Search NASA⌕ Search

SEARCH · Search NASA

Results for “Legendre expansions”

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.

14 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Analysis of sparse recovery for Legendre expansions using envelope bound

We provide novel sufficient conditions for the uniform recovery of sparse Legendre expansions using ℓ 1 minimization, where the sampling points are drawn according to orthogonalization (uniform) measure. So far, conditions of the form m ≳ Θ 2 s x log factors have been relied on to determine the minimum number of samples m that guarantees successful reconstruction of s-sparse vectors when the measurement matrix is associated to an orthonormal system. However, in case of sparse Legendre expansions, the uniform bound Θ of Legendre systems is so high that these conditions are unable to provide meaningful guarantees. Here, in this paper, we present an analysis which employs the envelop bound of all Legendre polynomials instead, and prove a new recovery guarantee for s-sparse Legendre expansions, m ≳ Θs 2 x log factors, which is independent of Θ. Arguably, this is the first recovery condition established for orthonormal systems without assuming the uniform boundedness of the sampling matrix. The key ingredient of our analysis is an extension of chaining arguments, recently developed in Bourgain and Chkifa et al., to handle the envelope bound. Furthermore, our recovery condition is proved via restricted eigenvalue property, a less demanding replacement of restricted isometry property which is perfectly suited to the considered scenario. Along the way, we derive simple criteria to detect good sample sets. Our numerical tests show that sets of uniformly sampled points that meet these criteria will perform better recovery on average.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

The Legendre Polynomial Axial Expansion Method

This work presents a new formulation of the axial expansion transport method explicitly using Legendre polynomials for arbitrarily high-order expansions. This new formulation also features an alternative method of axial leakage calculation to allow for nonextruded flat source region meshes. This alternative axial leakage is introduced alongside a balance equation requirement to ensure that neutron balance is preserved in the coarse mesh for a given axial leakage formulation, which allows for effective coarse mesh finite difference acceleration. A matrix exponential table method is derived to allow for fast computations of arbitrarily high-order matrix exponentials for this work and precludes the need for further research into matrix exponential calculations for this method. Numerical results are presented that demonstrate the stability of the axial expansion method in systems with voidlike regions, showcase the speedup from matrix exponential tables, and investigate the axial convergence of the method in terms of both expansion order and mesh size.

Herring, Nicholas↗
DOE OSTI2021

Accurate numerical, integral methods for computing drift-kinetic Trubnikov-Rosenbluth potentials

A novel numerical method is employed to compute the integral form of the axi-symmetric Trubnikov-Rosenbluth potentials. Two methods for quadrature in pitch-angle are described and their convergence properties are studied. Careful attention is given to quadrature over a singular Green's function. Here it is shown that an infinite series representation of the Green's function can be used more efficiently than its closed form involving complete elliptic integrals. Then a collocation method in speed, with its associated quadrature scheme, is laid out and its convergence properties are studied. Using the proposed scheme, accurate low-order moments of the field collision operator are obtained using relatively few velocity space degrees of freedom. The scheme is showcased by solving for the equilibrium, axi-symmetric bootstrap current in tokamaks. A C 0 Gauss-Lobatto-Legendre finite element pitch-angle basis with vertex nodes at the trapped/passing boundary is shown, in the context of the integral methods used, to be much more efficient than the more common Legendre polynomial expansion.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Exclusive π 0 p electroproduction off protons in the resonance region at photon virtualities 0.4 GeV 2 ≤ Q 2 ≤ 1 GeV 2

The limited electroproduction process e p → e ' p ' π 0 was measured in the range of photon virtualities Q 2 = 0.4 – 1.0 GeV 2 and the invariant mass range of the p π 0 system of W = 1.1 – 1.8 GeV. These kinematics are covered in exclusive π 0 electroproduction off the proton with nearly complete angular coverage in the p π 0 center-of-mass system and with high statistical accuracy. Nearly 36 000 cross-section points were measured, and the structure functions σ T + ε σ L , σ L T , and σ T T , were extracted via fitting the φ π 0 dependence of the cross section. A Legendre polynomial expansion analysis demonstrates the sensitivity of our data to high-lying N * and Δ * resonances with M > 1.6 GeV. As part of a broad effort to determine the electrocouplings of the N * and Δ * resonances using both single- and double-pion electroproduction, this dataset is key for the reliable extraction of the high-lying resonance electrocouplings from the combined isospin analysis of the N π and π + π - p channels.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Electrochemical Modeling and Experimental Verification of Lithiation Gradients in Oxide Cathodes of Lithium-Ion Cells

Lithiated nickel-cobalt-manganese oxides, such as NCM523, are used in the positive electrode (cathode) of Li-ion cells. Using operando X-ray diffraction profilometry, lithiation gradients in the cathode matrix can be observed and quantified by expansion into Legendre polynomials with time-dependent weights. These weights (referred to as gradients) increase in magnitude when electric current flows through the cell, decrease during potentiostatic hold and finally relax to zero when the current is interrupted during open circuit rest. Both physics-based electrochemical models and operando X-ray experiments suggest that the time constants for gradient growth and abatement are primarily determined by ionic diffusion in the oxide particles, which in turn depends on their lithium content. In contrast, the magnitude of gradients depends mainly on the applied current. The X-ray profilometry provides a way of directly probing the formation and disappearance of Li gradients across the cathode during fast cycling, which can help to diagnose the effects of material degradation in the cells.

