Search NASA⌕ Search

SEARCH · Search NASA

Results for “gradient calculation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS2017

Full Gradient Solution to Adaptive Hybrid Control

This paper focuses on the adaptation mechanisms in adaptive hybrid controllers. Most adaptive hybrid controllers update two filters individually according to the filtered reference least mean squares (FxLMS) algorithm. Because this algorithm was derived for feedforward control, it does not take into account the presence of a feedback loop in the gradient calculation. This paper provides a derivation of the proper weight vector gradient for hybrid (or feedback) controllers that takes into account the presence of feedback. In this formulation, a single weight vector is updated rather than two individually. An internal model structure is assumed for the feedback part of the controller. The full gradient is equivalent to that used in the standard FxLMS algorithm with the addition of a recursive term that is a function of the modeling error. Some simulations are provided to highlight the advantages of using the full gradient in the weight vector update rather than the approximation.

Bean, Jacob↗
DOE OSTI2022

Characterization and visualization of grain boundary disconnections

We introduce a method to visualize dislocations along grain boundaries at the atomic level. It uses an atomic-level Nye tensor, representing the dislocation density. To calculate the Nye tensor at grain boundaries, we extend the Hartley-Mishin strain gradient calculation to the displacement shift complete lattice. We show that the method is effective in visualizing disconnections and the dislocation content of grain boundary phase junctions in body-centered cubic tungsten, as well as face-centered cubic copper. In addition, we use the method to characterize the morphology of a two-dimensional grain boundary phase nucleus in a symmetric tilt grain boundary in tungsten. This method can be applied to both bulk dislocations and grain boundary disconnections, which makes it ideal for studying the interactions and reactions of bulk dislocations with grain boundaries, and grain boundary disconnections.

36 MATERIALS SCIENCE↗
DOE OSTI2014

Spin–orbit DFT with Analytic Gradients and Applications to Heavy Element Compounds

We have implemented the unrestricted DFT approach with one-electron spin–orbit operators in the massively parallel NWChem program. Also implemented is the analytic gradient in the DFT approach with spin–orbit interactions. The current capabilities include single-point calculations and geometry optimization. Vibrational frequencies can be calculated numerically from the analytically calculated gradients. The implementation is based on the spin–orbit interaction operator derived from the effective core potential approach. The exchange functionals used in the implementation are functionals derived for non-spin–orbit calculations, including GGA as well as hybrid functionals. Spin–orbit Hartree–Fock calculations can also be carried out. We have applied the spin–orbit DFT methods to the Uranyl aqua complexes. We have optimized the structures and calculated the vibrational frequencies of both (UO2 2+)aq and (UO2 +)aq with and without spin–orbit effects. The effects of the spin–orbit interaction on the structures and frequencies of these two complexes are discussed. We also carried out calculations for Th2, and several low-lying electronic states are calculated. Our results indicate that, for open-shell systems, there are significant effects due to the spin–orbit effects and the electronic configurations with and without spin–orbit interactions could change due to the occupation of orbitals of larger spin–orbit interactions.

Environmental Molecular Sciences Laboratory↗
DOE OSTI2015

Gas-phase oxygen abundances and radial metallicity gradients in the two nearby spiral galaxies NGC 7793 and NGC 4945

Gas-phase abundances in H ii regions of two spiral galaxies, NGC 7793 and NGC 4945, have been studied to determine their radial metallicity gradients. We used the strong-line method to derive oxygen abundances from spectra acquired with GMOS-S, the multi-object spectrograph on the 8 m Gemini South telescope. We found that NGC 7793 has a well-defined gas-phase radial oxygen gradient of −0.321 ± 0.112 dex R{sub 25}{sup −1} (or −0.054 ± 0.019 dex kpc{sup −1}) in the galactocentric range 0.17 < R{sub G}/R{sub 25} < 0.82, not dissimilar from gradients calculated with direct abundance methods in galaxies of similar mass and morphology. We also determined a shallow radial oxygen gradient in NGC 4945, −0.253 ± 0.149 dex R{sub 25}{sup −1} (or −0.019 ± 0.011 dex kpc{sup −1}) for 0.04 < R{sub G}/R{sub 25} < 0.51, where the larger relative uncertainty derives mostly from the larger inclination of this galaxy. NGC 7793 and NGC 4945 have been selected for this study because they are similar, in mass and morphology, to M33 and the Milky Way, respectively. Since at zeroth order we expect the radial metallicity gradients to depend on mass and galaxy type, we compared our galaxies in the framework of radial metallicity models best suited for M33 and the Galaxy. We found a good agreement between M33 and NGC 7793, pointing toward similar evolution for the two galaxies. We notice instead differences between NGC 4945 and the radial metallicity gradient model that best fits the Milky Way. We found that these differences are likely related to the presence of an active galactic nucleus combined with a bar in the central regions of NGC 4945, and to its interacting environment.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

