Results for “numerical differentiation” — Page 29

DOE OSTI2023

Control of Fractional Diffusion Problems via Dynamic Programming Equations

In this study, we explore the approximation of feedback control of integro-differential equations containing a fractional Laplacian term. To obtain feedback control for the state variable of this nonlocal equation, we use the Hamilton–Jacobi–Bellman equation. It is well known that this approach suffers from the curse of dimensionality, and to mitigate this problem we couple semi-Lagrangian schemes for the discretization of the dynamic programming principle with the use of Shepard approximation. This coupling enables approximation of high-dimensional problems. Numerical convergence toward the solution of the continuous problem is provided together with linear and nonlinear examples. The robustness of the method with respect to disturbances of the system is illustrated by comparisons with an open-loop control approach.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Multiscale modeling and nonlinear model predictive control for flue gas desulfurization

The primary source of sulfur dioxide (SO 2 ) emissions is flue gas from fossil fuels-based power plants. SO 2 emissions are known to not only cause health issues, but also have an adverse effect on the environment in various ways. Several Flue Gas Desulfurization (FGD) technologies have been incorporated in power plants. The most popular technology is Wet FGD, where a limestone slurry is used to absorb SO 2 from the flue gas. A detailed droplet scale model describing the instantaneous and finite rate chemistry is developed. The ill-posed Differential-Algebraic Equation (DAE) droplet model is reformulated to a well-posed index-1 DAE through index reduction. The droplet model is integrated with the bulk phase by incorporating gas-liquid mass transfer, and an oxidation reactor model to simulate the dynamic operation of the counter-current spray scrubber. As a result, the model has a well-conditioned Jacobian and overcomes the modeling challenges of previous works and enables numerical solution without requiring carefully selected initialization or specialized solution procedures. The model is successfully validated using power plant measurements, and nonlinear model predictive control (NMPC) studies are demonstrated to optimize recycle stream flowrates to minimize pumping costs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

The glass-ceiling convective regime and the origin and diversity of coronae on Venus

Venus and Earth are rocky planets of roughly the same size and bulk density, yet their surface volcanic and tectonic features appear substantially different. On Venus, the coexistence of large volcanic highlands—interpreted as the surface expression of long-lived mantle plumes—alongside coronae, smaller features thought to be caused by transient thermal diapirs, remains enigmatic. Using two-dimensional numerical models of mantle convection with sharp and broad mineral phase transitions for pyrolite, we show that both scales of upwellings can be generated in a stagnant lid planet with an interior temperature 250 to 400 K warmer than Earth’s. The smaller plumes originate from a ~600 km deep internal layer that exists as a consequence of the different sequence of mineral phase transitions that occur in warmer mantles less processed and differentiated by partial melting and volcanism. Future models that include melting will provide further tests of our hypothesis.

Science & Technology - Other Topics↗

DOE OSTI2020

Ice Nucleating Particles, Aerosols and Clouds over the Higher Latitude Southern Ocean

Improved understanding of aerosol-cloud-precipitation interactions is a key goal of the DOE-ASR program. Ice Nucleating Particles (INPs) are rare aerosol particles that trigger ice formation in clouds. By doing so they may initiate precipitation in clouds, and this impacts cloud lifetime. Nowhere may the role of INPs be manifested so dramatically as in the Southern Ocean (SO). This vast region was the focus of the DOE-ARM MARCUS (Measurement of Aerosols, Radiation and CloUds over the Southern Oceans) and MICRE (Macquarie Island Cloud and Radiation Experiment) campaigns. In the SO, a deficit of INPs has been a leading hypothesis for why liquid clouds persist, which underlays the bias of global climate models in predicting excess shortwave radiation reaching the ocean surface between 55°S and Antarctica. In this study, we used the 6-month acquisition of INP data on four MARCUS cruises from Hobart, Tasmania, to Antarctica, and the single-location full annual INP cycle observed in MICRE, to gain a holistic, quantitative picture of the INP number concentrations as a function of temperature and their variability over SO latitudes from 43°S to Antarctica. Augmenting initial processing of filter collections of aerosol particles during MARCUS and MICRE, additional freezing studies following thermal treatment (to removes INPs associated with microbes) and peroxide treatment (to remove all organics), as well as ionic chemistry and total organic carbon analysis, were applied to samples to improve time and space resolution, and differentiate marine from terrestrial contributions. We also performed Next Generation Sequencing to determine that bacterial composition can tag periods of enhanced and degraded marine organic INPs and the relative lack of occurrence of land-sourced bio-particles. Through these and other analyses, INP data were categorized into representative source types (e.g., marine versus terrestrially-influenced). The products of our analyses are being used to parameterize ice formation for use in numerical modeling studies to determine if, and when, ocean-derived INPs control the microphysical composition and radiative balance of SO clouds in a manner not presently captured by global models. The specific objectives and approaches proposed were largely accomplished, including: 1) Additional and value-added chemical and biological analyses of archived samples of aerosols were completed to categorize and quantify INP types and concentrations over the SO. 2) Analyses were completed to place INP data in meteorological and aerosol context. 3) Sea spray aerosol INPs were demonstrated to dominate the SO region in all but episodic events. 4) New parameterizations for INP sources relevant to the SO were constructed. 5) Collaborative modeling studies demonstrating the crucial role of INPs in determining cloud radiative and precipitation properties were conducted with MARCUS and MICRE partners. Collaborative publication submissions in this regard ensued within the two year study. This work will advance the science of interactions of aerosols, clouds and precipitation, to improve their representation in regional and global climate models. Results have informed representation of primary ice crystal nucleation, give inference to the role of secondary processes, and improve investigations of aerosol influences on clouds via ice nucleation over SO high latitudes and similar vast ocean regions. Application will improve representation of such interactions for clouds in both regional and global climate models. The data base and analyses methods applied will continue to serve research studies in the future.