25 ENERGY STORAGE↗
DOE OSTI2024

A New Galerkin Quadrature Method Not Requiring a Matrix Inverse

We derive a new Galerkin quadrature (GQ) method for S 𝑛 calculations that differs from the two methods preceding it in that a matrix inverse for an 𝑁 𝑑 × 𝑁 𝑑 matrix, where 𝑁𝑑 is the number of directions in the quadrature set, is no longer required. Galerkin quadrature methods are designed for calculations with highly anisotropic scattering. Such methods are not simply special angular quadratures but also are methods for representing the S 𝑛 scattering source that offers several advantages relative to the standard scattering source representation when highly truncated Legendre cross-section expansions must be used. Galerkin quadrature methods are also useful when the scattering is moderately anisotropic, but the quadrature being used is not sufficiently accurate for the order of the scattering source expansion that is required. Furthermore, we derive the new method and present computational results showing that its performance for two challenging problems is comparable to those of the two GQ methods that preceded it.

Galerkin quadrature↗
DOE OSTI2022

Extended kinetic lattice grand canonical Monte Carlo simulation method for transport of multicomponent ion mixtures through a model nanopore system

Abstract An extended version of the original kinetic lattice grand canonical Monte Carlo simulation method combined with mean field theory (KLGCMC/MF) ( J. Chem. Phys. 2007 , 127, 024706) is presented for the study of transport of multicomponent ion mixtures through a model nanopore. Comparison of the extended KLGCMC/MF (eKLGCMC/MF) simulation results with Poisson–Nernst–Planck (PNP) calculations is also made to confirm the validity of the extended simulation approach. Unlike the original version of KLGCMC/MF simulation method that treats only a binary ionic solution with one cation and one anion species, this extended version can deal with a system that includes ternary ion mixtures. A diffusion probability algorithm is also added to the extended version of the simulation method to describe the inhomogeneous diffusivity of ions that is often observed in the ion permeation through nanopores. Both Legendre and Chebyshev polynomials of the second kind were tested as a basis set for the basis set expansion (BSE) method with which to calculate the reaction field energy in the eKLGCMC/MF simulation. It turned out that the Legendre polynomials perform better than the Chebyshev polynomials, and as a result, the Legendre polynomials were implemented in the current version of eKLGCMC/MF simulation algorithm. The presented eKLGCMC/MF simulation method with new features finds its potential applications in nanopore systems where the correlation between ion species with the same sign of charges plays a key role such as oscillating ion currents or anomalous mole fraction effects.

Choi, Inhyeok↗
DOE OSTI2020

UQ Toolkit v 2.0

The Uncertainty Quantification (UQ) Toolkit is a software library for the characterizaton and propagation of uncertainties in computational models. For the characterization of uncertainties, Bayesian inference tools are provided to infer uncertain model parameters, as well as Bayesian compressive sensing methods for discovering sparse representations of high-dimensional input-output response surfaces, and also Karhunen-Loève expansions for representing stochastic processes. Uncertain parameters are treated as random variables and represented with Polynomial Chaos expansions (PCEs). The library implements several spectral basis function types (e.g. Hermite basis functions in terms of Gaussian random variables or Legendre basis functions in terms of uniform random variables) that can be used to represent random variables with PCEs. For propagation of uncertainty, tools are provided to propagate PCEs that describe the input uncertainty through the computational model using either intrusive methods (Galerkin projection of equations onto basis functions) or non-intrusive methods (perform deterministic operation at sampled values of the random values and project the obtained results onto basis functions).

Safta, Cosmin↗
DOE OSTI2024

A Characteristics Approach to the Finite Element Method

Herein, we present a new method for solving the linear Boltzmann transport equation. Two commonly used and well-understood methods for solving partial differential equations are the method of characteristics (MOC) and the finite element method (FEM). We propose a new method that combines the fundamental concept of the FEM with the analytic solution from the MOC to obtain coefficients for the FEM basis function expansion. Traditionally, coefficients for the FEM basis function expansion are obtained via matrix inversion. Instead, we solve for the coefficients with the MOC and represent the underlying fields with the basis function expansion using these coefficients. We provide a convergence study for our method with results from two sets of FEM basis functions: Gauss-Legendre and Gauss-Lobatto sets. We also compare two different variations of our method categorized as short characteristics and intermediate characteristics.

