Search NASA⌕ Search

SEARCH · Search NASA

Results for “eigenvalue problems”

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 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

A new method for multi-bit and qudit transfer based on commensurate waveguide arrays

The faithful state transfer is an important requirement in the construction of classical and quantum computers. While the high-speed transfer is realized by optical-fibre interconnects, its implementation in integrated optical circuits is affected by cross-talk. The cross-talk between densely packed optical waveguides limits the transfer fidelity and distorts the signal in each channel, thus severely impeding the parallel transfer of states such as classical registers, multiple qubits and qudits. Here, we leverage on the suitably engineered cross-talk between waveguides to achieve the parallel transfer on optical chip. Waveguide coupling coefficients are designed to yield commensurate eigenvalues of the array and hence, periodic revivals of the input state. While, in general, polynomially complex, the inverse eigenvalue problem permits analytic solutions for small number of waveguides. We present exact solutions for arrays of up to nine waveguides and use them to design realistic buses for multi-(qu)bit and qudit transfer. Advantages and limitations of the proposed solution are discussed in the context of available fabrication techniques.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form is based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Excited-state uncertainties in lattice-QCD calculations of multi-hadron systems

Excited-state effects lead to hard-to-quantify systematic uncertainties in lattice quantum chromodynamics (LQCD) spectroscopy calculations when computationally accessible imaginary times are smaller than inverse excitation gaps, as often arises for multi-hadron systems with signal-to-noise problems. Lanczos residual bounds address this by providing two-sided constraints on energies that do not require assumptions beyond Hermiticity, but often give very conservative systematic uncertainty estimates. Here, a more-constraining set of gap bounds is introduced for hadron spectroscopy. These bounds provide tighter constraints whose validity requires an explicit assumption about an energy gap. Exactly solvable lattice field theory correlators are used to test the utility of residual and gap bounds at finite and infinite statistics. Two-sided bounds and other analysis methods are then applied to a high-statistics LQCD calculation of nucleon-nucleon scattering at $m_π\sim 800$ MeV. Generalized eigenvalue problem (GEVP) and Lanczos energy estimators are compatible when applied to the same correlator data, but analyses including different interpolating operators show statistically significant inconsistencies. However, two-sided bounds from all operators are consistent. Under the assumption that the number of energy levels below $NΔ$ and $ΔΔ$ thresholds is the same as for non-interacting nucleons, gap bounds are sufficient to constrain nucleon-nucleon scattering amplitudes at phenomenologically relevant precision. Lanczos methods further reveal that energy-eigenstate estimates from previously studied asymmetric correlators have not converged over accessible imaginary times. Nevertheless, data-driven examples demonstrate why assumptions are required to draw conclusions about the natures of two-nucleon ground states at these masses.

Detmold, William [MIT, Cambridge, CTP]↗
DOE OSTI2017

Variance Estimation in Monte Carlo Eigenvalue Simulations Using Spectral Analysis Method

The Monte Carlo (MC) method is the 'gold standard' for performing nuclear reactor calculations and criticality safety analyses because of its ability to handle complex geometries and physics. Reactor and criticality safety analyses require solving eigenvalue problems in complex nuclear systems. Once a stationary fission source distribution is obtained in MC simulations, the sample mean of many stationary cycles is calculated. The MC simulation sample mean is typically referred to as a tally. Variance, or standard deviation, of the sample mean/tally is needed to indicate the level of statistical uncertainty in the simulation and to understand the convergence of the sample mean. Current MC codes typically use sample variance to estimate the statistical uncertainty of the simulation and assume that the MC stationary cycles are independent. However, it is known that cycles are correlated, and estimators of the variance that ignore these correlations systematically underestimate the variance. In this paper, we propose a novel method in the Fourier frequency domain to estimate the variance of the sample mean in MC eigenvalue calculations. Results indicate that the proposed estimator can be unbiased compared to the real variance. The proposed estimator requires minimal additional computing effort.

42 ENGINEERING↗
DOE OSTI2017

Self-similar analogues of Stark ladders: a path to fractal potentials

We treat the eigenvalue problem posed by self-similar potentials, i.e. homogeneous functions under a particular affine transformation, by means of symmetry techniques. We find that the eigenfunctions of such problems are localized, evenwhen the potential does not rise to infinity in every direction. It is shown that the logarithm of the energy displays levels contained in families that are analogous toWannier–Stark ladders. The position of each ladder is proved to be determined by the specific details of the potential and not by its transformation properties. This is done by direct computation of matrix elements. The results are compared with numerical solutions of the Schrödinger equation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

MPACT Standard Input User s Manual, Version 2.2.0

The MPACT (Michigan PArallel Charactistics based Transport) code is designed to perform high-fidelity light water reactor (LWR) analysis using whole-core pin-resolved neutron transport calculations on modern parallel-computing hardware. The code consists of several libraries which provide the functionality necessary to solve steady-state eigenvalue problems. Several transport capabilities are available within MPACT including both 2-D and 3-D Method of Characteristics (MOC). A three-dimensional whole core solution based on the 2D-1D solution method provides the capability for full core depletion calculations.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2016