58 GEOSCIENCES↗

DOE OSTI2021

A Continuous-Discontinuous Galerkin Method for Electromagnetic Simulations Based on an All-Frequency Stable Formulation

In this paper, a potential-based partial-differential formulation, called the all-frequency stable formulation, is presented for the accurate and robust simulation of electromagnetic problems at all frequencies. Due to its stability from (near) dc to microwave frequencies, this formulation can be applied to simulate wide-band and multiscale problems without encountering the infamous low-frequency breakdown issue or applying basis function decompositions such as the tree-cotree splitting technique. To provide both efficient and flexible numerical solutions to the electromagnetic formulation, a mixed continuous-discontinuous Galerkin (CDG) method is proposed and implemented. In regions with homogeneous media, the continuous Galerkin method is employed to avoid the introduction of duplicated degrees of freedom (DoFs) on the elemental interfaces, while on the interfaces of two different media, the discontinuous Galerkin method is applied to permit the jump of the normal components of the electromagnetic fields. Numerical examples are provided to validate and demonstrate the proposed numerical solver for problems in a wide electromagnetic spectrum.

Yan, Su↗

DOE OSTI2025

Power Flow Geometry and Approximation

Here, the power flow equations are important in numerous power systems problems of practical interest which consider alternating current power flow (ACPF) physics. Perhaps the most well studied being the alternating current optimal power flow problem (ACOPF), seeking to optimize the operation of an electric power system. Due to their non-linearity, problems which include the power flow equations are typically challenging, particularly in optimization. Interestingly, the set of solutions to the power flow equations forms a smooth manifold. As a result, differential geometry can be used to describe and analyze this set of equations. This approach has proven effective in several engineering applications (e.g., solving ACOPF and analyzing the solution space boundary). Central to the success of this approach is an understanding of the power flow manifold's geometry. In this work, we develop the geometric and topological properties of this manifold using concepts from differential geometry. After demonstrating the convenience of this manifold's representation as a function's graph, computational methods are emphasized: we develop retractions, error bounds for linear approximation, and formulas for evaluating the Riemannian metric (including associated objects such as geodesics and the curvature tensor). Scalar curvature and the second fundamental form play a new role in quantifying the quality of linear approximations, like the popular direct current approximation. All functions are implemented in Julia and available in an online repository. Proofs are included for completeness.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2024

Efficient general method for numerically modeling laser pulse propagation, overlap, and lifetime effects in amplifiers

An efficient numerical time-dependent general method is developed to address incoherent pulse overlap and lifetime effects in laser amplifiers. The alternating propagation-population laser energetics method (APPLE) has been validated against a semi-discrete coupled rate equation numerical method (SDRE) and analytic formalisms in bounding cases. APPLE is based on decoupled rates applied to a time-dependent framework where both space-time-dependent populations and pulse energetics are consistently updated in each time step. A significant advantage of APPLE lies in its conceptual simplicity, ease of implementation, and relatively small computational cost. SDRE tracks the populations through coupled rates and uses the method of lines to discretize the hyperbolic partial differential transport equations allowing for use of ordinary differential equation solvers. With reasonably sized mesh, we report both energetic and power pulse shape relative differences on the order of one percent between the models over a large range of initial conditions.

