Results for “numerical algorithm computing” — Page 6

NASA NTRS1995

Time-domain approach for the transient responses in stratified viscoelastic Earth models

We have developed the numerical algorithm for the computation of transient viscoelastic responses in the time domain for a radially stratified Earth model. Stratifications in both the elastic parameters and the viscosity profile have been considered. The particular viscosity profile employed has a viscosity maximum with a constrast of O(100) in the mid lower mantle. The distribution of relaxation times reveals the presence of a continuous spectrum situated between O(100) and O(exp 4) years. The principal mode is embedded within this continuous spectrum. From this initial-value approach we have found that for the low degree harmonics the non-modal contributions are comparable to the modal contributions. For this viscosity model the differences between the time-domain and normal-mode results are found to decrease strongly with increasing angular order. These calculations also show that a time-dependent effective relaxation time can be defined, which can be bounded by the relaxation times of the principal modes.

Hanyk, L.↗

NASA NTRS2000

VCSEL Applications and Simulation

This viewgraph presentation gives an overview of Vertical Cavity Surface Emitting Laser (VCSEL) simulation and its applications. Details are given on the optical interconnection in information technology of VCSEL, the formulation of the simulation, its numeric algorithm, and the computational results.

Cheung, Samson↗

DOE OSTI2025

Universal energy-speed-accuracy trade-offs in driven nonequilibrium systems

The connection between measure theoretic optimal transport and dissipative nonequilibrium dynamics provides a language for quantifying nonequilibrium control costs, leading to a collection of thermodynamic speed limits, which rely on the assumption that the target probability distribution is perfectly realized. This is almost never the case in experiments or numerical simulations, so here we address the situation in which the external controller is imperfect. We obtain a lower bound for the dissipated work in generic nonequilibrium control problems that (1) is asymptotically tight and (2) matches the thermodynamic speed limit in the case of optimal driving. Along with analytically solvable examples, we refine this imperfect driving notion to systems in which the controlled degrees of freedom are slow relative to the nonequilibrium relaxation rate, and identify independent energy contributions from fast and slow degrees of freedom. Furthermore, we develop a strategy for optimizing minimally dissipative protocols based on optimal transport flow matching, a generative machine learning technique. Furthermore, this latter approach ensures the scalability of both the theoretical and computational framework we put forth. Crucially, we demonstrate that we can compute the terms in our bound numerically using efficient algorithms from the computational optimal transport literature and that the protocols we learn saturate the bound.

59 BASIC BIOLOGICAL SCIENCES↗

NASA NTRS1991

The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations.

Beam, Richard M.↗

NASA NTRS1977

Numerical comparison of Kalman filter algorithms - Orbit determination case study

Numerical characteristics of various Kalman filter algorithms are illustrated with a realistic orbit determination study. The case study of this paper highlights the numerical deficiencies of the conventional and stabilized Kalman algorithms. Computational errors associated with these algorithms are found to be so large as to obscure important mismodeling effects and thus cause misleading estimates of filter accuracy. The positive result of this study is that the U-D covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs. Accuracies of the U-D filter using single precision arithmetic consistently match the double precision reference results. Numerical stability of the U-D filter is further demonstrated by its insensitivity to variations in the a priori statistics.

Bierman, G. J.↗

NASA NTRS1976

Numerical comparison of discrete Kalman filter algorithms - Orbit determination case study

Numerical characteristics of various Kalman filter algorithms are illustrated with a realistic orbit determination study. The case study of this paper highlights the numerical deficiencies of the conventional and stabilized Kalman algorithms. Computational errors associated with these algorithms are found to be so large as to obscure important mismodeling effects and thus cause misleading estimates of filter accuracy. The positive result of this study is that the U-D covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs. Accuracies of the U-D filter using single precision arithmetic consistently match the double precision reference results. Numerical stability of the U-D filter is further demonstrated by its insensitivity to variations in the a priori statistics.