A general approach to seismic inversion with automatic differentiation

Imaging Earth structure or seismic sources from seismic data involves minimizing a target misfit function, and is commonly solved through gradient-based optimization. The adjoint-state method has been developed to compute the gradient efficiently; however, its implementation can be time-consuming and difficult. We develop a general seismic inversion framework to calculate gradients using reverse-mode automatic differentiation. The central idea is that adjoint-state methods and reverse-mode automatic differentiation are mathematically equivalent. Here, the mapping between numerical PDE simulation and deep learning allows us to build a seismic inverse modeling library, ADSeismic, based on deep learning frameworks, which supports high performance reverse-mode automatic differentiation on CPUs and GPUs. We demonstrate the performance of ADSeismic on inverse problems related to velocity model estimation, rupture imaging, earthquake location, and source time function retrieval. ADSeismic has the potential to solve a wide variety of inverse modeling applications within a unified framework.

58 GEOSCIENCES↗
DOE OSTI2021

Improved Prediction of Cold-Air Pools in the Weather Research and Forecasting Model Using a Truly Horizontal Diffusion Scheme for Potential Temperature

The terrain-following vertical coordinate system used by many atmospheric models, including the Weather Research and Forecasting (WRF) Model, is prone to errors in regions of complex terrain. These errors stem, in part, from the calculation of horizontal gradients within the diffusion term of the momentum or scalar evolution equations. In WRF, such gradients can be calculated along coordinate surfaces, or using metric terms that help account for grid skewness. However, neither of these options ensures a truly horizontal gradient calculation, especially if a grid cell is skewed enough that the heights of the neighboring grid points used in the calculation fall outside the vertical range of the cell. In this work, an improved scheme that uses Taylor series approximations to vertically interpolate variables to the level necessary for a truly horizontal gradient calculation is implemented in WRF for the diffusion of potential temperature. The scheme is validated using an atmosphere-at-rest configuration, in which spurious flows develop only as a result of numerical errors and can thus be used as a proxy for model performance. Following validation, the method is applied to the simulation of cold-air pools (CAPs), which occur in regions of complex terrain and are characterized by strong near-surface temperature gradients. Using the truly horizontal scheme, idealized simulations demonstrate reduced numerical mixing in a quiescent CAP, and a realistic case study in the Columbia River basin shows a reduction in positive wind speed bias by up to roughly 20% compared to observations from the Second Wind Forecast Improvement Project.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1986

An analysis of mesoscale VAS retrievals using statistical structure functions

Mesoscale sounding data from a truth rawinsonde network and a VAS from March 6-7, 1982 are statistically studied. The parameters of temperature, geopotential height, and mixing ratio at 850, 700, 500, 300, and 200 mb levels, and thickness and precipitable water are examined. The VAS retrieval gradients derived from a physical algorithm and from a regression technique are analyzed; the analysis reveals that both techniques have similar error characteristics and capabilities for calculating gradients of mesoscale temperature and geopotential height. Ground-truth rawinsonde data and VAS retrievals are compared; it is observed that VAS data describes weaker gradients than rawinsonde data.

Fuelberg, H. E.↗
NASA NTRS2005

Traj_opt User's Guide

Trajectory optimization program Traj_opt was developed at Ames Research Center to help assess the potential benefits of ultrahigh temperature ceramic materials applied to reusable space vehicles with sharp noses and wing leading edges. Traj_opt loosely couples the Ames three-degrees-of-freedom trajectory package Traj (see NASA-TM-2004-212847) with the SNOPT optimization package (Stanford University Technical Report SOL 98-1). Traj_opt version January 22, 2003 is covered by this user guide. The program has been applied extensively to entry and ascent abort trajectory calculations for sharp and blunt crew transfer vehicles. The main optimization variables are control points for the angle of attack and bank angle time histories. No propulsion options are provided, but numerous objective functions may be specified and the nonlinear constraints implemented include a distributed surface heating constraint capability. Aero-capture calculations are also treated with an option to minimize orbital eccentricity at apoapsis. Traj_opt runs efficiently on a single processor, using forward or central differences for the gradient calculations. Results may be displayed conveniently with Gnuplot scripts. Control files recommended for five standard reentry and ascent abort trajectories are included along with detailed descriptions of the inputs and outputs.