47 OTHER INSTRUMENTATION↗

DOE OSTI2021

xsacrcfrspccopol (a1)

ARM's scanning cloud radars are fully coherent dual-frequency, dual-polarization Doppler radars mounted on a common scanning pedestal. Each pedestal includes a Ka-band radar (2kW peak power) and the deployment location determines whether the second radar is a W-band (WSACR; 1.7 kW peak power) or X-band (XSACR; 20 kW peak power). At ARM's tropical sites, X-band radars are paired with the Ka-band because they are better suited for the atmospheric attenuation in this region. Beamwidth for the X-band is approximately 1 degree, and the Ka-band beamwidth is roughly 0.3 degrees. Due to the narrow antenna beamwidth, ARM’s scanning cloud radars use scanning strategies unlike typical weather radars. Rather than focusing on plan position indicator, or PPI, scans, the XSACR uses range height indicator, or RHI, scans at numerous azimuths to obtain cloud volume data. For the second ARM Mobile Facility (AMF2), the XSACR can be dismounted from the pedestal it shares with the KASACR and mounted on a separate pedestal. This allows the XSACR to operate more like a weather (precipitation) radar in deployments where such local coverage is lacking. Measurements collected with the XSACR are copolar and cross-polar radar reflectivity, Doppler velocity, spectra width and spectra when not scanning, differential Reflectivity (Zdr), correlation coefficient (rho-hv), and specific differential phase (phi-dp). XSACR data from the 2018–2019 Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign in Argentina are now available as b1-level products. Building on the original CACTI operational data, the b1-level products feature improved data quality resulting from extensive analyses and corrections. The data are cross-calibrated to a common point, datastreams are corrected for operational issues that occurred during the campaign, and several data quality masks and basic derived products are incorporated. For more information, read the CACTI radar b1-level processing report.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

xsacrcfrspcxpol (a1)

ARM's scanning cloud radars are fully coherent dual-frequency, dual-polarization Doppler radars mounted on a common scanning pedestal. Each pedestal includes a Ka-band radar (2kW peak power) and the deployment location determines whether the second radar is a W-band (WSACR; 1.7 kW peak power) or X-band (XSACR; 20 kW peak power). At ARM's tropical sites, X-band radars are paired with the Ka-band because they are better suited for the atmospheric attenuation in this region. Beamwidth for the X-band is approximately 1 degree, and the Ka-band beamwidth is roughly 0.3 degrees. Due to the narrow antenna beamwidth, ARM’s scanning cloud radars use scanning strategies unlike typical weather radars. Rather than focusing on plan position indicator, or PPI, scans, the XSACR uses range height indicator, or RHI, scans at numerous azimuths to obtain cloud volume data. For the second ARM Mobile Facility (AMF2), the XSACR can be dismounted from the pedestal it shares with the KASACR and mounted on a separate pedestal. This allows the XSACR to operate more like a weather (precipitation) radar in deployments where such local coverage is lacking. Measurements collected with the XSACR are copolar and cross-polar radar reflectivity, Doppler velocity, spectra width and spectra when not scanning, differential Reflectivity (Zdr), correlation coefficient (rho-hv), and specific differential phase (phi-dp). XSACR data from the 2018–2019 Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign in Argentina are now available as b1-level products. Building on the original CACTI operational data, the b1-level products feature improved data quality resulting from extensive analyses and corrections. The data are cross-calibrated to a common point, datastreams are corrected for operational issues that occurred during the campaign, and several data quality masks and basic derived products are incorporated. For more information, read the CACTI radar b1-level processing report.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

xsacrcfrspccross (a1)