Bierman, G. J.↗

NASA NTRS1988

Exact computation of the unwrapped phase of a finite-length time series

McGowan and Kuc (1982) showed that a direct relationship between a time series and its unwrapped phase exists. They proposed an algorithm for computing the unwrapped phase by counting the number of sign changes in a Sturm sequence generated from the real and imaginary parts of the discrete Fourier transform. Their algorithm is limited to relatively short sequences by numerical accuracy. An extension of their algorithm is proposed which, by using all-integer arithmetic, permits exact computation of the number of multiples of pi required to determine the unwrapped phase for rational-valued time sequences of arbitrary length. Since the computation is exact, the extended numerical algorithm should be of interest when accurate phase unwrapping is required.

Long, David G.↗

DOE OSTI2015

Image reconstruction of single photon emission computed tomography (SPECT) on a pebble bed reactor (PBR) using expectation maximization and exact inversion algorithms: Comparison study by means of numerical phantom

Single Photon Emission Computed Tomography (SPECT) is a well-known imaging technique used in medical application, and it is part of medical imaging modalities that made the diagnosis and treatment of disease possible. However, SPECT technique is not only limited to the medical sector. Many works are carried out to adapt the same concept by using high-energy photon emission to diagnose process malfunctions in critical industrial systems such as in chemical reaction engineering research laboratories, as well as in oil and gas, petrochemical and petrochemical refining industries. Motivated by vast applications of SPECT technique, this work attempts to study the application of SPECT on a Pebble Bed Reactor (PBR) using numerical phantom of pebbles inside the PBR core. From the cross-sectional images obtained from SPECT, the behavior of pebbles inside the core can be analyzed for further improvement of the PBR design. As the quality of the reconstructed image is largely dependent on the algorithm used, this work aims to compare two image reconstruction algorithms for SPECT, namely the Expectation Maximization Algorithm and the Exact Inversion Formula. The results obtained from the Exact Inversion Formula showed better image contrast and sharpness, and shorter computational time compared to the Expectation Maximization Algorithm.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2013

Fluid-Structure Interaction Modeling of High-Aspect Ratio Nuclear Fuel Plates Using COMSOL

The High Flux Isotope Reactor at the Oak Ridge National Lab is in the research stage of converting its fuel from high-enriched uranium to low-enriched uranium. Due to different physical properties of the new fuel and changes to the internal fuel plate design, the current safety basis must be re-evaluated through rigorous computational analyses. One of the areas being explored is the fluid-structure interaction phenomenon due to the interaction of thin fuel plates (50 mils thickness) and the cooling fluid (water). Detailed computational fluid dynamics and fluid-structure interaction simulations have only recently become feasible due to improved numerical algorithms and advancements in computing technology. For many reasons including the already built-in fluid-structure interaction module, COMSOL has been chosen for this complex problem. COMSOL's ability to solve multiphysics problems using a fully-coupled and implicit solution algorithm is crucial in obtaining a stable and accurate solution. Our initial findings show that COMSOL can accurately model such problems due to its ability to closely couple the fluid dynamics and the structural dynamics problems.

Fluid-Structure Interaction↗

NASA NTRS1991

A special purpose silicon compiler for designing supercomputing VLSI systems

Design of general/special purpose supercomputing VLSI systems for numeric algorithm execution involves tackling two important aspects, namely their computational and communication complexities. Development of software tools for designing such systems itself becomes complex. Hence a novel design methodology has to be developed. For designing such complex systems a special purpose silicon compiler is needed in which: the computational and communicational structures of different numeric algorithms should be taken into account to simplify the silicon compiler design, the approach is macrocell based, and the software tools at different levels (algorithm down to the VLSI circuit layout) should get integrated. In this paper a special purpose silicon (SPS) compiler based on PACUBE macrocell VLSI arrays for designing supercomputing VLSI systems is presented. It is shown that turn-around time and silicon real estate get reduced over the silicon compilers based on PLA's, SLA's, and gate arrays. The first two silicon compiler characteristics mentioned above enable the SPS compiler to perform systolic mapping (at the macrocell level) of algorithms whose computational structures are of GIPOP (generalized inner product outer product) form. Direct systolic mapping on PLA's, SLA's, and gate arrays is very difficult as they are micro-cell based. A novel GIPOP processor is under development using this special purpose silicon compiler.

