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Results for “QUADRATURE APPROXIMATION”

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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1977

An improved digital algorithm for fast amplitude approximations of quadrature pairs

A computationally fast algorithm for approximating the amplitude (A) of a quadrature pair (I,Q) has been discovered. The piecewise linear formula, the unbiased estimate of A=X+1/8y when X or = 3Y and 7/8X + 1/2Y when X or = 3Y applies where X is congruent to the maximum absolute value of (I,Q) and Y is congruent to the minimum absolute value of (I,Q). This algorithm has been proven to be far more accurate then currently used approximations. An immediate application is the wideband digital spectrum analyzer under development for monitoring radio frequency interference at Deep Space Network stations.

Levitt, B. K.↗
NASA NTRS1997

Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids

This paper is concerned with two important elements in the high-order accurate spatial discretization of finite volume equations over arbitrary grids. One element is the integration of basis functions over arbitrary domains, which is used in expressing various spatial integrals in terms of discrete unknowns. The other consists of quadrature approximations to those integrals. Only polynomial basis functions applied to polyhedral and polygonal grids are treated here. Non-triangular polygonal faces are subdivided into a union of planar triangular facets, and the resulting triangulated polyhedron is subdivided into a union of tetrahedra. The straight line segment, triangle, and tetrahedron are thus the fundamental shapes that are the building blocks for all integrations and quadrature approximations. Integrals of products up to the fifth order are derived in a unified manner for the three fundamental shapes in terms of the position vectors of vertices. Results are given both in terms of tensor products and products of Cartesian coordinates. The exact polynomial integrals are used to obtain symmetric quadrature approximations of any degree of precision up to five for arbitrary integrals over the three fundamental domains. Using a coordinate-free formulation, simple and rational procedures are developed to derive virtually all quadrature formulas, including some previously unpublished. Four symmetry groups of quadrature points are introduced to derive Gauss formulas, while their limiting forms are used to derive Lobatto formulas. Representative Gauss and Lobatto formulas are tabulated. The relative efficiency of their application to polyhedral and polygonal grids is detailed. The extension to higher degrees of precision is discussed.

Liu, Yen↗
DOE OSTI2024

Quadrature Based Neural Network Learning of Stochastic Hamiltonian Systems

Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs’ solutions, where we denoise the loss-by-moment calculations of the solutions. The integral pattern of the models transforms the source of the essential learning error from the discrepancy between the modified Hamiltonian and the true Hamiltonian in the classical HNN models into that between the integrals and their quadrature approximations. This transforms the challenging task of deriving the relation between the modified and the true Hamiltonians from the (stochastic) Hamilton–Jacobi PDEs, into the one that only requires invoking results from the numerical quadrature theory. Meanwhile, denoising via moments calculations gives a simpler data fitting method than, e.g., via probability density fitting, which may imply better generalization ability in certain circumstances. Numerical experiments validate the proposed learning strategy on several concrete Hamiltonian systems. The experimental results show that both the learned Hamiltonian function and the predicted solution of our quadrature-based model are more accurate than that of the corrected symplectic HNN method on a harmonic oscillator, and the three-point Gaussian quadrature-based model produces higher accuracy in long-time prediction than the Kramers–Moyal method and the numerics-informed likelihood method on the stochastic Kubo oscillator as well as other two stochastic systems with non-polynomial Hamiltonian functions. Moreover, the Hamiltonian learning error εH arising from the Gaussian quadrature-based model is lower than that from Simpson’s quadrature-based model. These demonstrate the superiority of our approach in learning accuracy and long-time prediction ability compared to certain existing methods and exhibit its potential to improve learning accuracy via applying precise quadrature formulae.

Mathematics↗
NASA NTRS1977

A simple technique for high resolution time domain phase noise measurement

A new time domain phase comparator is described. The device uses a novel technique to allow time domain phase measurements to be made with period and time interval counters without the use of offset reference oscillators. The device uses a single reference oscillator and allows measurements with a phase resolution greater than the noise floor of the reference. Data is presented showing a phase resolution of 0.02ps at 5 MHz with a crystal reference. The device has application in measuring the phase stability of systems where approximate phase quadrature can be maintained.

Reinhardt, V. S.↗
NASA NTRS2001

Forced and Free Intra-Seasonal Variability Over the South Asian Monsoon Region Simulated by 10 AGCMs

This study examines intra-seasonal (20-70 day) variability in the South Asian monsoon region during 1997/98 in ensembles of 10 simulations with 10 different atmospheric general circulation models. The 10 ensemble members for each model are forced with the same observed weekly sea surface temperature (SST) but differ from each other in that they are started from different initial atmospheric conditions. The results show considerable differences between the models in the simulated 20-70 day variability, ranging from much weaker to much stronger than the observed. A key result is that the models do produce, to varying degrees, a response to the imposed weekly SST. The forced variability tends to be largest in the Indian and western Pacific Oceans where, for some models, it accounts for more than 1/4 of the 20-70 day intra-seasonal variability in the upper level velocity potential during these two years. A case study of a strong observed MJO (intraseasonal oscillation) event shows that the models produce an ensemble mean eastward propagating signal in the tropical precipitation field over the Indian Ocean and western Pacific, similar to that found in the observations. The associated forced 200 mb velocity potential anomalies are strongly phase locked with the precipitation anomalies, propagating slowly to the east (about 5 m/s) with a local zonal wave number two pattern that is generally consistent with the developing observed MJO. The simulated and observed events are, however, approximately in quadrature, with the simulated response 2 leading by 5-10 days. The phase lag occurs because, in the observations, the positive SST anomalies develop upstream of the main convective center in the subsidence region of the MJO, while in the simulations, the forced component is in phase with the SST. For all the models examined here, the intraseasonal variability is dominated by the free (intra-ensemble) component. The results of our case study show that the free variability has a predominately zonal wave number one pattern, and has propagation speeds (10 - 15 m/s) that are more typical of observed MJO behavior away from the convectively active regions. The free variability appears to be synchronized with the forced response, at least, during the strong event examined here. The results of this study support the idea that coupling with SSTs plays an important, though probably not dominant, role in the MJO. The magnitude of the atmospheric response to the SST appears to be in the range of 15% - 30% of the 20-70 day variability over much of the tropical eastern Indian and western Pacific Oceans. The results also highlight the need to use caution when interpreting atmospheric model simulations in which the prescribed SST resolve MJO time scales.