ARM's scanning cloud radars are fully coherent dual-frequency, dual-polarization Doppler radars mounted on a common scanning pedestal. Each pedestal includes a Ka-band radar (2kW peak power) and the deployment location determines whether the second radar is a W-band (WSACR; 1.7 kW peak power) or X-band (XSACR; 20 kW peak power). At ARM's tropical sites, X-band radars are paired with the Ka-band because they are better suited for the atmospheric attenuation in this region. Beamwidth for the X-band is approximately 1 degree, and the Ka-band beamwidth is roughly 0.3 degrees. Due to the narrow antenna beamwidth, ARM’s scanning cloud radars use scanning strategies unlike typical weather radars. Rather than focusing on plan position indicator, or PPI, scans, the XSACR uses range height indicator, or RHI, scans at numerous azimuths to obtain cloud volume data. For the second ARM Mobile Facility (AMF2), the XSACR can be dismounted from the pedestal it shares with the KASACR and mounted on a separate pedestal. This allows the XSACR to operate more like a weather (precipitation) radar in deployments where such local coverage is lacking. Measurements collected with the XSACR are copolar and cross-polar radar reflectivity, Doppler velocity, spectra width and spectra when not scanning, differential Reflectivity (Zdr), correlation coefficient (rho-hv), and specific differential phase (phi-dp). XSACR data from the 2018–2019 Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign in Argentina are now available as b1-level products. Building on the original CACTI operational data, the b1-level products feature improved data quality resulting from extensive analyses and corrections. The data are cross-calibrated to a common point, datastreams are corrected for operational issues that occurred during the campaign, and several data quality masks and basic derived products are incorporated. For more information, read the CACTI radar b1-level processing report.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Solving high-dimensional partial integral differential equations: The finite expression method

Partial integro-differential equations (PIDEs) have broad applications in the sciences, from electro-magnetism to options pricing. Here, in this paper, we introduce a new finite expression method (FEX) to solve PIDEs. This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel method of parameter grouping is proposed to reduce the number of coefficients in high-dimensional function approximation; 2) A Taylor series approximation method is implemented to significantly improve the computational efficiency and accuracy of the evaluation of the integral terms of PIDEs. The new FEX based method, denoted FEX-PG to indicate the addition of the parameter grouping (PG) step to the algorithm, provides both high accuracy and interpretable numerical solutions, with the outcome being an explicit equation that facilitates intuitive understanding of the underlying solution structures. These features are often absent in traditional methods, such as finite element methods (FEM) and finite difference methods, as well as in deep learning-based approaches. To benchmark our method against recent advances, we apply the new FEX-PG to solve benchmark PIDEs in the literature. In high-dimensional settings, FEX-PG exhibits strong and robust performance, achieving relative errors on the order of single precision machine epsilon, significantly outperforming existing approaches based on neural networks.

Combinatorial optimization↗

DOE OSTI2021

Model-based and Model-free Designs for an Extended Continuous-time LQR with Exogenous Inputs

We present an extended linear quadratic regulator (LQR) design for continuous-time linear time-invariant (LTI) systems in the presence of exogenous inputs. We first propose a model-based solution with cost minimization guarantees for states and inputs using dynamic programming (DP). The control law consists of a combination of the optimal state feedback and an additional optimal term dependent on the exogenous inputs. The control gains for the two components are obtained by solving a set of matrix differential equations. We provide these solutions for both finite horizons and steady-state cases. In the second part of the paper, we formulate a reinforcement learning (RL) based algorithm which does not need any model information except the input matrix, and can compute an approximate steady-state LQR gain using measurements of the states, the control inputs, and the exogenous inputs. Both model-based and data-driven optimal control algorithms are tested with a numerical example under different exogenous inputs showcasing the effectiveness of the designs.

Mukherjee, Sayak↗

DOE OSTI2024

Direct, simple and efficient computation of all components of the virtual-casing magnetic field in axisymmetric geometries with Kapur–Rokhlin quadrature

In a recent publication (Toler et al., J. Plasma Phys., vol. 89, issue 2, 2023, p. 905890210), we demonstrated that for axisymmetric geometries, the Kapur–Rokhlin quadrature rule provided an efficient and high-order accurate method for computing the normal component, on the plasma surface, of the magnetic field due to the toroidal current flowing in the plasma, via the virtual-casing principle. The calculation was indirect, as it required the prior computation of the magnetic vector potential from the virtual-casing principle, followed by the computation of its tangential derivative by Fourier differentiation, to obtain the normal component of the magnetic field. Our approach did not provide the other components of the virtual-casing magnetic field. In this letter, we show that a more direct and more general approach is available for the computation of the virtual-casing magnetic field. The Kapur–Rokhlin quadrature rule accurately calculates the principal value integrals in the expression for all the components of the magnetic field on the plasma boundary, and the numerical error converges at a rate nearly as high as the indirect method we presented previously.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

A Generalized Stress Inversion Approach With Application to Residual Stress Estimation

In this work develop a generalized stress inversion technique (or the generalized inversion method) capable of recovering stresses in linear elastic bodies subjected to arbitrary cuts. Specifically, given a set of displacement measurements found experimentally from digital image correlation (DIC), we formulate a stress estimation inverse problem as a partial differential equation-constrained optimization problem. We use gradient-based optimization methods, and we accordingly derive the necessary gradient and Hessian information in a matrix-free form to allow for parallel, large-scale operations. By using a combination of finite elements, DIC, and a matrix-free optimization framework, the generalized inversion method can be used on any arbitrary geometry, provided that the DIC camera can view a sufficient part of the surface. We present numerical simulations and experiments, and we demonstrate that the generalized inversion method can be applied to estimate residual stress.