Venkateswaran, N.↗

DOE OSTI2014

Numerical simulation of the dynamics of a liquid crystal in the case of plane strain using GPUs

Parallel computational algorithm is worked out for numerical implementation of two-dimensional dynamic model of a liquid crystal, which takes into account the mutual influence of three physical effects – transfer of acoustic energy due to translational motion, viscoelastic rotation of the particles under the action of tangential stresses, anisotropic thermal expansion and compression. The algorithm is based on the Godunov gap decay method, the Ivanov scheme with controlled dissipation of energy and the splitting method with respect to spatial variables. The CUDA technology for computer systems with graphic accelerators is used. Results of computations demonstrating the efficiency of proposed method and algorithm are represented.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

GentenMPI: Distributed Memory Sparse Tensor Decomposition

GentenMPl is a toolkit of sparse canonical polyadic (CP) tensor decomposition algorithms that is designed to run effectively on distributed-memory high-performance computers. Its use of distributed-memory parallelism enables it to efficiently decompose tensors that are too large for a single compute node's memory. GentenMPl leverages Sandia's decades-long investment in the Trilinos solver framework for much of its parallel-computation capability. Trilinos contains numerical algorithms and linear algebra classes that have been optimized for parallel simulation of complex physical phenomena. This work applies these tools to the data science problem of sparse tensor decomposition. In this report, we describe the use of Trilinos in GentenMPl, extensions needed for sparse tensor decomposition, and implementations of the CP-ALS (CP via alternating least squares) and GCP-SGD (generalized CP via stochastic gradient descent) sparse tensor decomposition algorithms. We show that GentenMPl can decompose sparse tensors of extreme size, e.g., a 12.6-terabyte tensor on 8192 computer cores. We demonstrate that the Trilinos backbone provides good strong and weak scaling of the tensor decomposition algorithms.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1990

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Yee, H. C.↗

DOE OSTI2022

Landau discriminants

Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2012

Discontinuous Galerkin Dynamical Core in HOMME

Atmospheric numerical modeling has been going through radical changes over the past decade. One major reason for this trend is due to the recent paradigm change in scientific computing , triggered by the arrival of petascale computing resources with core counts in the tens of thousands to hundreds of thousands range. Modern atmospheric modelers must adapt grid systems and numerical algorithms to facilitate an unprecedented levels of scalability on these modern highly parallel computer architectures. The numerical algorithms which can address these challenges should have the local properties such as high on-processor floating-point operation count to bytes moved and minimum parallel communication overhead.

17 WIND ENERGY↗

NASA NTRS1983

Improvement of mixed time implicit-explicit algorithms for thermal analysis of structures

Computer implementation aspects and numerical evaluation of the recently introduced mixed time implicit-explicit algorithms in thermal analysis of structures are presented. A computationally useful method of estimating the critical time step for a linear quadrilateral element is given herein for the methods introduced by Liu and co-workers. Numerical tests confirm the stability criterion and accuracy characteristics of the methods. The superiority of these mixed time methods to the fully implicit method or the fully explicit method is also demonstrated.

Liu, W. K.↗

NASA NTRS1990

Verification of floating-point software

Floating point computation presents a number of problems for formal verification. Should one treat the actual details of floating point operations, or accept them as imprecisely defined, or should one ignore round-off error altogether and behave as if floating point operations are perfectly accurate. There is the further problem that a numerical algorithm usually only approximately computes some mathematical function, and we often do not know just how good the approximation is, even in the absence of round-off error. ORA has developed a theory of asymptotic correctness which allows one to verify floating point software with a minimum entanglement in these problems. This theory and its implementation in the Ariel C verification system are described. The theory is illustrated using a simple program which finds a zero of a given function by bisection. This paper is presented in viewgraph form.

Hoover, Doug N.↗

NASA NTRS1991

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I - The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Yee, H. C.↗