42 ENGINEERING↗
DOE OSTI2024

Galaxy bispectrum in the spherical Fourier-Bessel basis

The bispectrum, the three-point correlation in Fourier space, is a crucial statistic for studying many effects targeted by the next-generation galaxy surveys, such as primordial non-Gaussianity (PNG) and general relativistic (GR) effects on large scales. In this work we develop a formalism for the bispectrum in the spherical Fourier-Bessel (SFB) basis—a natural basis for computing correlation functions on the curved sky, as it diagonalizes the Laplacian operator in spherical coordinates. Working in the SFB basis allows for line-of-sight effects such as redshift space distortions and GR to be accounted for exactly, i.e., without having to resort to perturbative expansions to go beyond the plane-parallel approximation. Only analytic results for the SFB bispectrum exist in the literature given the intensive computations needed. We numerically calculate the SFB bispectrum for the first time, enabled by a few techniques: We implement a template decomposition of the redshift-space kernel Z 2 into Legendre polynomials, and separately treat the PNG and velocity-divergence terms. We derive an identity to integrate a product of three spherical harmonics connected by a Dirac delta function as a simple sum and use it to investigate the limit of a homogeneous and isotropic Universe. Furthermore, we present a formalism for convolving the signal with separable window functions and use a toy spherically symmetric window to demonstrate the computation and give insights into the properties of the observed bispectrum signal. While our implementation remains computationally challenging, it is a step toward a feasible full extraction of information on large scales via a SFB bispectrum analysis.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

First look at neutron emission shape characteristics of ignition hotspots at the National Ignition Facility (invited)

The nuclear imaging system has been capturing neutron images of inertial confinement fusion (ICF) driven implosions for over a decade at the National Ignition Facility. This imaging system has evolved from one to three nearly orthogonal lines-of-sight, allowing for the study of three-dimensional shape characteristics of ignition shots. Limited-view tomography algorithms help visualize the burning hotspot in 3D and assess neutron source geometry using Legendre mode parameters. With its neutron, gamma-ray, and x-ray image reconstruction capabilities, NIS has provided critical insight into mechanisms that have limited implosion performance, such as fill tube diameter for ignition-type targets. This comprehensive diagnostic suite opens a window into the shape characteristics of ignition shots and how symmetry affects ICF implosion performance. In more recent ignition shots, neutron yields have visibly increased. Analyzing the shape and size of the reconstructed neutron source has shown an expansion of the burn volume, which is indicative of more efficient alpha heating during the implosion process.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Redshift space power spectrum beyond Einstein-de Sitter kernels

Abstract We develop a framework to compute the redshift space power spectrum (PS), with kernels beyond Einstein-de Sitter (EdS), that can be applied to a wide variety of generalized cosmologies. We build upon a formalism that was recently employed for standard cosmology in Chen, Vlah & White (2020), and utilize an expansion of the density-weighted velocity moment generating function that explicitly separates the magnitude of the k-modes and their angle to the line-of-sight direction dependencies. We compute the PS for matter and biased tracers to 1-loop Perturbation Theory (PT) and show that the expansion has a correct infrared and ultraviolet behavior, free of unwanted divergences. We also add Effective Field Theory (EFT) counterterms, necessary to account for small-scale contributions to PT, and employ an IR-resummation prescription to properly model the smearing of the BAO due to large scale bulk flows within Standard-PT. To demonstrate the applicability of our formalism, we apply it on the ΛCDM and the Hu-Sawicki f(R) models, and compare our numerical results against the elephant suite of N-body simulations, finding very good agreement up to k = 0.27 Mpc -1 h at z = 0.5 for the first three non-vanishing Legendre multipoles of the PS. To our knowledge, the model presented in this work is the most accurate theoretical EFT-PT for modified gravity to date, being the only one that accounts for beyond linear local biasing in redshift-space. Hence, we argue our RSD modeling is a promising tool to construct theoretical templates in order to test deviations from ΛCDM using real data obtained from the next stage of cosmological surveys such as DESI and LSST.

Astronomy & Astrophysics↗
DOE OSTI2023

Proof-of-Principle Experiment on the Dynamic Shell Formation for Inertial Confinement Fusion

In the dynamic-shell (DS) concept [Goncharov et al., Phys Rev. Lett. 125, 065001 (2020)] for laser-driven inertial confinement fusion the deuterium-tritium fuel is initially in the form of a homogeneous liquid inside a wetted-foam spherical shell. This fuel is ignited using a conventional implosion, which is preceded by a initial compression of the fuel followed by its expansion and dynamic formation of a high-density fuel shell with a low-density interior. This letter reports on a scaled-down, proof-of-principle experiment on the OMEGA laser demonstrating, for the first time, the feasibility of DS formation. A shell is formed by convergent shocks launched by laser pulses at the edge of a plasma sphere, with the plasma itself formed as a result of laser-driven compression and relaxation of a surrogate plastic-foam ball target. Three x-ray diagnostics, namely, 1-D spatially resolved self-emission streaked imaging, 2-D self-emission framed imaging, and backlighting radiography, have shown good agreement with the predicted evolution of the DS and its stability to low Legendre mode perturbations introduced by laser irradiation and target asymmetries.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