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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 OSTI2020

Validation of Thermal Scattering Laws for Light Water at Elevated Temperatures with Diffusion Experiments

Light water is the most important neutron moderator in many reactor applications. Thermal neutron scattering kernels for H bound in H 2 O are represented by ENDF-format thermal scattering laws (TSLs) evaluated at discrete temperatures T . Nuclear data evaluations are conventionally validated utilizing a combination of critical benchmarks and experimental cross section data. Existing public critical benchmarks may be inadequate for testing water TSLs over the full range of T of interest in reactor applications, and experimental thermal scattering cross section data for water is sparse at elevated T . In this work, MC21 is used to simulate the decay of the equilibrium thermal neutron flux in light-water spheres of several radii. The fundamental-mode time eigenvalue $\mathcal{a}$ is calculated at 22 °C and 227 °C (at saturation pressure) for each sphere using three different H-H 2 O TSLs. Fitting $\mathcal{a}$ to a polynomial function of geometric buckling allows calculation of the thermal neutron diffusion length L . Extrapolation lengths are treated as a function of the transport mean free path and geometry. Experimental L data from 25 publications at 49 temperatures (from 10 °C to 295 °C), computed by several different time-dependent and space-dependent decay methods, is used to develop an empirical fit of L vs. T . The MC21-calculated L for the TSLs tested are within ±1% of the predicted values at 22 °C and 227 °C. This validation approach, which may be repeated at arbitrary T , constitutes an integral benchmark specific to and characterizing the detailed physics of the thermal scattering kernel applied.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2023

A Refinement-by-Superposition -Method for (curl)- and (div)-Conforming Discretizations

Here, we present refinement-by-superposition (RBS) hp-refinement infrastructure for computational electromagnetics (CEMs), which permits exponential rates of convergence. In contrast to dominant approaches to hp-refinement for continuous Galerkin methods, which rely on explicit constraint equations, the multilevel strategy presented drastically reduces the implementation complexity. Through the RBS methodology, enforcement of continuity occurs by construction, enabling arbitrary levels of refinement with ease, and without the practical (but not theoretical) limitations of constrained-node refinement. We outline the construction of the RBS hp-method for refinement with H (curl)- and H (div)-conforming finite cells. Numerical simulations for the 2-D finite element method (FEM) solution of the Maxwell eigenvalue problem demonstrate the effectiveness of RBS hp-refinement. As an additional goal of this work, we aim to promote the use of mixed-order (low- and high-order) elements in practical CEM applications.

42 ENGINEERING↗

DOE OSTI2018

Distributed Link Removal Using Local Estimation of Network Topology

In this paper, we consider the problem of network structure manipulation in the absence of information on the global network topology. In particular, the problem of removing some of links is investigated in order to slow or stop the spread of disease in a network while preserving its connectivity. Existing methods solve this combinatorial problem in a centralized manner and they require the global information of network structure. In this paper, we propose a distributed design algorithm to compute a suboptimal solution to this problem efficiently by mimicking gradient based method, namely by iteratively removing one or multiple links at a time from the network. Specifically, using matrix perturbation analysis we formulate an optimization problem involving the eigenvector associated with the largest eigenvalue of the adjacency matrix and whose solution is equal to a suboptimal solution to the original problem. This strategy also enables us to overcome the combinatorial issue of the problem. Distributed algorithms to estimate the eigenvector and to verify network’s connectivity are then proposed which facilitate us to solve the new optimization problem. In addition, topological insights into the proposed algorithm and optimality of its solution are also discussed. Lastly, the proposed distributed design method is demonstrated and evaluated via several numerical examples.

24 POWER TRANSMISSION AND DISTRIBUTION↗

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 OSTI2018

Coherent Transport of Electron Excitations in Organic Solar Cells

The mechanisms of formation and coherent transport of free and bound electron excited states in organic solar cells are considered. In the model of a photocell (one-dimensional chain of photosensitive molecules in a uniform electric field of the p–n junction), the energy eigenvalues and the eigenfunctions of molecular excitons, charge-transfer excitons (CTE), and electron–hole pairs are determined. It is assumed that processes of transport between adjacent sites dominate in the case of the Coulomb interaction between the electron and the hole constituting a CTE. With decreasing Coulomb coupling energy, the CTE wavefunctions become superpositions of localized functions of the increasing number of sites. The decay time determined by independent transitions of the electron and the hole in this case becomes shorter than the transport time of the CTE as a whole. It is shown that autoionization of molecular excitons and small-radius CTEs in a strong electric field of a nanosize chain induces the mixing of states of these excitons as well as of electron–hole pairs, which substantially increases the quantum yield of the photoeffect.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2018