SENSMG: First-Order Sensitivities of Neutron Reaction Rates, Reaction-Rate Ratios, Leakage, k eff , and α Using PARTISN

SENSMG is a tool for computing first-order sensitivities of neutron reaction rates, reaction-rate ratios, leakage, k eff , and α using the PARTISN multigroup discrete-ordinates code. SENSMG computes sensitivities to all of the transport cross sections and data (total, fission, nu, chi, and all scattering moments), two edit cross sections (absorption and capture), and the density for every isotope and energy group. It also computes sensitivities to the mass density for every material and derivatives with respect to all interface locations. The tool can be used for one-dimensional spherical (r) and two-dimensional cylindrical (r-z) geometries. The tool can be used for fixed-source and eigenvalue problems. The tool implements Generalized Perturbation Theory (GPT) as discussed by Williams and Stacey. Section II of this report describes the theory behind adjoint-based sensitivities, gives the equations that SENSMG solves, and defines the sensitivities that are output. Section III describes the user interface, including the input file and command line options. Section IV describes the output. Section V gives some notes about the coding that may be of interest. Section VI discusses verification, which is ongoing. Section VII lists needs and ideas for future work. Appendix A lists all of the input files whose results are presented in Sec. VI.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Planetary normal mode computation: Parallel algorithms, performance and reproducibility

This report is an extension of work entitled “Computing planetary interior normal modes with a highly parallel polynomial filtering eigensolver.” by Shi et al., [1] originally presented at the SC18 conference. A highly parallel polynomial filtered eigensolver was developed and exploited to calculate the planetary normal modes. The proposed method is ideally suited for computing interior eigenpairs for large-scale eigenvalue problems as it greatly enhances memory and computational efficiency. In this article, the second-order finite element method is used to further improve the accuracy as only the first-order finite element method was deployed in the previous work. The parallel algorithm, its parallel performance up to 20k processors, and the great computational accuracy are illustrated. The reproducibility of the previous work was successfully performed on the Student Cluster Competition at the SC19 conference by several participant teams using a completely different Mars-model dataset on different clusters. Both weak and strong scaling performances of the reproducibility by the participant teams were impressive and encouraging. The analysis and reflection of their results are demonstrated and future direction is discussed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Hybrid eigensolvers for nuclear configuration interaction calculations

We examine and compare several iterative methods for solving large-scale eigenvalue problems arising from nuclear structure calculations. In particular, we discuss the possibility of using block Lanczos method, a Chebyshev filtering based subspace iterations and the residual minimization method accelerated by direct inversion of iterative subspace (RMM-DIIS) and describe how these algorithms compare with the standard Lanczos algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm. Although the RMM-DIIS method does not exhibit rapid convergence when the initial approximations to the desired eigenvectors are not sufficiently accurate, it can be effectively combined with either the block Lanczos or the LOBPCG method to yield a hybrid eigensolver that has several desirable properties. We will describe a few practical issues that need to be addressed to make the hybrid solver efficient and robust.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Fragmentation of a Filamentary Cloud Permeated by a Perpendicular Magnetic Field

We examine the linear stability of an isothermal filamentary cloud permeated by a perpendicular magnetic field. Our model cloud is assumed to be supported by gas pressure against self-gravity in the unperturbed state. For simplicity, the density distribution is assumed to be symmetric around the axis. Also for simplicity, the initial magnetic field is assumed to be uniform, and turbulence is not taken into account. The perturbation equation is formulated to be an eigenvalue problem. The growth rate is obtained as a function of the wavenumber for fragmentation along the axis and the magnetic field strength. The growth rate depends critically on the outer boundary. If the displacement vanishes in regions very far from the cloud axis (fixed boundary), cloud fragmentation is suppressed by a moderate magnetic field, which means the plasma beta is below 1.67 on the cloud axis. If the displacement is constant along the magnetic field in regions very far from the cloud, the cloud is unstable even when the magnetic field is infinitely strong. The cloud is deformed by circulation in the plane perpendicular to the magnetic field. The unstable mode is not likely to induce dynamical collapse, since it is excited even when the whole cloud is magnetically subcritical. For both boundary conditions, the magnetic field increases the wavelength of the most unstable mode. We find that the magnetic force suppresses compression perpendicular to the magnetic field especially in regions of low density.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Numerical investigation of non-perturbative kinetic effects of energetic particles on toroidicity-induced Alfvén eigenmodes in tokamaks and stellarators

