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DOE OSTI2018

A low-order block preconditioner for saddle point linear systems

A preconditioner is proposed for the large and sparse linear saddle point problems, which is based on a low-order three-by-three block saddle point form. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial for the preconditioned matrix are discussed. Numerical results show that the optimal convergence behavior can be achieved when the new preconditioner is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Rationally extended many-body truncated Calogero–Sutherland model

We construct a rational extension of the truncated Calogero–Sutherland model by Pittman et al. The exact solution of this rationally extended model is obtained analytically and it is shown that while the energy eigenvalues remain unchanged, however the eigenfunctions are completely different and written in terms of exceptional X{sub 1} Laguerre orthogonal polynomials. The rational model is further extended to a more general X{sub m} case by introducing m dependent interaction term. As expected, in the special case of m=0, the extended model reduces to the conventional model of Pittman et al. In the two appropriate limits, we thereby obtain rational extensions of the celebrated Calogero–Sutherland as well as Jain–Khare models. The multi-index extension of the model is also discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Ab-initio simulation of spin-vibronic spectra of methoxy radical

Despite the fact that experimental and theoretical work on the spectrum of methoxy has stretched from the microwave to the ultraviolet and proceeded for nearly 50 years, parts of the spectrum have remained a challenge to simulate theoretically and make reliable line-by-line assignments. The spectral complexity arises because the radical has a non-zero electron spin and significant vibronic coupling between the two elec- tronic components of the ground state due to the presence of a conical intersection. This work describes a completely ab initio effort to understand and assign the spin- vibronic levels of the X 2E state from 0 to above 3000 cm−1, a region that includes the fundamental transitions of the C-H symmetric and asymmetric stretches that have not previously been identified uniquely. A potential energy surface for methoxy was calculated at the EOM-CCSDT/ANO1 level of theory. Subsequently this potential energy surface was fit to a quartic power series expansion of all nine vibrational nor- mal coordinates (as determined at the minimum of the conical intersection) by the use of a machine-learning-based algorithm. After the addition of spin-orbit coupling, the spin-vibronic problem was solved using both the Krylov-Schur and Lanczos algorithms with the SOCJT3 software to converge eigenvalues up to 3500 cm−1 and their eigen- vectors. The latter were used, in conjunction with the calculated dipole moment and its derivatives (calculated using finite differences at EOM-CCSDT/ANO1 level), to determine spectral intensities for the spin-vibronic spectra. The calculated transition frequencies and intensities were used to simulate and assign the observed transitions of the spin-vibronic spectra of the radical. The credibility of the assignments and their significance is discussed in detail.

Sharma, Ketan [University of Florida, Gainesville,↗

DOE OSTI2016

Asymptotic Representation for the Eigenvalues of a Non-selfadjoint Operator Governing the Dynamics of an Energy Harvesting Model

We consider a well known model of a piezoelectric energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler–Bernoulli model for the transverse vibrations of the beam and the second one represents the Kirchhoff’s law for the electric circuit. Both equations are coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. Our main result is an explicit asymptotic formula for the eigenvalues of this generator, i.e., we perform the modal analysis for electrically loaded (not short-circuit) system. We show that the spectrum splits into an infinite sequence of stable eigenvalues that approaches a vertical line in the left half plane and possibly of a finite number of unstable eigenvalues. This paper is the first in a series of three works. In the second one we will prove that the generalized eigenvectors of the dynamics generator form a Riesz basis (and, moreover, a Bari basis) in the energy space. In the third paper we will apply the results of the first two to control problems for this model.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Comparison of two Galerkin quadrature methods

Here, we compare two methods for generating Galerkin quadratures. In method 1, the standard S N method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard S N method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwise sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding S N equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2016

COMPARISONS OF THE FINITE-ELEMENT-WITH-DISCONTIGUOUS-SUPPORT METHOD TO CONTINUOUS-ENERGY MONTE CARLO FOR PIN-CELL PROBLEMS

