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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

CLPNets: Coupled Lie–Poisson neural networks for multi-part Hamiltonian systems with symmetries

To accurately compute data-based prediction of Hamiltonian systems, it is essential to utilize methods that preserve the structure of the equations over time. We consider a particularly challenging case of systems with interacting parts that do not reduce to pure momentum evolution. Such systems are essential in scientific computations, such as discretization of a continuum elastic rod, which can be viewed as the group of rotations and translations $SE(3)$. The evolution involves not only the momenta but also the relative positions and orientations of the particles. The presence of Lie group-valued elements, such as relative positions and orientations, poses a problem for applying previously derived methods for data-based computing. We develop a novel method of data-based computation and complete phase space learning of such systems. We follow the original framework of SympNets (Jin et al., 2020) and LPNets (Eldred et al., 2024), building the neural network from phase space mappings that preserve the Lie–Poisson structure. We derive a novel system of mappings that are built into neural networks describing the evolution of such systems. We call such networks Coupled Lie–Poisson Neural Networks, or CLPNets. We consider increasingly complex examples for the applications of CLPNets, starting with the rotation of two rigid bodies about a common axis, progressing to the free rotation of two rigid bodies, and finally to the evolution of two connected and interacting $SE(3)$ components, describing the discretization of an elastic rod into two elements. Our method preserves all Casimir invariants to machine precision, preserves energy to high accuracy, and shows good resistance to the curse of dimensionality, requiring only a few thousand data points for all cases studied (three to eighteen dimensions). Additionally, the method is highly economical in memory requirements, requiring only about 200 parameters for the most complex case considered.

Data-based modeling↗
DOE OSTI2020

Combined Neyman–Pearson chi-square: An improved approximation to the Poisson-likelihood chi-square

We describe an approximation to the widely-used Poisson-likelihood chi-square using a linear combination of Neyman’s and Pearson’s chi-squares, namely “combined Neyman–Pearson chi-square” ($X^2_{CNP}$). Through analytical derivations and toy model simulations, we show that leads to a significantly smaller bias on the best-fit model parameters compared to those using either Neyman’s or Pearson’s chi-square. When the computational cost of using the Poisson-likelihood chi-square is high, $X^2_{CNP}$ provides a good alternative given its natural connection to the covariance matrix formalism.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2018

The Young Modulus and the Poisson Coefficient of Two-Dimensionally Extended Columnar Graphene

The Young modulus and the Poisson coefficient of two-dimensionally extended columnar graphene are theoretically studied in the uniaxial tension. The effect of length and diameter of nanotubes constituting the composite is considered. The numerical experiments are implemented using the minimum structural link and the periodical boundary conditions via the density functional theory method. The Young moduli of the composite are evaluated upon its extension along the normal to the graphene and along the graphene directions with increasing lengths of carbon nanotubes (CNTs) in the composite. The Poisson coefficient for this type of composites is found to be 0.025.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2020

Bond-Based Peridynamics: a Tale of Two Poisson’s Ratios

This paper explores the restrictions imposed by bond-based peridynamics, particularly with respect to plane strain and plane stress models. We begin with a review of the derivations in Gerstle et al. (2005) wherein for isotropic materials a Poisson’s ratio restriction of $\frac{1}{4}$ for plane strain and $\frac{1}{3}$ for plane stress is deduced. Next, we show Cauchy’s relations are an intrinsic limitation of bond-based peridynamics and specialize these restrictions to plane strain and plane stress models. This generalizes the results from Gerstle et al. (2005) and demonstrates that the Poisson’s ratio restrictions described in Gerstle et al. (2005) are merely a consequence of Cauchy’s relations for isotropic materials. We conclude with a discussion of the validity of peridynamic plane strain and plane stress models formulated from two-dimensional bond-based peridynamic models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Analysis of the weighted shifted boundary method for the Poisson and Stokes problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.

Approximate domain boundaries↗
DOE OSTI2020

Deformation of Lie–Poisson algebras and chirality

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, -λ and $\barλ$ (hence, -$\barλ$) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra s o ( 3 ) , yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Poisson_CCD: A dedicated simulator for modeling CCDs

A dedicated simulator, Poisson_CCD, has been constructed which models astronomical CCDs by solving Poisson’s equation numerically and simulating charge transport within the CCD. The potentials and free carrier densities within the CCD are self-consistently solved for, giving realistic results for the charge distribution within the CCD storage wells. The simulator has been used to model the CCDs that are being used to construct the LSST digital camera. The simulator output has been validated by comparing its predictions with several different types of CCD measurements, including astrometric shifts, brighter-fatter induced pixel–pixel covariances, saturation effects, and diffusion spreading. The code is open source and freely available.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Multipoint Correlations in Poisson Media

Multipoint correlations in randomly heterogeneous composite media are determined by the probability that a set of points belong to specific phases. They determine a wide range of macroscopic transport properties such as conductivity, dielectric constant, diffusion coefficient, and transmittance. The Poisson model—a random tesselation of space by hyperplanes—provides realistic descriptions of heterogeneous media in, e.g., radiation-transport applications; yet, until now, it has lacked closed-form expressions for its multipoint correlations. We resolve this problem by presenting an exact solution for the multipoint correlations in the Poisson model. By comparing it to Monte Carlo simulations of four-point correlations in three dimensions, we demonstrate the accuracy of our solution. In conclusion, we visualize the multipoint correlations and discuss their features.

