containing the flow increases. Subtle changes in the relative permeability curve can lead to significant variations in the trajectories. The ordinary differential equation for the trajectory provides an expression for the travel time along the path. The expression produces a semi-analytical approximation to the model parameter sensitivities, the partial derivatives of the travel times with respect to changes in the permeability model. The semi-analytical trajectory-based sensitivities generally agree with those computed using a numerical reservoir simulator and a perturbation approach. The sensitivities are useful in tomographic imaging algorithms designed to estimate the spatial variation in permeability within a porous medium using multiphase observations.

58 GEOSCIENCES↗
DOE OSTI2024

Accurate data-driven surrogates of dynamical systems for forward propagation of uncertainty

Stochastic collocation (SC) is a well-known non-intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full-field uncertainty propagation that characterizes the distributions of the high-dimensional solution fields of a model with stochastic input parameters. However, due to the highly nonlinear nature of the parameter-to-solution map in even the simplest dynamical systems, the constructed SC surrogates are often inaccurate. Here, this work presents an alternative approach, where we apply the SC approximation over the dynamics of the model, rather than the solution. By combining the data-driven sparse identification of nonlinear dynamics framework with SC, we construct dynamics surrogates and integrate them through time to construct the surrogate solutions. We demonstrate that the SC-over-dynamics framework leads to smaller errors, both in terms of the approximated system trajectories as well as the model state distributions, when compared against full-field SC applied to the solutions directly. We present numerical evidence of this improvement using three test problems: a chaotic ordinary differential equation, and two partial differential equations from solid mechanics.

42 ENGINEERING↗
DOE OSTI2022

A Realistic Theory of Quantum Measurement

Abstract We propose that the ontic understanding of quantum mechanics can be extended to a fully realistic theory that describes the evolution of the wavefunction at all times, including during a measurement. In such an approach the wave equation should reduce to the standard wave equation when there is no measurement, and describe state reduction when the system is measured. The general wave equation must be nonlinear and nonlocal, and we require it to be time-symmetric; consequently, this approach is not a new interpretation but a new theory. The wave equation is an integrodifferential equation (IDE). The time symmetry requirement leads to a retrocausal approach, in which the wave equation is solved subject to initial and final conditions to determine history at intermediate times. We propose that different outcomes from (apparently) identically prepared experiments may result from uncontrolled parameters; both the nonlocality and the retrocausality of the theory imply that Bell’s Theorem cannot rule out such “hidden variables.” Beginning with Hamilton’s principle, we demonstrate the construction of such a theory by replacing the action with a functional designed to give rise to a nonlinear, nonlocal IDE as the wave equation. This IDE reduces to the standard wave equation (a differential equation) in the absence of a measurement, but exhibits state reduction to a single eigenvalue when the system interacts with another system with the properties of a measurement apparatus. We demonstrate several desirable features of this theory; for other properties we indicate their plausibility and possible avenues to a proof.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Predicting rare events using neural networks and short-trajectory data

Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental dynamics, accurate prediction from direct observations becomes challenging. In such cases a more effective approach is to cast statistics of interest as solutions to Feynman-Kac equations (partial differential equations). Here, we develop an approach to solve Feynman-Kac equations by training neural networks on short-trajectory data. Our approach is based on a Markov approximation but otherwise avoids assumptions about the underlying model and dynamics. This makes it applicable to treating complex computational models and observational data. Additionally, we illustrate the advantages of our method using a low-dimensional model that facilitates visualization, and this analysis motivates an adaptive sampling strategy that allows on-the-fly identification of and addition of data to regions important for predicting the statistics of interest. Finally, we demonstrate that we can compute accurate statistics for a 75-dimensional model of sudden stratospheric warming. This system provides a stringent test bed for our method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Kinematic flow from the flow of cuts

The wavefunction coefficients of conformally coupled scalars in power-law FRW cosmologies satisfy differential equations governed by a set of simple combinatorial rules known as the kinematic flow. In this paper we derive the kinematic flow, expressed using a set of differential forms referred to as the cut basis, from a geometric perspective, relying solely on the cosmological hyperplane arrangement and without invoking bulk physics. Each element of the cut basis corresponds to the positive geometry associated to an independent cut of the physical FRW-form and can be labeled by decorating (minors of) the truncated Feynman graph with an acyclic orientation. We provide a straightforward prescription to associate a logarithmic differential form to each element of the cut basis by considering its corresponding decorated graph. Moreover, we show that the residues of the physical FRW-form are canonical forms of certain graphical zonotopes labeled by the same set of decorated graphs. These zonotopes control the cut combinatorics -- flow of cuts -- of the physical FRW-form and the cut basis (by construction). Using the theory of relative twisted cohomology and intersection theory, we derive a closed form formula for the differential equations of the cut basis. We also introduce combinatorial rules that compute the kinematic differential of any basis element without explicit calculation. The combinatorics of our differential equations is a natural consequence of the flow of cuts and is equivalent (up to rescaling) to the kinematic flow for the recently studied time integral basis. In particular, our differential equations decouple into exponentially many sectors, one for each way of cutting a subset of edges of the graph.