Saunders, David A.↗
DOE OSTI2017

Is it possible that MHD instability triggers a transition into the improved confinement regime of toroidal plasmas?

Analyzing the experimental data from L-2M stellarator and performing appropriate theoretical analysis, the authors affirmatively answer the question raised in the title of the article. The external peeling mode is proposed as the instability triggering the transition. Peeling mode stability in magnetic-hill Mercierstable plasmas is analyzed analytically. It is shown that correctly taking into account vacuum region forbids internal peeling modes, while the external peeling mode has a threshold with respect to the plasma pressure gradient. Calculations and experimental data are in reasonable agreement.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

When ancient numerical demons meet physics-informed machine learning: adjoint-based gradients for implicit differentiable modeling

Recent advances in differentiable modeling, a genre of physics-informed machine learning that trains neural networks (NNs) together with process-based equations, have shown promise in enhancing hydrological models' accuracy, interpretability, and knowledge-discovery potential. Current differentiable models are efficient for NN-based parameter regionalization, but the simple explicit numerical schemes paired with sequential calculations (operator splitting) can incur numerical errors whose impacts on models' representation power and learned parameters are not clear. Implicit schemes, however, cannot rely on automatic differentiation to calculate gradients due to potential issues of gradient vanishing and memory demand. Here we propose a “discretize-then-optimize” adjoint method to enable differentiable implicit numerical schemes for the first time for large-scale hydrological modeling. The adjoint model demonstrates comprehensively improved performance, with Kling–Gupta efficiency coefficients, peak-flow and low-flow metrics, and evapotranspiration that moderately surpass the already-competitive explicit model. Therefore, the previous sequential-calculation approach had a detrimental impact on the model's ability to represent hydrological dynamics. Furthermore, with a structural update that describes capillary rise, the adjoint model can better describe baseflow in arid regions and also produce low flows that outperform even pure machine learning methods such as long short-term memory networks. The adjoint model rectified some parameter distortions but did not alter spatial parameter distributions, demonstrating the robustness of regionalized parameterization. Despite higher computational expenses and modest improvements, the adjoint model's success removes the barrier for complex implicit schemes to enrich differentiable modeling in hydrology.

58 GEOSCIENCES↗
DOE OSTI2017

Accuracy of finite-difference harmonic frequencies in density functional theory

Analytic Hessians are often viewed as essential for the calculation of accurate harmonic frequencies, but the implementation of analytic second derivatives is nontrivial and solution of the requisite coupled-perturbed equations engenders a sizable memory footprint for large systems, given that these equations are not required for energy and gradient calculations in density functional theory. Here, we benchmark the alternative approach to harmonic frequencies based on finite differences of analytic first derivatives, a procedure that is amenable to large-scale parallelization. Not only for absolute frequencies but also for isotopic and conformer-dependent frequency shifts in flexible molecules, we find that the finite-difference approach exhibits mean errors < 0.1 cm -1 as compared to results based on an analytic Hessian. For very small frequencies corresponding to nonbonded vibrations in noncovalent complexes (for which the harmonic approximation is questionable anyway), the finite-difference error can be larger, but even in these cases the errors can be reduced below 0.1 cm -1 by judicious choice of the displacement step size and a higher-order finite-difference approach. Finally, the surprising accuracy and robustness of the finite-difference results suggests that availability of the analytic Hessian is not so important in today's era of commodity processors that are readily available in large numbers.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2018