On the Theory of Elastic Scattering of Particles

The problem of elastic scattering of particles has been considered by the potential quantization method. The properties of the eigenfunctions ϕ{sub ν}{sup (+)}(r) and eigenvalues α{sub ν}{sup (+)} of this method for states with positive energies (continuous spectrum) have been studied. Under the assumption of the completeness of the set {ϕ_ν"("+")(r)}, the problem has been solved by expanding the desired wavefunction in this set. The proposed approach reproduces known results in all considered limiting cases. At the same time, it gives a more detailed description of the scattering amplitude than the standard phase theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2019

Green’s Function of the Schrödinger Equation in the Potential Quantization Method

The problem for determining Green’s function G(r, r') for the time-independent Schrödinger equation is considered using the potential quantization method. The expression for G(r, r') is obtained in terms of the eigenfunctions and eigenvalues of this method. The equivalence of the used and traditional methods for solving the given problem has been proven.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

On Inhomogeneous Polarized States near the Phase Transition Point in a Thin Ferroelectric Film

It is shown in terms of the phenomenological Landau theory of phase transitions that a phase transition to an inhomogeneous polar phase preceding in temperature a phase transition to a homogeneous polar state is possible. As a result of solving a boundary eigenvalue problem for the polarization equilibrium equation and electrostatics equations, wave vector k{sub ⊥} characterizing the inhomogeneous phase has been determined and the temperature boundaries of its existence in the dependence on the film thickness and its surface properties have been found.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

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 OSTI2018

A Robust and Efficient Implementation of LOBPCG

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. Here, we also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Polynomial Preconditioned Arnoldi with Stability Control

Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization; for difficult problems, it can also reduce the number of matrix-vector products. Parallel computations can particularly benefit from the reduction of communication-intensive operations. Additoinally, the GMRES algorithm provides a simple and effective way of generating the preconditioning polynomial. For some problems high degree polynomials are especially effective, but they can lead to stability problems that must be mitigated. A two-level “double polynomial preconditioning” strategy provides an effective way to generate high-degree preconditioners.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Performing k eff Validation of As-Loaded Criticality Safety Calculations Using UNF-ST&DARDS: Sensitivity Calculations

The general method for performing validation of as loaded criticality safety calculations using UNF ST&DARDS is presented in a paper by Clarity, which includes a description of the UNF-ST&DARDS system. Proof-of-principle analyses were performed in the summer of 2019 for MPC-32 dual purpose canisters (DPCs) containing pressurized water reactor (PWR) fuel assemblies. Summaries of these results are presented in this and a companion paper for this conference. The current paper describes the TSUNAMI-3D calculations performed to generate sensitivity data, and the companion paper discusses the selection of critical experiments applicable for validation of the 11 MPC-32 DPCs considered. The generation of sensitivity data for as-loaded spent nuclear fuel (SNF) DPCs is a challenge given the detailed model of the fuel compositions generated by UNF ST&DARDS. Each fuel assembly is modeled with its own irradiation history in 18 axial nodes, unless the fuel assembly is damaged and thus considered as fresh by design basis. This results in a set of 576 fuel compositions, each of which must be processed separately in a multigroup (MG) calculation. Therefore, a continuous-energy (CE) TSUNAMI-3D method was chosen to alleviate this challenge. Two CE TSUNAMI-3D methods are available in SCALE 6.2.3: the iterated fission probability (IFP) and contribution-linked eigenvalue sensitivity/uncertainty estimation via track-length importance characterization (CLUTCH). Since the IFP method is not feasible because of memory requirements associated with its implementation in SCALE, the CLUTCH method was selected for these calculations. CLUTCH has been implemented in SCALE in parallel, allowing long calculations to be performed in reasonable timeframes. The two primary user inputs necessary for CLUTCH calculations are the F*(r) mesh and the number of latent generations used in determining the F*(r) function. This F*(r) function is used as the importance function for fission chains originating in a given volume element (voxel), and it is calculated using the IFP method in the skipped generations. A large number of skipped generations is thus required to ensure accurate calculation of this importance function. In these calculations, 500 generations were used to calculate the F*(r) function. For more information regarding the calculation of F*(r), see Jones [4]. The remainder of this paper is focused on the selection of the F*(r) mesh and the number of latent generations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2020