Amorphous materials↗
DOE OSTI2026

Block encoding of the three-dimensional heterogeneous Poisson equation with application to fracture flow

Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as state preparation, data loading, and efficient information extraction. In this work, we study the feasibility of applying QLS algorithms to solve discretized three-dimensional (3D) heterogeneous Poisson equations, with specific examples relating to groundwater flow through geologic fracture networks. We explicitly construct a block encoding for the 3D heterogeneous Poisson matrix by leveraging the sparse local structure of the discretized operator. While classical solvers benefit from preconditioning, we show that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS run-time. This differs from classical approaches where the preconditioner and the system matrix can often be implemented independently. Nevertheless, due to the structure of the problem in three dimensions, the quantum algorithm achieves a run-time of 𝑂⁡(𝑁 2/3 polylog 𝑁 ⋅log (1/𝜖)), outperforming the best classical methods (with run times of 𝑂⁡(𝑁⁢log 𝑁 ⋅log (1/𝜖))) and offering exponential memory savings. These results highlight both the promise and limitations of QLS algorithms for practical scientific computing, and point to effective condition-number reduction as a key barrier in achieving quantum advantages.

58 GEOSCIENCES↗
DOE OSTI2022

Quantum-inspired method for solving the Vlasov-Poisson equations

Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high-resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. Here, in this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations for systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

A Quantum Random Number Generator Based on the 100-Mbit/s Poisson Photocount Statistics

A quantum random number generator is experimentally realized. The initial randomness source is a sequence of photocounts from quasi-single-photon radiation detected with a SiPM (Silicon Photo Multiplier) silicon avalanche detector array. The use of a SiPM provides a reliable control of the quantum nature of the Poisson photocount statistics. A special algorithm with nonexponential complexity allows one to extract from the Poisson process all the randomness contained in it, namely, a random uniform sequence of 0 and 1.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Stability and Conservation properties of Hermite-based approximations of the Vlasov-Poisson System

Spectral approximation based on Hermite-Fourier expansion of the Vlasov-Poisson model for a collisionless plasma in the electro-static limit is provided by adding high-order artificial collision operators of Lenard-Bernstein type. These differential operators are suitably designed in order to preserve the physically-meaningful invariants (number of particles, momentum, energy). In view of time-discretization, stability results in appropriate norms are presented. In this study, necessary conditions link the magnitude of the artificial collision term, the number of spectral modes of the discretization, as well as the time-step. The analysis, carried out in full for the Hermite discretization of a simple linear problem in one-dimension, is then partly extended to cover the complete nonlinear Vlasov-Poisson model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

On the prescription of boundary conditions for nonlocal Poisson's and peridynamics models

We introduce a technique to automatically convert local boundary conditions into nonlocal volume constraints for nonlocal Poisson’s and peridynamic models. The proposed strategy is based on the approximation of nonlocal Dirichlet or Neumann data with a local solution obtained by using available boundary, local data. The corresponding nonlocal solution converges quadratically to the local solution as the nonlocal horizon vanishes, making the proposed technique asymptotically compatible. The proposed conversion method does not have any geometry or dimensionality constraints and its computational cost is negligible, compared to the numerical solution of the nonlocal equation. The consistency of the method and its quadratic convergence with respect to the horizon is illustrated by several two-dimensional numerical experiments conducted by meshfree discretization for both the Poisson’s problem and the linear peridynamic solid model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Poisson Equation for a (General) Homogeneous d-Dimensional Ellipsoid with Applications to Beam Envelope Tracking

This note describes the solution of the free-space Poisson equation in the interior of a $d$-dimensional homogeneous ellipsoid, and the associated space charge fields. An explicit formula (\ref{Sformula}) is provided that relates the $d\times d$ matrix describing the space charge (quadratic) potential to the $d\times d$ covariance matrix of the ellipsoid. For the cases $d=2$ and $d=3$, this result is used to determine the linear map corresponding to a space charge kick, that may be used to push the beam $6\times 6$ covariance matrix during envelope tracking. The treatment of upright ellipsoids for $d=2$ and $d=3$ is well-represented in the literature. However, the approach taken here emphasizes a general ellipsoid with arbitrary correlations in any dimension. The Appendix provides a general solution of the free-space Poisson equation in dimension $d$ for a source distribution with ellipsoidal symmetry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Pressure dependence of Poisson’s ratio of glassy Baltic amber studied by Brillouin scattering spectroscopy

Amber is a typical natural glass with very long aging time. Elastic properties of amber at high pressures have been studied using Brillouin scattering and a diamond anvil cell. Both longitudinal and transverse acoustic modes have been observed up to 12 GPa. The pressure dependences of longitudinal, shear, Young’s, and bulk moduli, compressibility, and Poisson’s ratio were determined. The longitudinal, shear, Young’s, bulk moduli show the remarkable increase, and compressibility shows a marked decrease with increasing pressure. However, it is found that the pressure dependence of Poisson’s ratio is very small. The mechanism of this small pressure dependence was discussed. Finally, the Cauchy type relation between longitudinal and shear moduli was examined. Its coefficients show the deviation from the Cauchy relation owing to the coexistence of different intermolecular interactions in amber.

07 ISOTOPE AND RADIATION SOURCES↗
NASA NTRS1972

A Bayesian approach to parameter and reliability estimation in the Poisson distribution.

For life testing procedures, a Bayesian analysis is developed with respect to a random intensity parameter in the Poisson distribution. Bayes estimators are derived for the Poisson parameter and the reliability function based on uniform and gamma prior distributions of that parameter. A Monte Carlo procedure is implemented to make possible an empirical mean-squared error comparison between Bayes and existing minimum variance unbiased, as well as maximum likelihood, estimators. As expected, the Bayes estimators have mean-squared errors that are appreciably smaller than those of the other two.

Canavos, G. C.↗