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DOE OSTI2020

A robust and efficient generalized real cascade model

This paper describes a new separation cascade model that allows for arbitrary cascade connectivity and includes the effects of mixing, material losses and generalized separation element performance. The model also accommodates mixtures with an arbitrary number of component species. The mathematical and numerical framework of this model includes both time-dependent and steady state cascade simulation. Robust numerical approximations of the continuous mathematical model are accomplished using the Sundials suite of nonlinear algebraic and differential-algebraic equation solvers. Verification of the model is accomplished using manufactured solutions while validation is demonstrated by comparison to published experimental results and benchmarks.

36 MATERIALS SCIENCE↗

DOE OSTI2017

Parallel Asynchronous Distributed-Memory Maximal Independent Set Algorithm with Work Ordering

The maximal independent set (MIS) graph problem arises in many applications such as computer vision, information theory, molecular biology, and process scheduling. The growing scale of graph data suggests the use of distributed memory hardware as a cost-effective approach to providing necessary compute and memory resources. Existing distributed memory parallel MIS algorithms rely on synchronous communication and use techniques such as subgraph computations. In this paper, we present an asynchronous distributed-memory parallel graph algorithm that relies on a virtual directed acyclic graph (DAG) that is created during the algorithm execution. We introduce two additional algorithms that save computations by ordering generated work. The first algorithm applies ordering globally to reduce computations, and the second algorithm applies ordering locally at the level of threads to minimize the synchronization overhead. We use two different implementations of Luby’s algorithm variants as baseline to compare the performance of the presented algorithms: (1) vertex-centric Luby A and Luby B implementations, and (2) the CombBLAS linear-algebra Luby A implementation. Results show that proposed algorithms outperform both implementations of Luby algorithms, especially in distributed execution. Furthermore, we show that for low-diameter graphs the algorithm that applies global ordering scales better than other algorithms and for high diameter graphs the original asynchronous algorithm and thread-level ordering algorithm show better performance.

maximal independent set (MIS), distributed memory ↗

DOE OSTI2015

On the Design, Development, and Analysis of Optimized Matrix-Vector Multiplication Routines for Coprocessors

The manycore paradigm shift, and the resulting change in modern computer architectures, has made the development of optimal numerical routines extremely challenging. In this work, we target the development of numerical algorithms and implementations for Xeon Phi coprocessor architecture designs. In particular, we examine and optimize the general and symmetric matrix-vector multiplication routines (gemv/symv), which are some of the most heavily used linear algebra kernels in many important engineering and physics applications. We describe a successful approach on how to address the challenges for this problem, starting with our algorithm design, performance analysis and programing model and moving to kernel optimization. Our goal, by targeting low-level and easy to understand fundamental kernels, is to develop new optimization strategies that can be effective elsewhere for use on manycore coprocessors, and to show significant performance improvements compared to existing state-of-the-art implementations. Therefore, in addition to the new optimization strategies, analysis, and optimal performance results, we finally present the significance of using these routines/strategies to accelerate higher-level numerical algorithms for the eigenvalue problem (EVP) and the singular value decomposition (SVD) that by themselves are foundational for many important applications.

Kabir, Khairul↗

DOE OSTI2023

Mixed Precision Numerical Linear Algebra (Final Report)

The objective of this subcontract was to identify opportunities for the use of mixed precision within iterative solvers and to explore these opportunities both theoretically and experimentally. All quarterly milestones were achieved. Quarterly achievements are summarized in sections.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Diffusion Synthetic Acceleration for High Order S{sub N} Transport on Meshes with Curved Surfaces

Thermal radiation transport is integral to multiphysics applications such as inertial confinement fusion and astrophysics. Materials in these problems can be exceptionally optically thick and diffusive. A transport solver must be capable of accurate and efficient solutions to such problems. Previously, Woods et al. established a proof-of-concept for high order (HO) discontinuous Galerkin finite element (DGFEM) transport on meshes with curved surfaces. They numerically demonstrated the spatial convergence is order p + 1. Remarkably, these convergence rates are relatively invariant to the amount of curvature of the spatial mesh. We present here an extension of this work that demonstrates a rapidly-convergent iterative solver for the FEM transport equations on meshes with curved sides. We use the modified interior penalty (MIP) diffusion synthetic acceleration (DSA) established by Wang and Ragusa. The partially consistent MIP equations stem from the symmetric interior penalty (IP) derivation of the diffusion equation. The IP method is not stable for optically thick cells so it was combined with the diffusion conforming form (DCF), a discretized diffusion equation derived from the discretized S N equations that is not stable for intermediate optical thicknesses. The MIP equations adapt a 'switch' to the IP method in optically thinner regions and the DCF method in optically thicker regions. The literature does not reveal any Fourier analyses performed for iterative transport on meshes with curved surfaces. Wang and Ragusa performed a Fourier analysis for orthogonal meshes and our results appear to be as expected. Previous literature has reported on some of the sensitivities that affect the spectral radius of DSA schemes: polynomial order, and scattering ratios. Presently, we focus on the sensitivity of the spectral radius to polynomial order, scattering ratio, the constant C, and mesh curvature for varying cell thicknesses. We also numerically study the behavior of the spectral radius to verify our DSA implementation. A sequence of problems is designed to exercise the acceleration. As the problems become more optically thick and diffusive, the accelerated solution continues to converge toward the analytic solution. We employ the Modular Finite Element Methods (MFEM) package, developed at Lawrence Livermore National Laboratory, to generate the linear algebraic system of equations for each angle in the discrete ordinates quadrature set, and use UMFPack to solve this system of equations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2017

Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners

This study presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits domain decomposition (DD) and low-rank corrections. The DD approach decouples the matrix and, once inverted, a low-rank approximation is applied by exploiting the Sherman--Morrison--Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisons with pARMS, a DD-based parallel incomplete LU (ILU) preconditioning method, are presented for solving Poisson's equation and linear elasticity problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

“Gray Zone” Simulations Using a Three-Dimensional Planetary Boundary Layer Parameterization in the Weather Research and Forecasting Model

Abstract Generating accurate weather forecasts of planetary boundary layer (PBL) properties is challenging in many geographical regions, oftentimes due to complex topography or horizontal variability in, for example, land characteristics. While recent advances in high-performance computing platforms have led to an increase in the spatial resolution of numerical weather prediction (NWP) models, the horizontal gridcell spacing (Δ x ) of many regional-scale NWP models currently fall within or are beginning to approach the gray zone (i.e., Δ x ≈ 100–1000 m). At these gridcell spacings, three-dimensional (3D) effects are important, as the most energetic turbulent eddies are neither fully parameterized (as in traditional mesoscale simulations) nor fully resolved [as in traditional large-eddy simulations (LES)]. In light of this modeling challenge, we have implemented a 3D PBL parameterization for high-resolution mesoscale simulations using the Weather Research and Forecasting Model. The PBL scheme, which is based on the algebraic model developed by Mellor and Yamada, accounts for the 3D effects of turbulence by calculating explicitly the momentum, heat, and moisture flux divergences in addition to the turbulent kinetic energy. In this study, we present results from idealized simulations in the gray zone that illustrate the benefit of using a fully consistent turbulence closure framework under convective conditions. While the 3D PBL scheme reproduces the evolution of convective features more appropriately than the traditional 1D PBL scheme, we highlight the need to improve the turbulent length scale formulation. Significance Statement The spatial resolution of weather models continues to increase at a rapid rate in accordance with the enhancement of computing power. As a result, smaller-scale atmospheric features become more explicitly resolved. However, most numerical models still ignore the impact of horizontal weather variations on boundary layer flows, which becomes more important at these smaller spatial scales. To address this issue, we have implemented a new modeling approach, using fundamental principles, which accounts for horizontal variability. Our results show that including three-dimensional effects of turbulence is necessary to achieve realistic boundary layer characteristics. This novel technique may be useful for many applications including complex terrain flows, pollutant dispersion, and surface–atmosphere interaction studies.

Juliano, Timothy W.↗

DOE OSTI2021

Hyperbolic three-string vertex

We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

The influence of interval arithmetic on the shape of uncertainly defined domains modelled by closed curves

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C{sup 2} continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Analytical transport network theory to guide the design of 3-D microstructural networks in energy materials: Part 1. Flow without reactions

Here we present a fully analytical, heuristic model – the “Analytical Transport Network Model” – for steady-state, diffusive, potential flow through a 3-D network. Employing a combination of graph theory, linear algebra, and geometry, the model explicitly relates a microstructural network's topology and the morphology of its channels to an effective material transport coefficient (a general term meant to encompass, e.g., conductivity or diffusion coefficient). The model's transport coefficient predictions agree well with those from electrochemical fin (ECF) theory and finite element analysis (FEA), but are computed 0.5–1.5 and 5–6 orders of magnitude faster, respectively. In addition, the theory explicitly relates a number of morphological and topological parameters directly to the transport coefficient, whereby the distributions that characterize the structure are readily available for further analysis. Furthermore, ATN's explicit development provides insight into the nature of the tortuosity factor and offers the potential to apply theory from network science and to consider the optimization of a network's effective resistance in a mathematically rigorous manner. The ATN model's speed and relative ease-of-use offer the potential to aid in accelerating the design (with respect to transport), and thus reducing the cost, of energy materials.