Experiments at Sandia to Measure the Effect of Temperature on Critical Systems

Estimation of the uncertainty in a critical experiment attributable to uncertainties in the measured experiment temperature is done by calculating the variation of the eigenvalue of a benchmark configuration as a function of temperature. In the low-enriched water-moderated critical experiments performed at Sandia, this is done by 1) estimating the effects of changing the water temperature while holding the UO 2 fuel temperature constant, 2) estimating the effects of changing the UO 2 temperature while holding the water temperature constant, and 3) combining the two results. This assumes that the two effects are separable. The results of such an analysis are nonintuitive and need experimental verification. Critical experiments are being planned at Sandia National Laboratories (Sandia) to measure the effect of temperature on critical systems and will serve to test the methods used in estimating the temperature effects in critical experiments.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Sampling electronic structure quadratic unconstrained binary optimization problems (QUBOs) with Ocean and Mukai solvers

The most advanced D-Wave Advantage quantum annealer has 5000+ qubits, however, every qubit is connected to a small number of neighbors. As such, implementation of a fully-connected graph results in an order of magnitude reduction in qubit count. To compensate for the reduced number of qubits, one has to rely on special heuristic software such as qbsolv, the purpose of which is to decompose a large quadratic unconstrained binary optimization (QUBO) problem into smaller pieces that fit onto a quantum annealer. In this work, we compare the performance of the open-source qbsolv which is a part of the D-Wave Ocean tools and a new Mukai QUBO solver from Quantum Computing Inc. (QCI). The comparison is done for solving the electronic structure problem and is implemented in a classical mode (Tabu search techniques). The Quantum Annealer Eigensolver is used to map the electronic structure eigenvalue-eigenvector equation to a QUBO problem, solvable on a D-Wave annealer. We find that the Mukai QUBO solver outperforms the Ocean qbsolv with one to two orders of magnitude more accurate energies for all calculations done in the present work, both the ground and excited state calculations. This work stimulates the further development of software to assist in the utilization of modern quantum annealers.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Experiments at Sandia to Measure the Effect of Temperature on Critical Systems [Slides]

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2020

Validation of Recent Changes in Actinide Evaluations U-235, U-233, Pu-239 [Slides]

This presentation discusses U-235 updates, including fluctuations in the fission cross section in the URR range that were refined to represent measured data following the Paradela et al. evaluation. There were small changes to the low-energy resonances to improve the fit to measured capture data from RPI (capture reduced by 5% from 0.06-7.8 eV, by 7.7% from 7.8-11 eV) and there was a very small change to the thermal nu-bar within uncertainties. Discrete level data are stored in MF=6 (format requirement, no impact on calculations) and issues leading to negative eigenvalues in cross-covariances in the RR have been sorted out. In conclusion, many improvements were made to evaluated nuclear data of actinides in the resonance region and above, such as minor changes to U-235 including assessment of changes that depend on the finalization of the O-16 evaluation. U-233 improvements show significant progress in performance and some further improvement might be needed at intermediate energies. Pu-239 ORNL/IAEA evaluation is promising; however, more work is needed.

07 ISOTOPE AND RADIATION SOURCES↗

DOE OSTI2020

Covariance Testing at BNL [Slides]

This presentation discusses testing implemented in FUDGE & ADVANCE in regard to proper ENDF formatting: STAN, STANEF, CHECKR, FUDGE, NJOY, convert to BOXR format, plotting, endfcovreport.py which is an overview of ENDF library’s covariance contents, and covariance Matrix class implements that are the core of all GNDS covariance. Several checks are performed, including symmetric, real, positive definite, condition # of Eigenvalues, and too big uncertainties. The presentation states that it can fix too big and/or small uncertainties and that it can plot matrices. It states that further testing is recommended by CSEWG but has not yet been implemented and suggests methods such as documentation which needs a human to execute, completeness which is partially complete, and reasonableness which has not been started. The presentation concludes by elaborating on reasonableness and states that only cross section is documented, that similar limits on CP reactions cross sections are needed, that maximum limits are needed too which is tricky since ENDF assumes Normal PDFs, and that there may be other observables.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