Vertical Gradients in Fluidized Sand Columns

This study evaluated the influence of diameter and grain size on the initiation and continuation phase of piping erosion. Twelve tests were performed in a laboratory setting in which sand was fluidized in a controlled setting and hydraulic gradients within the sand, flow rates through the sand, and expansion of the sand columns were measured as the sand was fluidized. This study was conducted to improve the understanding of the resistance a vertical column of sand offers to piping erosion. From the fluidization tests conducted in two permeameters with different diameters, it cannot be concluded that permeameter diameter has a significant influence on a specimen’s resistance to piping erosion. Although one test conducted in the smaller diameter permeameter did record hydraulic gradients lower than those recorded in tests conducted in the larger diameter permeameter, this was not a consistent observation for all tests run in the smaller diameter permeameter. Further analysis ought to involve more tests run in each permeameter as well as incorporating additional permeameters with both larger and smaller diameters than those used in this study. From the fluidization tests conducted using the well-graded, coarse-, medium-, and fine-grained sands, it appears that resistance to piping is dependent on grain size and grain size distribution. The coarse-grained sand offered the most resistance to piping, the fine-grained sand offered the least, and the medium-grained and well-graded sands offered similar resistances between those of the coarse-grained and fine-grained sands. Gradients at which sand boils were observed, maximum gradients reached before the sand fluidized, and the minimum gradients reached at the conclusion of each test also appear to be influenced by grain size and grain size distribution. Tests conducted on the coarse-grained sand recorded the highest and most closely spaced sand boil, maximum, and minimum gradients, indicating that it offered the highest and most constant resistance to piping, while the fine-grained sand tests recorded the lowest sand boil, maximum, and minimum gradients, indicating that it offered the least resistance to piping and that resistance decreased as the tests progressed. The medium-grained sand tests recorded sand boil, maximum, and minimum gradients between those of the coarse-grained and fine-grained sands. The well-graded sand tests recorded the lowest sand boil gradient, which could be indicative of the occurrence of suffusion erosion occurring in conjunction with piping erosion, whereby the finer particles were washed out of the sand matrix at relatively low gradients, forming sand boils while the relatively coarser matrix remained resistant to piping. All tests recorded maximum gradients that were lower than the theoretical critical gradients calculated using Equation 4. This is likely due to the changes in the internal configuration of the sand as the piping process 33 initiates. In order to better understand the influence of gradation on resistance to piping, more sand gradations should be tested in future work. Measured hydraulic gradients in the lower portion of the fluidized sands were higher than those measured in higher portions. This could be explained if the density of the fluidized sand was not uniform, with more material near the base of the column than near the top, which is contrary to the findings of Sellmeijer. Furthermore, the tests were terminated when the maximum height of the upstream reservoir was reached but did not give any evidence that gradients had leveled off to a uniform “final gradient”. It is likely that gradients would have continued to dissipate had more hydraulic head been applied to the upstream reservoir. In order to better understand this, additional testing should be conducted using an upstream reservoir capable of simulating higher hydraulic heads.

13 HYDRO ENERGY↗
DOE OSTI2020

Efficient Ensemble-Based Stochastic Gradient Methods for Optimization Under Geological Uncertainty

Ensemble-based stochastic gradient methods, such as the ensemble optimization (EnOpt) method, the simplex gradient (SG) method, and the stochastic simplex approximate gradient (StoSAG) method, approximate the gradient of an objective function using an ensemble of perturbed control vectors. These methods are increasingly used in solving reservoir optimization problems because they are not only easy to parallelize and couple with any simulator but also computationally more efficient than the conventional finite-difference method for gradient calculations. In this work, we show that EnOpt may fail to achieve sufficient improvement of the objective function when the differences between the objective function values of perturbed control variables and their ensemble mean are large. On the basis of the comparison of EnOpt and SG, we propose a hybrid gradient of EnOpt and SG to save on the computational cost of SG. We also suggest practical ways to reduce the computational cost of EnOpt and StoSAG by approximating the objective function values of unperturbed control variables using the values of perturbed ones. We first demonstrate the performance of our improved ensemble schemes using a benchmark problem. Results show that the proposed gradients saved about 30–50% of the computational cost of the same optimization by using EnOpt, SG, and StoSAG. As a real application, we consider pressure management in carbon storage reservoirs, for which brine extraction wells need to be optimally placed to reduce reservoir pressure buildup while maximizing the net present value. Results show that our improved schemes reduce the computational cost significantly.