Wu, Man Li C.↗
DOE OSTI2025

Optimized Auxiliary Functions for Robust Mitigation of Finite-Size Errors in Periodic Hybrid Density Functional Theory

When calculating properties of periodic systems at the thermodynamic limit (TDL), the dominant source of finite size error (FSE) arises from the long-range Coulomb interaction, and can manifest as a slowly converging quadrature error when approximating an integral in the reciprocal space by a finite sum. The singularity subtraction (SS) method offers a systematic approach for reducing this quadrature error and thus the FSE. Here, in this work, we first investigate the performance of the SS method in the simplest setting, aiming at reducing the FSE in exact exchange calculations by subtracting the Coulomb contribution with a single, adjustable Gaussian auxiliary function. We demonstrate that a simple fitting method can robustly estimate the optimal Gaussian width and leads to rapid convergence toward the TDL. Furthermore, we suggest new forms of the auxiliary function, whose optimal parameters could also be determined through least-squares fitting. For a range of semiconductors and insulators, the proposed auxiliary functions achieve robust, millihartree-level accuracy in hybrid density functional theory calculations, including cases with sparse k-meshes and large basis sets.

Quiton, Stephen Jon [University of California, Ber↗
DOE OSTI2024

Tensor network representation of non-abelian gauge theory coupled to reduced staggered fermions

We show how to construct a tensor network representation of the path integral for reduced staggered fermions coupled to a non-abelian gauge field in two dimensions. The resulting formulation is both memory and computation efficient because reduced staggered fermions can be represented in terms of a minimal number of tensor indices while the gauge sector can be approximated using Gaussian quadrature with a truncation. Numerical results obtained using the Grassmann TRG algorithm are shown for the case of SU(2) lattice gauge theory and compared to Monte Carlo results.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Calibrating two jets at once

Jet-energy calibration is an important aspect of many measurements and searches at the LHC. Currently, these calibrations are performed on a per-jet basis, i.e., agnostic to the properties of other jets in the same event. In this work, we propose taking advantage of the correlations induced by momentum conservation between jets in order to improve their jet-energy calibration. By fitting the p T asymmetry of dijet events in simulation, while remaining agnostic to the p T spectra themselves, we are able to obtain correlation-improved maximum likelihood estimates. This approach is demonstrated with simulated jets from the CMS detector, yielding a 3%–5% relative improvement in the jet-energy resolution, corresponding to a quadrature improvement of approximately 35%. Published by the American Physical Society 2024

Gambhir, Rikab (ORCID:0000000251080448)↗
NASA NTRS1982

Expansion series of integral functions occurring in unsteady aerodynamics

Attention is given to two real integral functions which occur in the kernel of singular integral equations for subsonic unsteady lifting surfaces. The arguments k, r, and X of the functions correspond to the reduced frequency, spanwise distance, and modified coordinate in the flow direction, respectively. The value of the parameter nu in the functions depends on geometrical conditions. The considered investigation has the objective to present a series for general values of the nonnegative integer nu in order to compute efficiently the integral functions. The approach makes it possible to avoid any approximation or numerical quadrature.

Ueda, T.↗
NASA NTRS1990

Quadrature imposition of compatibility conditions in Chebyshev methods

Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula can not be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula, however it is shown that this is the wrong choice and it may lead to divergence if time dependent methods are used to march the solution to steady state. The correct quadrature formula is developed for these problems. This formula takes into account the degree of the polynomials involved. It is shown that this formula leads to a well conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.

Gottlieb, D.↗
NASA NTRS1990

Quadrature imposition of compatibility conditions in Chebyshev methods

This paper handles a problem encountered in the solution by way of Chebyshev spectral collocation discretization of the Poisson equation with homogeneous Neumann boundary conditions. When pseudospectral Chebyshev methodology is utilized to discretize the partial differential equation these integrals have to be approximated by an appropriate quadrature formula. The correct quadrature formula is developed for these problems, taking into account the degree of the polynomials involved. It is shown that this formula leads to a well-conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.

Gottlieb, D.↗
NASA NTRS1990

The Osher scheme for real gases

An extension of Osher's approximate Riemann solver to include gases with an arbitrary equation of state is presented. By a judicious choice of thermodynamic variables, the Riemann invariats are reduced to quadratures which are then approximated numerically. The extension is rigorous and does not involve any further assumptions or approximations over the ideal gas case. Numerical results are presented to demonstrate the feasibility and accuracy of the proposed method.

Suresh, Ambady↗