The standard multigroup (MG) method for energy discretization of the transport equation can be sensitive to approximations in the weighting spectrum chosen for cross-section averaging. As a result, MG often inaccurately treats important phenomena such as self-shielding variations across a material. From a finite-element viewpoint, MG uses a single fixed basis function (the pre-selected spectrum) within each group, with no mechanism to adapt to local solution behavior. In this work, we introduce the Finite-Element-with-Discontiguous-Support (FEDS) method, whose only approximation with respect to energy is that the angular flux is a linear combination of unknowns multiplied by basis functions. A basis function is non-zero only in the discontiguous set of energy intervals associated with its energy element. Discontiguous energy elements are generalizations of bands and are determined by minimizing a norm of the difference between snapshot spectra and their averages over the energy elements. We begin by presenting the theory of the FEDS method. We then compare to continuous-energy Monte Carlo for one-dimensional slab and two-dimensional pin-cell problem. We find FEDS to be accurate and efficient at producing quantities of interest such as reaction rates and eigenvalues. Results show that FEDS converges at a rate that is approximately first-order in the number of energy elements and that FEDS is less sensitive to weighting spectrum than standard MG.

Cross sections↗

DOE OSTI2016

An Asynchronous Task-based Fan-Both Sparse Cholesky Solver

Systems of linear equations arise at the heart of many scientific and engineering applications. Many of these linear systems are sparse; i.e., most of the elements in the coefficient matrix are zero. Direct methods based on matrix factorizations are sometimes needed to ensure accurate solutions. For example, accurate solution of sparse linear systems is needed in shift-invert Lanczos to compute interior eigenvalues. The performance and resource usage of sparse matrix factorizations are critical to time-to-solution and maximum problem size solvable on a given platform. In many applications, the coefficient matrices are symmetric, and exploiting symmetry will reduce both the amount of work and storage cost required for factorization. When the factorization is performed on large-scale distributed memory platforms, communication cost is critical to the performance of the algorithm. At the same time, network topologies have become increasingly complex, so that modern platforms exhibit a high level of performance variability. This makes scheduling of computations an intricate and performance-critical task. In this paper, we investigate the use of an asynchronous task paradigm, one-sided communication and dynamic scheduling in implementing sparse Cholesky factorization (symPACK) on large-scale distributed memory platforms. Our solver symPACK relies on efficient and flexible communication primitives provided by the UPC++ library. Performance evaluation shows good scalability and that symPACK outperforms state-of-the-art parallel distributed memory factorization packages, validating our approach on practical cases.

Jacquelin, Mathias↗

DOE OSTI2022

Investigation of methods for targeted search of dominant higher modes in subcritical systems

The reactor physics modeling of subcritical systems requires a substantially different approach from that of a critical reactor due to the presence of the so-called higher modes. Earlier investigations showed that the dominant higher modes lie in the inner part of the eigenvalue spectrum, making the conventional eigenvalue searches unfeasible and calling for a targeted search. This paper investigates the possibility of targeted and multitudinous eigenvalue calculation by testing the Krylov-Schur and the Dynamic Modes Decomposition (DMD) method for simple, analytically tractable problems. While the DMD method clearly showed its efficiency in a one-group homogeneous problem, modal analysis of a three-dimensional reflected reactor in two-groups approximation demonstrated that the increasing complexity poses challenges for both methods. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2014

On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations

The Karhunen–Lòeve (KL) decomposition provides a low-dimensional representation for random fields as it is optimal in the mean square sense. Although for many stochastic systems of practical interest, described by stochastic partial differential equations (SPDEs), solutions possess this low-dimensional character, they also have a strongly time-dependent form and to this end a fixed-in-time basis may not describe the solution in an efficient way. Motivated by this limitation of standard KL expansion, Sapsis and Lermusiaux (2009) [26] developed the dynamically orthogonal (DO) field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty ‘lives’ but also the stochastic characteristics of uncertainty. Recently, Cheng et al. (2013) [28] introduced an alternative approach, the bi-orthogonal (BO) method, which performs the exact same tasks, i.e. it evolves the spatial basis and the stochastic characteristics of uncertainty. In the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence, in the sense that one method is an exact reformulation of the other. We show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the BO and the DO solutions. We also examine a pathology of the BO equations that occurs when two eigenvalues of the solution cross, resulting in an instantaneous, infinite-speed, internal rotation of the computed spatial basis. We demonstrate that despite the instantaneous duration of the singularity this has important implications on the numerical performance of the BO approach. On the other hand, it is observed that the BO is more stable in nonlinear problems involving a relatively large number of modes. Several examples, linear and nonlinear, are presented to illustrate the DO and BO methods as well as their equivalence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Data-driven probability concentration and sampling on manifold