The resonant interaction of shear Alfvén waves with energetic particles is investigated numerically in tokamak and stellarator geometry using a non-perturbative MHD-kinetic hybrid approach. The focus lies on toroidicity-induced Alfvén eigenmodes (TAEs), which are most easily destabilized by a fast-particle population in fusion plasmas. While the background plasma is treated within the framework of an ideal-MHD theory, the drive of the fast particles, as well as Landau damping of the background plasma, is modelled using the drift-kinetic Vlasov equation without collisions. Building on analytical theory, a fast numerical tool, STAE-K, has been developed to solve the resulting eigenvalue problem using a Riccati shooting method. The code, which can be used for parameter scans, is applied to tokamaks and the stellarator Wendelstein 7-X. High energetic-ion pressure leads to large growth rates of the TAEs and to their conversion into kinetically modified TAEs and kinetic Alfvén waves via continuum interaction. To better understand the physics of this conversion mechanism, the connections between TAEs and the shear Alfvén wave continuum are examined. It is shown that, when energetic particles are present, the continuum deforms substantially and the TAE frequency can leave the continuum gap. The interaction of the TAE with the continuum leads to singularities in the eigenfunctions. To further advance the physical model and also to eliminate the MHD continuum together with the singularities in the eigenfunctions, a fourth-order term connected to radiative damping has been included. The radiative damping term is connected to non-ideal effects of the bulk plasma and introduces higher-order derivatives to the model. Thus, it has the potential to substantially change the nature of the solution. For the first time, the fast-particle drive, Landau damping, continuum damping, and radiative damping have been modelled together in tokamak- as well as in stellarator geometry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Boson star normal modes

Boson stars are gravitationally bound objects that arise in ultralight dark matter models and form in the centers of galactic halos or axion miniclusters. We systematically study the excitations of a boson star, taking into account the mixing between positive and negative frequencies introduced by gravity. We show that the spectrum contains zero-energy modes in the monopole and dipole sectors resulting from spontaneous symmetry breaking by the boson star background. We analyze the general properties of the eigenmodes and derive their orthogonality and completeness conditions which have non-standard form due to the positive-negative frequency mixing. The eigenvalue problem is solved numerically for the first few energy levels in different multipole sectors and the results are compared to the solutions of the Schrödinger equation in fixed boson star gravitational potential. The two solutions differ significantly for the lowest modes, but get close for higher levels. We further confirm the normal mode spectrum in 3D wave simulations where we inject perturbations with different multipoles. As an application of the normal mode solutions, we compute the matrix element entering the evaporation rate of a boson star immersed in a hot axion gas. The computation combines the use of exact wavefunctions for the low-lying bound states and of the Schrödinger approximation for the high-energy excitations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

The continuum and leading twist limits of parton distribution functions in lattice QCD

In this study, we present continuum limit results for the unpolarized parton distribution function of the nucleon computed in lattice QCD. This study is the first continuum limit using the pseudo-PDF approach with Short Distance Factorization for factorizing lattice QCD calculable matrix elements. Our findings are also compared with the pertinent phenomenological determinations. Inter alia, we are employing the summation Generalized Eigenvalue Problem (sGEVP) technique in order to optimize our control over the excited state contamination which can be one of the most serious systematic errors in this type of calculations. A crucial novel ingredient of our analysis is the parameterization of systematic errors using Jacobi polynomials to characterize and remove both lattice spacing and higher twist contaminations, as well as the leading twist distribution. This method can be expanded in further studies to remove all other systematic errors.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

A Non-perturbative Approach to Computing Seismic Normal Modes in Rotating Planets

In this work, a continuous Galerkin method based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard “radial” algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

58 GEOSCIENCES↗
DOE OSTI2025

Accelerating eigenvalue computation for nuclear structure calculations via perturbative corrections

Subspace projection methods utilizing perturbative corrections have been proposed for computing the lowest few eigenvalues and corresponding eigenvectors of large Hamiltonian matrices. In this paper, we build upon these methods and introduce the term Subspace Projection with Perturbative Corrections (SPPC) method to refer to this approach. We tailor the SPPC for nuclear many-body Hamiltonians represented in a truncated configuration interaction subspace, i.e., the no-core shell model (NCSM). We use the hierarchical structure of the NCSM Hamiltonian to partition the Hamiltonian as the sum of two matrices. The first matrix corresponds to the Hamiltonian represented in a small configuration space, whereas the second is viewed as the perturbation to the first matrix. Eigenvalues and eigenvectors of the first matrix can be computed efficiently. Because of the split, perturbative corrections to the eigenvectors of the first matrix can be obtained efficiently from the solutions of a sequence of linear systems of equations defined in the small configuration space. These correction vectors can be combined with the approximate eigenvectors of the first matrix to construct a subspace from which more accurate approximations of the desired eigenpairs can be obtained. We show by numerical examples that the SPPC method can be more efficient than conventional iterative methods for solving large-scale eigenvalue problems such as the Lanczos, block Lanczos and the locally optimal block preconditioned conjugate gradient (LOBPCG) method. The method can also be combined with other methods to avoid convergence stagnation.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

The MOOSE electromagnetics module

The Multiphysics Object-Oriented Simulation Environment (MOOSE) electromagnetics module has been developed to increase MOOSE physics module capabilities, enabling standalone and coupled computational electromagnetics within the MOOSE multiphysics ecosystem. The module is actively being utilized in the areas of plasma physics and advanced manufacturing, and it currently provides initial demonstrated capability in multi-dimensional, complex-valued electromagnetic wave propagation, electrostatic contact, reflection and transmission, and electromagnetic eigenvalue problems. Two-dimensional wave propagation and one-dimensional wave reflection and transmission are showcased as examples in this work. The modularity, parallelism, and plug-in infrastructure for custom future development is inherited from MOOSE itself, and the module can be used with both MOOSE-based and external codes, giving great flexibility.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