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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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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton–Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Recursive Factorization of the Inverse Overlap Matrix in Linear Scaling Quantum Molecular Dynamics Simulations

We present a reduced complexity algorithm to compute the inverse overlap factors required to solve the generalized eigenvalue problem in a quantum-based molecular dynamics (MD) simulation. Our method is based on the recursive iterative re nement of an initial guess Z of the inverse overlap matrix S. The initial guess of Z is obtained beforehand either by using an approximate divide and conquer technique or dynamically, propagated within an extended Lagrangian dynamics from previous MD time steps. With this formulation, we achieve long-term stability and energy conservation even under incomplete approximate iterative re nement of Z. Linear scaling performance is obtained using numerically thresholded sparse matrix algebra based on the ELLPACK-R sparse matrix data format, which also enables e cient shared memory parallelization. As we show in this article using selfconsistent density functional based tight-binding MD, our approach is faster than conventional methods based on the direct diagonalization of the overlap matrix S for systems as small as a few hundred atoms, substantially accelerating quantum-based simulations even for molecular structures of intermediate size. For a 4,158 atom water-solvated polyalanine system we nd an average speedup factor of 122 for the computation of Z in each MD step.

36 MATERIALS SCIENCE↗
DOE OSTI2016

PROTEUS-SN User Manual

PROTEUS-SN is a three-dimensional, highly scalable, high-fidelity neutron transport code developed at Argonne National Laboratory. The code is applicable to all spectrum reactor transport calculations, particularly those in which a high degree of fidelity is needed either to represent spatial detail or to resolve solution gradients. PROTEUS-SN solves the second order formulation of the transport equation using the continuous Galerkin finite element method in space, the discrete ordinates approximation in angle, and the multigroup approximation in energy. PROTEUS-SN’s parallel methodology permits the efficient decomposition of the problem by both space and angle, permitting large problems to run efficiently on hundreds of thousands of cores. PROTEUS-SN can also be used in serial or on smaller compute clusters (10’s to 100’s of cores) for smaller homogenized problems, although it is generally more computationally expensive than traditional homogenized methodology codes. PROTEUS-SN has been used to model partially homogenized systems, where regions of interest are represented explicitly and other regions are homogenized to reduce the problem size and required computational resources. PROTEUS-SN solves forward and adjoint eigenvalue problems and permits both neutron upscattering and downscattering. An adiabatic kinetics option has recently been included for performing simple time-dependent calculations in addition to standard steady state calculations. PROTEUS-SN handles void and reflective boundary conditions. Multigroup cross sections can be generated externally using the MC2-3 fast reactor multigroup cross section generation code or internally using the cross section application programming interface (API) which can treat the subgroup or resonance table libraries. PROTEUS-SN is written in Fortran 90 and also includes C preprocessor definitions. The code links against the PETSc, METIS, HDF5, and MPICH libraries. It optionally links against the MOAB library and is a part of the SHARP multi-physics suite for coupled multi-physics analysis of nuclear reactors. This user manual describes how to set up a neutron transport simulation with the PROTEUS-SN code. A companion methodology manual describes the theory and algorithms within PROTEUS-SN.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Deterministically estimated fission source distributions for Monte Carlo k-eigenvalue problems

The standard Monte Carlo (MC) k-eigenvalue algorithm involves iteratively converging the fission source distribution using a series of potentially time-consuming inactive cycles before quantities of interest can be tallied. One strategy for reducing the computational time requirements of these inactive cycles is the Sourcerer method, in which a deterministic eigenvalue calculation is performed to obtain an improved initial guess for the fission source distribution. This method has been implemented in the Exnihilo software suite within SCALE using the SPNSPN or SNSN solvers in Denovo and the Shift MC code. The efficacy of this method is assessed with different Denovo solution parameters for a series of typical k-eigenvalue problems including small criticality benchmarks, full-core reactors, and a fuel cask. Here it is found that, in most cases, when a large number of histories per cycle are required to obtain a detailed flux distribution, the Sourcerer method can be used to reduce the computational time requirements of the inactive cycles.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2018

Orbital-enriched flat-top partition of unity method for the Schrödinger eigenproblem

Quantum mechanical calculations require the repeated solution of a Schrödinger equation for the wavefunctions of the system, from which materials properties follow. Recent work has shown the effectiveness of enriched finite element type Galerkin methods at significantly reducing the degrees of freedom required to obtain accurate solutions. However, time to solution has been adversely affected by the need to solve a generalized rather than standard eigenvalue problem and the ill-conditioning of associated system matrices. In this work, we address both issues by proposing a stable and efficient orbital-enriched partition of unity method to solve the Schrödinger boundary-value problem in a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In the proposed partition of unity method, the three-dimensional domain is covered by overlapping patches, with a compactly-supported weight function associated with each patch. A key ingredient in our approach is the use of non-negative weight functions that possess the flat-top property, i.e., each weight function is identically equal to unity over some finite subset of its support. This flat-top property provides a pathway to devise a stable approximation over the whole domain. On each patch, we use $p$th degree orthogonal (Legendre) polynomials that ensure $p$th order completeness, and in addition include eigenfunctions of the radial Schrödinger equation. Furthermore, we adopt a variational lumping approach to construct a (block-)diagonal overlap matrix that yields a standard eigenvalue problem for which there exist efficient eigensolvers. The accuracy, stability, and efficiency of the proposed method is demonstrated for the Schrödinger equation with a harmonic potential as well as a localized Gaussian potential. In conclusion, we show that the proposed approach delivers optimal rates of convergence in the energy, and the use of orbital enrichment significantly reduces the number of degrees of freedom for a given desired accuracy in the energy eigenvalues while the stability of the enriched approach is fully maintained.