A new methodology is proposed for generating realizations of a random vector with values in a finite-dimensional Euclidean space that are statistically consistent with a dataset of observations of this vector. The probability distribution of this random vector, while a priori not known, is presumed to be concentrated on an unknown subset of the Euclidean space. A random matrix is introduced whose columns are independent copies of the random vector and for which the number of columns is the number of data points in the dataset. The approach is based on the use of (i) the multidimensional kernel-density estimation method for estimating the probability distribution of the random matrix, (ii) a MCMC method for generating realizations for the random matrix, (iii) the diffusion-maps approach for discovering and characterizing the geometry and the structure of the dataset, and (iv) a reduced-order representation of the random matrix, which is constructed using the diffusion-maps vectors associated with the first eigenvalues of the transition matrix relative to the given dataset. The convergence aspects of the proposed methodology are analyzed and a numerical validation is explored through three applications of increasing complexity. The proposed method is found to be robust to noise levels and data complexity as well as to the intrinsic dimension of data and the size of experimental datasets. Both the methodology and the underlying mathematical framework presented in this paper contribute new capabilities and perspectives at the interface of uncertainty quantification, statistical data analysis, stochastic modeling and associated statistical inverse problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Tunable phononic bandgap materials designed via topology optimization

Topology optimization is used to design phononic bandgap materials that are tunable by mechanical deformation. A periodic media is considered, which due to the assumption of length scale separation, allows the dispersion relations to be obtained by analyzing a single unit cell subjected to Floquet–Bloch boundary conditions. A finite macroscopic deformation is applied to the unit cell to affect its geometry and hence dispersion. We tune the dispersion–deformation relation to our liking by solving a topology optimization problem using nonlinear programming. The adjoint method is employed to compute the sensitivities, and the non-differentiability of degenerate eigenvalues is avoided using symmetric polynomials. Several tunable phononic crystal designs are presented. Also, a verification analysis is performed, wherein the optimized design is interpreted and analyzed using a conforming finite element mesh.

42 ENGINEERING↗

DOE OSTI2026

Analog and symbolic computation through the Koopman framework

We develop a Koopman operator framework for studying the computational structure of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ‘Koopman halting problem’ that is recursively enumerable in general. For symbolic systems, such as those defined on Cantor space, this operator formulation captures reachability between clopen sets, while for equicontinuous systems we prove that the Koopman halting problem is decidable. Our framework demonstrates that absorbing (halting) states in coarse-grained finite automata correspond to Koopman eigenfunctions with eigenvalue one, while cycles in the transition graph impose spectral constraints associated with periodic dynamics. These results provide a unifying perspective on computation in symbolic and analog systems, showing how computational universality is reflected in operator spectra, invariant subspaces, and algebraic structures. Beyond symbolic dynamics, this operator-theoretic lens opens pathways to analyze the computational properties of a broader class of dynamical systems, including polynomial and analog models, and suggests that computational hardness may admit dynamical signatures in terms of Koopman spectral structure.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Improved Fast Reactor Capability of Griffin in FY23

Griffin is a MOOSE based reactor multiphysics analysis application jointly developed by Idaho National Laboratory and Argonne National Laboratory under the DOE-NE NEAMS program. In FY23, we enhanced capabilities required for fast reactor analysis. This effort included primarily updating the cross-section generation workflow using MC2-3 for various reactor configurations, such as homogeneous, duct-heterogeneous, ring-heterogeneous, and fully-heterogeneous geometries. In addition, we initiated the implementation of a multi-cycle depletion and shuffling capability. To support fast reactor simulation capabilities, we significantly improved the performance of the DFEM-SN-based R-Z transport solver to efficiently solve ultrafine group (over 1000 groups) transport problems. Additionally, the performance of HFEM-PN was improved by introducing red-black iteration, the cmfd acceleration technique, and various optimizations. We also completed the pin power reconstruction capability to support multiphysics simulations while identifying and addressing issues associated with SPH equivalence parameter approach. These enhanced capabilities for fast reactor core simulations, specially HFEM-PN and pin power reconstruction features, were applied to benchmark problems involving ABTR and ABR-1000. These applications showcased excellent agreement with Monte Carlo and other code solutions in terms of eigenvalue, control rod worth, and assembly and pin powers.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2019