58 GEOSCIENCES↗
DOE OSTI2024

Qualitative and quantitative enhancement of parameter estimation for model-based diagnostics using automatic differentiation with an application to inertial fusion

Parameter estimation using observables is a fundamental concept in the experimental sciences. Mathematical models that represent the physical processes can enable reconstructions of the experimental observables and greatly assist in parameter estimation by turning it into an optimization problem which can be solved by gradient-free or gradient-based methods. In this work, the recent rise in flexible frameworks for developing differentiable scientific computing programs is leveraged in order to dramatically accelerate data analysis of a common experimental diagnostic relevant to laser–plasma and inertial fusion experiments, Thomson scattering. A differentiable Thomson-scattering data analysis tool is developed that uses reverse-mode automatic differentiation (AD) to calculate gradients. By switching from finite differencing to reverse-mode AD, three distinct outcomes are achieved. First, gradient descent is accelerated dramatically to the extent that it enables near real-time usage in laser–plasma experiments. Second, qualitatively novel quantities which require $\mathcal{O}(10^3)$ parameters can now be included in the analysis of data which enables unprecedented measurements of small-scale laser–plasma phenomena. Third, uncertainty estimation approaches that leverage the value of the Hessian become accurate and efficient because reverse-mode AD can be used for calculating the Hessian.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

Multidisciplinary design optimization using genetic algorithms

Multidisciplinary design optimization (MDO) is an important step in the conceptual design and evaluation of launch vehicles since it can have a significant impact on performance and life cycle cost. The objective is to search the system design space to determine values of design variables that optimize the performance characteristic subject to system constraints. Gradient-based optimization routines have been used extensively for aerospace design optimization. However, one limitation of gradient based optimizers is their need for gradient information. Therefore, design problems which include discrete variables can not be studied. Such problems are common in launch vehicle design. For example, the number of engines and material choices must be integer values or assume only a few discrete values. In this study, genetic algorithms are investigated as an approach to MDO problems involving discrete variables and discontinuous domains. Optimization by genetic algorithms (GA) uses a search procedure which is fundamentally different from those gradient based methods. Genetic algorithms seek to find good solutions in an efficient and timely manner rather than finding the best solution. GA are designed to mimic evolutionary selection. A population of candidate designs is evaluated at each iteration, and each individual's probability of reproduction (existence in the next generation) depends on its fitness value (related to the value of the objective function). Progress toward the optimum is achieved by the crossover and mutation operations. GA is attractive since it uses only objective function values in the search process, so gradient calculations are avoided. Hence, GA are able to deal with discrete variables. Studies report success in the use of GA for aircraft design optimization studies, trajectory analysis, space structure design and control systems design. In these studies reliable convergence was achieved, but the number of function evaluations was large compared with efficient gradient methods. Applicaiton of GA is underway for a cost optimization study for a launch-vehicle fuel-tank and structural design of a wing. The strengths and limitations of GA for launch vehicle design optimization is studied.

Unal, Resit↗
NASA NTRS1971

Study to determine and improve design for lithium-doped solar cells

Solar cell experiments show that a single lithium density parameter, the lithium density gradient, calculated from nondestructive capacitance measurements, provides the basis for accurate predictions of lithium cell behavior in a 1-MeV electron environment for fluences ranging between 3 X 10 to the 13th power e/sq cm and 3 X 10 to the 15th power/e sq cm. The oxygen-rich (quartz crucible) lithium cell with phosphorous starting dopant and lithium gradient between approximately 5 X 10 to the 18th power and 1.5 x 10 to the 19th power/cm to the 4th power was found superior in performance to the commercial 10 ohm-cm n/p control cells. Post-recovery stability of oxygen-rich cells was satisfactory. An average post-recovery current drop of approximately 1 mA was observed for 70 crucible cells after 1 year-equivalent storage time at 80 C. In contrast the oxygen-poor (float zone and Lopex) lithium cells displayed spotty initial performance and stability problems at room temperature.

Brucker, G.↗
NASA NTRS1990

Shape design sensitivities using fully automatic 3-D mesh generation

Previous work in three dimensional shape optimization involved specifying design variables by associating parameters directly with mesh points. More recent work has shown the use of fully-automatic mesh generation based upon a parameterized geometric representation. Design variables have been associated with a mathematical model of the part rather than the discretized representation. The mesh generation procedure uses a nonuniform grid intersection technique to place nodal points directly on the surface geometry. Although there exists an associativity between the mesh and the geometrical/topological entities, there is no mathematical functional relationship. This poses a problem during certain steps in the optimization process in which geometry modification is required. For the large geometrical changes which occur at the beginning of each optimization step, a completely new mesh is created. However, for gradient calculations many small changes must be made and it would be too costly to regenerate the mesh for each design variable perturbation. For that reason, a local remeshing procedure has been implemented which operates only on the specific edges and faces associated with the design variable being perturbed. Two realistic design problems are presented which show the efficiency of this process and test the accuracy of the gradient computations.

Botkin, M. E.↗