42 ENGINEERING↗
DOE OSTI2016

Space-Dependent Wielandt Shift Methods for Multigroup Diffusion Eigenvalue Problems

In the calculation of neutronic eigenfunctions and eigenvalues, it is often desirable to approximate the neutron transport equation by a diffusion equation that requires significantly less computational resources to solve. Even when a full transport solution is needed, the solution of the diffusion system can be used to accelerate iterative methods for calculating the transport eigenfunction and eigenvalue. The convergence rate of the basic power iteration method for eigenvalue problems is governed by the dominance ratio; realistic reactor core problems often have high dominance ratios and the power iteration method can converge slowly as a result. A common technique for accelerating the convergence of power iteration is the Wielandt shift (WS) method, which reduces the dominance ratio by 'shifting' the eigenvalue spectrum using an estimate of the desired eigenvalue. WS provides a significant speedup over power iteration if the eigenvalue estimate is close to the true eigenvalue, but it can be difficult to obtain such an estimate. In practice, the shift in the WS method can be dynamically updated in the iteration scheme, in a way that depends on the eigenvalue estimates from the most recent iterations. We present a physically-motivated Space-Dependent Wielandt Shift (SDWS) for solving multigroup diffusion eigenvalue problems. The space-dependence of the eigenvalue shift in SDWS enables a more effective acceleration of the power iteration scheme than WS at the beginning of the iteration scheme, when the current estimate of the eigenvalue is not close to the actual eigenvalue. Several variants of SDWS are discussed here, including an SDWS method that combines the strengths of SDWS (in the early iterations) with those of standard WS (in the later iterations). Numerical results from a 1-D multigroup diffusion problem demonstrate an order of magnitude speedup over power iteration and a 41% speedup over a standard WS technique. This speedup is problem-dependent and appears to be greater for 'more difficult' problems. In this summary, the term 'speedup' refers to a reduction in the number of iterations required to converge to the correct eigenvalue and eigenfunction. Because the scattering is treated fully implicitly (i.e., it is part of the left-hand side of the diffusion system and directly inverted), the computational cost of a single iteration does not vary significantly among the methods discussed in this summary. The number of iterations is therefore representative of the total computational cost for a method. We acknowledge that certain iterative solvers for linear systems (e.g., GMRES) require more computational time to converge when the system is nearly singular (i.e., when a very aggressive Wielandt shift is applied). Nonetheless, when using SDWS in the manner described in this summary, we have seen that the reduction in computational time from reducing the number of power iterations generally outweighs the extra computational time required by iterative solvers such as GMRES. (authors)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2019

Photonic Crystals Based on Media with Arbitrary Anisotropy of Permittivity and Permeability

A general approach to analyzing eigenmodes in anisotropic and gyrotropic 3D photonic crystals based on dielectric and magnetic media is proposed. This approach is based on the representation of stationary macroscopic Maxwell equations in the operator form corresponding to the quantum-mechanical equation for a photon with spin s = 1. In these equations, the strengths of electric and magnetic fields are put into correspondence with state vectors in the complex Hilbert space. The permittivity and permeability serve as operators acting on these vectors. It is shown that the problem of determining the eigenmodes of a photonic crystal is reduced to searching for eigenvectors and eigenvalues of the Hermitian operator characterizing the spin–orbit interaction of a photon in the periodic anisotropic medium under study. It is proposed to use photon states with a certain wave vector (certain momentum) and a certain linear or circular spin polarization as a basis for the representation operator equations. One-dimensional photonic crystals are considered as an example. The influence of anisotropy and gyrotropy on the dispersion of eigenmodes in these crystals is investigated. The group velocity of the eigenmodes, momentum transferred by them, and spin angular momentum are analyzed for the case of gyrotropic media.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Applying Nonlinear Diffusion Acceleration to Fixed-Source Problems with Anisotropic Scattering - Paper 15