42 ENGINEERING↗

DOE OSTI2023

Catalyst: Fast and flexible modeling of reaction networks

We introduce Catalyst.jl, a flexible and feature-filled Julia library for modeling and high-performance simulation of chemical reaction networks (CRNs). Catalyst supports simulating stochastic chemical kinetics (jump process), chemical Langevin equation (stochastic differential equation), and reaction rate equation (ordinary differential equation) representations for CRNs. Through comprehensive benchmarks, we demonstrate that Catalyst simulation runtimes are often one to two orders of magnitude faster than other popular tools. More broadly, Catalyst acts as both a domain-specific language and an intermediate representation for symbolically encoding CRN models as Julia-native objects. This enables a pipeline of symbolically specifying, analyzing, and modifying CRNs; converting Catalyst models to symbolic representations of concrete mathematical models; and generating compiled code for numerical solvers. Leveraging ModelingToolkit.jl and Symbolics.jl, Catalyst models can be analyzed, simplified, and compiled into optimized representations for use in numerical solvers. Finally, we demonstrate Catalyst’s broad extensibility and composability by highlighting how it can compose with a variety of Julia libraries, and how existing open-source biological modeling projects have extended its intermediate representation.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2024

Short-term Electricity Price Forecasting with Constrained Regressors

The volatility of electricity price presents a challenge to market participants as their decision-making process are highly depend on the accuracy of price forecasts. However, there is growing empirical evidence of increasing price volatility and price spikes in electricity markets as a result of variable renewable energy generation, extreme weather events, and other factors. The distribution shift caused by spikes in electricity price data differentiates the forecasting tasks from other renewable energy sources. Moreover, the observations may be compromised by cyberattacks and thus not available in the testing phase. To this end, we propose a Similarity-Enhanced Electricity Decomposition Forecasting model (SEED-Forecaster) to address the missing response problem and spikes capturing in short-term electricity price forecasting. The effectiveness of the proposed framework is tested on real-world electricity price data from California Independent System Operator (CAISO). Numerical results of case studies show that the proposed SEED-Forecsater can enhance forecasting performance, particularly in capturing electricity spikes, even under conditions without regressors during testing stage.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2026

Approximation of refrigerant thermophysical properties using neural networks to speed up transient thermofluid simulations

Accurate and efficient evaluations of refrigerant thermophysical properties and their partial derivatives are essential for transient simulations of thermofluid systems, where several computations need to be executed at each integration time step. Since the utilization of an Equation of State for retrieving properties based on a pair of independent inputs typically involves numerical iterations in solution procedures, when the input variables differ from the refrigerant state variables employed in dynamic models, a variety of approaches including lookup table interpolation and curve fitting have been developed to explicitly approximate these properties based on the state variables, and consequently eliminate internal iterations. This paper presents an alternative method that exploits derivative-informed neural networks to model refrigerant properties explicitly from inputs of pressure and enthalpy, while ensuring consistent partial derivatives generated by differentiating the neural networks. Computational speed and accuracy of the proposed approach are demonstrated via transient simulations of a discretized heat exchanger model in Modelica, and comparisons against other property evaluation routines. Simulation results indicate that the proposed approach can realize a significant speedup with negligible discrepancies in predicted transients. The method is implemented in an open-source Modelica library.

Ma, Jiacheng↗

DOE OSTI2022

Gaussian processes meet NeuralODEs: a Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data

We present a machine learning framework (GP-NODE) for Bayesian model discovery from partial, noisy and irregular observations of nonlinear dynamical systems. The proposed method takes advantage of differentiable programming to propagate gradient information through ordinary differential equation solvers and perform Bayesian inference with respect to unknown model parameters using Hamiltonian Monte Carlo sampling and Gaussian Process priors over the observed system states. This allows us to exploit temporal correlations in the observed data, and efficiently infer posterior distributions over plausible models with quantified uncertainty. The use of the Finnish Horseshoe as a sparsity-promoting prior for free model parameters also enables the discovery of parsimonious representations for the latent dynamics. A series of numerical studies is presented to demonstrate the effectiveness of the proposed GP-NODE method including predator–prey systems, systems biology and a 50-dimensional human motion dynamical system. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

Science & Technology - Other Topics↗