Micro Reactor Simulation Using the PROTEUS Suite in FY19

Two micro reactor cores were simulated using the MOC solver of PROTEUS: MegaPower for a heat pipe micro reactor and Holos-Quad for a high-temperature gas-cooled micro reactor. Verification of PROTEUS standalone was performed by simulating 2D and 3D core benchmark problems developed based on MegaPower and Holos-Quad. Simulation results indicated that PROTEUS successfully modelled 2D and 3D cores of the MegaPower and Holos-Quad micro reactors and PROTEUS eigenvalues agreed well with the SERPENT Monte Carlo solutions. For a heat pipe cooled micro reactor, multiphysics simulations were performed using PROTEUS / FLUENT / ANLHTP. The coupling system of the three codes were developed using Python-based drivers, which exchange the data (power, temperature, and heat rate) required for the coupling and control the boundary conditions of FLUENT and ANLHTP. The coupled system was qualitatively verified using a 3D unit assembly problem composed of six fuel rods and seven heat pipes, which was developed based on the MegaPower compositions. The steady-state problem was solved using FLUENT / ANLHTP with given asymmetric fuel powers, demonstrating the reasonable convergence of temperatures of 7 heat pipes. Transient simulations were also performed by making one out of 7 heat pipes fail. For the transient, the coupled system of PROTEUS / FLUENT / ANLHTP demonstrated reasonable changes of total power and heat pipe temperatures with time, accounting for temperature feedback effects. In this simulation, no effect of structural change was accounted for, which will be performed using ANSYS-mechanical in future along with further verification with larger or whole-core 3D problems.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2015

Complex network approach to fractional time series

In order to extract correlation information inherited in stochastic time series, the visibility graph algorithm has been recently proposed, by which a time series can be mapped onto a complex network. We demonstrate that the visibility algorithm is not an appropriate one to study the correlation aspects of a time series. We then employ the horizontal visibility algorithm, as a much simpler one, to map fractional processes onto complex networks. The degree distributions are shown to have parabolic exponential forms with Hurst dependent fitting parameter. Further, we take into account other topological properties such as maximum eigenvalue of the adjacency matrix and the degree assortativity, and show that such topological quantities can also be used to predict the Hurst exponent, with an exception for anti-persistent fractional Gaussian noises. To solve this problem, we take into account the Spearman correlation coefficient between nodes' degrees and their corresponding data values in the original time series.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Computational Aspects of Single-Molecule Kinetics for Coupled Catalytic Cycles: A Spectral Analysis

Catalysis from single active sites is analyzed using methods developed from single molecule kinetics. Using a stochastic Markov state description, the observable properties of general catalytic networks of reactions are expressed using an eigenvalue decomposition of the transition matrix for the Markov process. By the use of a sensitivity analysis, the necessary eigenvalues and eigenvectors are related to the energies of controlling barriers and wells located along the reaction routes. A generalization of the energetic span theory allows the eigenvalues to be computed from several activation energies corresponding to distinct barrier-well pairings. The formalism is demonstrated accurately for model problems for a physically realistic mechanism for an alkene hydrogenation reaction on a single atom catalyst.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2015

Quantum chi-squared and goodness of fit testing

A quantum mechanical hypothesis test is presented for the hypothesis that a certain setup produces a given quantum state. Although the classical and the quantum problems are very much related to each other, the quantum problem is much richer due to the additional optimization over the measurement basis. A goodness of fit test for i.i.d quantum states is developed and a max-min characterization for the optimal measurement is introduced. We find the quantum measurement which leads both to the maximal Pitman and Bahadur efficiencies, and determine the associated divergence rates. We discuss the relationship of the quantum goodness of fit test to the problem of estimating multiple parameters from a density matrix. These problems are found to be closely related and we show that the largest error of an optimal strategy, determined by the smallest eigenvalue of the Fisher information matrix, is given by the divergence rate of the goodness of fit test.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2015

Chaotic dynamics and diffusion in a piecewise linear equation

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