Work over the last several years has demonstrated the impressive performance gains that can be attained by using Moment-Based Acceleration (MBA), or High-Order/Low-Order (HOLO), methods in a variety of transport applications. In the field of neutronics, Nonlinear Diffusion Acceleration (NDA) has been used to accelerate the solution to the fixed-source problem and the k-eigenvalue problem. These methods efficiently move a major portion of the computational work out of the full phase-space (high-order (HO)) domain into a reduced phase-space (low-order (LO)) setting. In previous implementations, NDA has only been applied to isotropic scattering problems. In this paper, we will extend this methodology to anisotropic scattering. This initial work on anisotropic scattering will focus on the 1-D multi-group neutron transport equation. This summary focuses on a fixed-source problem, but the extension to criticality problems is straight-forward via NDA-NCA. Within this paper, we have described a method for extending Nonlinear Diffusion Acceleration to solve problems with anisotropic scattering. We demonstrated that a consistent low-order system, resembling the low-order system from, could be constructed. We demonstrated that the NDA method performed exceptionally in the presence of anisotropic scattering. NDA reduced the number of iterations required to meet the convergence tolerance by a factor of 500 over source iteration for a one fixed-source problem and by 3.5-50 x for the Henyey-Greenstein problems. This method is equally well-suited for eigenvalue problems. The NDA method allows us to transition the eigenvalue iteration to the low-order domain.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2014

Influence of annular boundaries on Thomson's vortex polygon stability

The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle inside an annulus is presented. The problem is reduced to one of the equilibrium stability of the Hamiltonian system with a cyclic variable. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The stability of the stationary motion is interpreted as a Routh stability. The exponential instability is shown always to take place if N ≥ 7. For N = 2, 4, and 6, the parameter space is divided in two: a Routh stability domain in an exact nonlinear setting and an exponential instability domain. For N = 3 and 5, the parameter space consists of three domains. The stability of the third in an exact nonlinear setting is sandwiched between the Routh and exponential domains. Its analysis remains an open problem with its solution requiring nonlinear analysis.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

This work discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2018

Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2017

Model Order Reduction Algorithm for Estimating the Absorption Spectrum

The ab initio description of the spectral interior of the absorption spectrum poses both a theoretical and computational challenge for modern electronic structure theory. Due to the often spectrally dense character of this domain in the quantum propagator’s eigenspectrum for medium-to-large sized systems, traditional approaches based on the partial diagonalization of the propagator often encounter oscillatory and stagnating convergence. Electronic structure methods which solve the molecular response problem through the solution of spectrally shifted linear systems, such as the complex polarization propagator, offer an alternative approach which is agnostic to the underlying spectral density or domain location. This generality comes at a seemingly high computational cost associated with solving a large linear system for each spectral shift in some discretization of the spectral domain of interest. In this work, we present a novel, adaptive solution to this high computational overhead based on model order reduction techniques via interpolation. Model order reduction reduces the computational complexity of mathematical models and is ubiquitous in the simulation of dynamical systems and control theory. The efficiency and effectiveness of the proposed algorithm in the ab initio prediction of X-ray absorption spectra is demonstrated using a test set of challenging water clusters which are spectrally dense in the neighborhood of the oxygen K-edge. On the basis of a single, user defined tolerance we automatically determine the order of the reduced models and approximate the absorption spectrum up to the given tolerance. We also illustrate that, for the systems studied, the automatically determined model order increases logarithmically with the problem dimension, compared to a linear increase of the number of eigenvalues within the energy window. Furthermore, we observed that the computational cost of the proposed algorithm only scales quadratically with respect to the problem dimension.

IDREAM, K-Edge, theoretical, computational, electr↗
DOE OSTI2017

Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems

We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Relativistic levels of mesic atoms

We revisit the derivation of the covariant two-body scalar-fermion equation with a Coulomb interaction, presented in a previous paper. We show that it can be given the formal aspect of a Dirac equation, but for the fact that the eigenvalue is also contained in one of the coefficients and thus it is not linearly included. The discussion of the boundary value problem is therefore different, although some properties of the Dirac equation can be recovered in an approximation bringing back to the concept of reduced mass. We discuss a mixed analytic-numerical method of solution which allows to obtain very accurate results and we calculate the lowest levels, states and QED corrections for pionic and kaonic atoms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Reduction of the molecular hamiltonian matrix using quantum community detection

Abstract Quantum chemistry is interested in calculating ground and excited states of molecular systems by solving the electronic Schrödinger equation. The exact numerical solution of this equation, frequently represented as an eigenvalue problem, remains unfeasible for most molecules and requires approximate methods. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix in Slater determinant basis without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants (as off-diagonal elements) is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. A gauge metric based on perturbation theory is used to determine the lowest energy cluster. This cluster or sub-matrix of Slater determinants is used to calculate approximate ground state and excited state energies within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and bond dissociation cases. These examples provide proof-of-principle results for approximate solution of the electronic structure problem using quantum computing. This approach is general and shows potential to reduce the computational complexity of post-Hartree–Fock methods as future advances in quantum hardware become available.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Colloquium: Eigenvector continuation and projection-based emulators

Eigenvector continuation is a computational method for parametric eigenvalue problems that uses subspace projection with a basis derived from eigenvector snapshots from different parameter sets. It is part of a broader class of subspace-projection techniques called reduced-basis methods. In this Colloquium, the development, theory, and applications of eigenvector continuation and projection-based emulators are presented. In conclusion, the basic concepts are introduced, the underlying theory and convergence properties are discussed, and recent applications for quantum systems and future prospects are presented.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