25 ENERGY STORAGE↗

DOE OSTI2014

An extension of the OpenModelica compiler for using Modelica models in a discrete event simulation

In this article, a new back-end and run-time system is described for the OpenModelica compiler. This new back-end transforms a Modelica model into a module for the adevs discrete event simulation package, thereby extending adevs to encompass complex, hybrid dynamical systems. The new run-time system that has been built within the adevs simulation package supports models with state-events and time-events and that comprise differential-algebraic systems with high index. Finally, although the procedure for effecting this transformation is based on adevs and the Discrete Event System Specification, it can be adapted to any discrete event simulation package.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Pu239 Cross-Section Variations Based on Experimental Uncertainties and Covariances

Algorithms and software have been developed for producing variations in plutonium-239 neutron cross sections based on experimental uncertainties and covariances. The varied cross-section sets may be produced as random samples from the multi-variate normal distribution defined by an experimental mean vector and covariance matrix, or they may be produced as Latin-Hypercube/Orthogonal-Array samples (based on the same means and covariances) for use in parametrized studies. The variations obey two classes of constraints that are obligatory for cross-section sets and which put related constraints on the mean vector and covariance matrix that detemine the sampling. Because the experimental means and covariances do not obey some of these constraints to sufficient precision, imposing the constraints requires modifying the experimental mean vector and covariance matrix. Modification is done with an algorithm based on linear algebra that minimizes changes to the means and covariances while insuring that the operations that impose the different constraints do not conflict with each other.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Threaded Multi-Core GEMM with MoA and Cache-Blocking: Preprint

A threaded multi-core implementation of the high performance dense linear algebra matrix-matrix multiply GEMM kernel is described. This kernel is widely implemented by vendors in the basic linear algebra subroutine BLAS library. The mathematics of arrays (MoA) paradigm due to Mullin (1988) results in contiguous memory accesses by employing outer-product forms. Our performance studies demonstrate that the MoA implementation of double precision DGEMM combined with optimal cache-blocking strategies results in at least a 25% performance gain on the Intel Xeon Skylake processor over the vendor supplied Intel MKL basic linear algebra libraries. Results are presented for the NREL Eagle supercomputer. The multi-core DGEMM achieves over 100 GigaFlops/sec with eight openMP threads.

cache-blocking↗

DOE OSTI2021

Mixed Precision Numerical Linear Algebra (Final Report)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Mixed Precision Numerical Linear Algebra (Final Report)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

SCF Framework, HF Stability, and RPA Correlation for Jordan–Wigner-Transformed Spin Hamiltonians on Arbitrary Coupling Topologies

Mapping spins to fermions via the Jordan–Wigner (JW) transformation can render mean-field (Hartree–Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytical orbital Hessian to diagnose HF stability and compute the ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J 1 –J 2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Window convolution of the galaxy clustering bispectrum

In galaxy survey analysis, the observed clustering statistics do not directly match theoretical predictions but rather have been processed by a window function that arises from the survey geometry including the sky footprint, redshift-dependent background number density and systematic weights. While window convolution of the power spectrum is well studied, for the bispectrum with a larger number of degrees of freedom, it poses a significant numerical and computational challenge. In this work, we consider the effect of the survey window in the tripolar spherical harmonic decomposition of the bispectrum and lay down a formal procedure for their convolution via a series expansion of configuration-space three-point correlation functions, which was first proposed by Sugiyama et al. (2019). We then provide a linear algebra formulation of the full window convolution, where an unwindowed bispectrum model vector can be directly premultiplied by a window matrix specific to each survey geometry. To validate the pipeline, we focus on the Dark Energy Spectroscopic Instrument (DESI) Data Release 1 (DR1) luminous red galaxy (LRG) sample in the South Galactic Cap (SGC) in the redshift bin 0.4 ≤ z ≤ 0.6. We first perform convergence checks on the measurement of the window function from discrete random catalogues, and then investigate the convergence of the window convolution series expansion truncated at a finite of number of terms as well as the performance of the window matrix. This work highlights the differences in window convolution between the power spectrum and bispectrum, and provides a streamlined pipeline for the latter for current surveys such as DESI and the Euclid mission.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

A generalized Selberg zeta function for flat space cosmologies

Flat space cosmologies (FSCs) are time dependent solutions of three-dimensional (3D) gravity with a vanishing cosmological constant. They can be constructed from a discrete quotient of empty 3D flat spacetime and are also called shifted-boost orbifolds. Using this quotient structure, we build a new and generalized Selberg zeta function for FSCs, and show that it is directly related to the scalar 1-loop partition function. We then propose an extension of this formalism applicable to more general quotient manifolds $\mathcal{M}$/ℤ, based on representation theory of fields propagating on this background. Our prescription constitutes a novel and expedient method for calculating regularized 1-loop determinants, without resorting to the heat kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg zeta function, and match them to known results.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