General Relativity and Quantum Cosmology↗
DOE OSTI2025

Kinematic flow for cosmological loop integrands

Recently, an interesting pattern was found in the differential equations satisfied by the Feynman integrals describing tree-level correlators of conformally coupled scalars in a power-law FRW cosmology [1, 2]. It was proven that simple and universal graphical rules predict the equations for arbitrary graphs as a flow in kinematic space. In this note, we show that the same rules — with one small addition — also determine the differential equations for loop integrands. We explain that both the basis of master integrals and the singularities of the differential equations can be represented by tubings of marked graphs. An important novelty in the case of loops is that some basis functions can vanish, and we present a graphical rule to identify these vanishing functions. Taking this into account, we then demonstrate that the kinematic flow correctly predicts the differential equations for all loop integrands.

Cosmological models↗
DOE OSTI2021

Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully-resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary application areas – subsurface transport, turbulence, and anomalous materials – in which fractional-order differential equations provide accurate and predictive models. For each area, we report on the evidence of anomalous behavior that justifies the use of fractional-order models, and survey both foundational models as well as more expressive state-of-the-art models. We also propose avenues for future research, including more advanced and physically sound models, as well as tools for calibration and discovery of fractional-order models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Optimal sensor network design for multi-scale, time-varying differential algebraic equation systems: Application to an entrained-flow gasifier refractory brick

An algorithm for optimal sensor network design for multi-scale, time-varying differential algebraic equation systems with non-separable dynamics is presented. As the process is time-varying, an integral normalized posterior error covariance of a multi-scale filter is minimized to obtain the optimal sensor locations. For reducing the computational cost, an adaptive sampling rate approach is considered for the slowly-varying variables. The algorithm is applied to a smart refractory brick with embedded sensors as part of an entrained-flow gasifier. Thermistors and interdigital capacitors are considered as candidate measurement technologies for estimating temperature and slag penetration profile along the gasifier wall. Finally, when the optimal set of sensors obtained from the algorithm is used for estimating temperature and slag penetration profiles in a multi-scale Kalman filter framework, satisfactory estimates are obtained despite high measurement noise and model mismatch.

42 ENGINEERING↗
DOE OSTI2021

TINES - Time Integration, Newton and Eigen Solver v. 1.0

SAND2021-1505 O. TINES is an open source software providing math infrastructure for solving many stiff time ordinary differential equations (ODEs) and/or differential algebraic equations (DAEs) using a batch hierarchical parallelism. The code is written using a parallel programming model (i.e., Kokkos) to future-proof the next generation parallel computing platforms such as GPU accelerators. This code is developed to support Exascale Catalytic Chemistry (ECC) Project. The code provides fundamental math helpers that can aid other research projects. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Kim, Kyungjoo↗
DOE OSTI2023

Quantum algorithm for the linear Vlasov equation with collisions

The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. Further, this representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Efficient Bayesian inference with latent Hamiltonian neural networks in No-U-Turn Sampling

When sampling for Bayesian inference, one popular approach in the computational field is to use Hamiltonian Monte Carlo (HMC) and specifically the No-U-Turn Sampler (NUTS), which automatically decides the end time of the Hamiltonian trajectory. However, HMC and NUTS can require numerous numerical gradients of the target density and can prove slow in practice when relying on computationally expensive forward models. We propose Latent Hamiltonian neural networks (L-HNNs) with HMC and NUTS for solving Bayesian inference problems. Once trained, L-HNNs do not require numerical gradients of the target density during sampling, and hence numerous evaluations of the forward computational model. Moreover, L-HNNs satisfy important properties such as perfect time reversibility and Hamiltonian conservation, making them well-suited for use within HMC and NUTS because stationarity can be shown. We also propose the integration of L-HNNs in an online error monitoring scheme, in which numerical gradients of the target density are used for a few samples whenever the L-HNNs prediction errors are large. This online error monitor scheme prevents sample degeneracy in regions of low probability density and ensures robust uncertainty quantification. We demonstrate L-HNNs in NUTS with online error monitoring on several analytical examples involving complex, heavy-tailed, and high-local-curvature probability densities. We then demonstrate the applicability of L-HNNs in NUTS to two computational case studies, namely the Allen-Cahn stochastic partial differential equation and an elliptic partial differential equation with 25 and 50 inference parameters, respectively. Overall, the L-HNNs in NUTS with online error monitoring satisfactorily inferred these probability densities. In conclusion, compared to traditional NUTS, L-HNNs in NUTS with online error monitoring required 1–2 orders of magnitude fewer numerical gradients of the target density and improved the effective sample size (ESS) per gradient (which is a measure of both the sampling quality and the computational expense) by an order of magnitude.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Adaptive Space-Time Methods for Large Scale Optimal Design

When modeling complex physical systems with advanced dynamics, such as shocks and singularities, many classic methods for solving partial differential equations can return inaccurate or unusable results. One way to resolve these complex dynamics is through r-adaptive refinement methods, in which a fixed number of mesh points are shifted to areas of high interest. The mesh refinement map can be found through the solution of the Monge-Ampére equation, a highly nonlinear partial differential equation. Due to its nonlinearity, the numerical solution of the Monge-Ampére equation is nontrivial and has previously required computationally expensive methods. In this report, we detail our novel optimization-based, multigrid-enabled solver for a low-order finite element approximation of the Monge-Ampére equation. This fast and scalable solver makes r-adaptive meshing more readily available for problems related to large-scale optimal design. Beyond mesh adaptivity, our report discusses additional applications where our fast solver for the Monge-Ampére equation could be easily applied.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