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A method of solving a class of linear matrix equations over various rings is proposed, using results from linear geometric control theory. An algorithm, successfully implemented, is presented, along with non-trivial numerical examples. Applications of the method to the algebraic control system design methodology are discussed.
Parallel or distributed processing is key to getting highest performance workstations. However, designing and implementing efficient parallel algorithms is difficult and error-prone. It is even more difficult to write code that is both portable to and efficient on many different computers. Finally, it is harder still to satisfy the above requirements and include the reliability and ease of use required of commercial software intended for use in a production environment. As a result, the application of parallel processing technology to commercial software has been extremely small even though there are numerous computationally demanding programs that would significantly benefit from application of parallel processing. This paper describes DSSLIB, which is a library of subroutines that perform many of the time-consuming computations in engineering and scientific software. DSSLIB combines the high efficiency and speed of parallel computation with a serial programming model that eliminates many undesirable side-effects of typical parallel code. The result is a simple way to incorporate the power of parallel processing into commercial software without compromising maintainability, reliability, or ease of use. This gives significant advantages over less powerful non-parallel entries in the market.
The development and testing of a parallel unstructured agglomeration multigrid algorithm for steady-state aerodynamic flows is discussed. The agglomeration multigrid strategy uses a graph algorithm to construct the coarse multigrid levels from the given fine grid, similar to an algebraic multigrid approach, but operates directly on the non-linear system using the FAS (Full Approximation Scheme) approach. The scalability and convergence rate of the multigrid algorithm are examined on the SGI Origin 2000 and the Cray T3E. An argument is given which indicates that the asymptotic scalability of the multigrid algorithm should be similar to that of its underlying single grid smoothing scheme. For medium size problems involving several million grid points, near perfect scalability is obtained for the single grid algorithm, while only a slight drop-off in parallel efficiency is observed for the multigrid V- and W-cycles, using up to 128 processors on the SGI Origin 2000, and up to 512 processors on the Cray T3E. For a large problem using 25 million grid points, good scalability is observed for the multigrid algorithm using up to 1450 processors on a Cray T3E, even when the coarsest grid level contains fewer points than the total number of processors.
Summary With discrete Intel GPUs entering the high‐performance computing landscape, there is an urgent need for production‐ready software stacks for these platforms. In this article, we report how we enable the Ginkgo math library to execute on Intel GPUs by developing a kernel backed based on the DPC++ programming environment. We discuss conceptual differences between the CUDA and DPC++ programming models and describe workflows for simplified code conversion. We evaluate the performance of basic and advanced sparse linear algebra routines available in Ginkgo's DPC++ backend in the hardware‐specific performance bounds and compare against routines providing the same functionality that ship with Intel's oneMKL vendor library.
In this study, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O ( p 2d ) storage and O ( p 3d ) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O ( p d+1 ) storage, O ( p d+1 ) work in two spatial dimensions, and O ( p d+2 ) work in three spatial dimensions. Combined with a matrix-free Newton–Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O ( p 9 ) to O ( p 5 ) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier–Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p .
Memory effects are often presented during coarse-graining of a complex dynamical system. In particular, a generalized Langevin equation (GLE) for the coarse-grained (CG) system arises in the context of Mori–Zwanzig formalism. Upon a pairwise decomposition, GLE can be reformulated into its pairwise version, i.e., non-Markovian dissipative particle dynamics (DPD). GLE models the dynamics of a single coarse particle, while DPD considers the dynamics of many interacting CG particles, with both CG systems governed by non-Markovian interactions. We compare two different methods for the practical implementation of the non-Markovian interactions in GLE and DPD systems. In particular, a direct evaluation of the non-Markovian (NM) terms is performed in LE-NM and DPD-NM models, which requires the storage of historical information that significantly increases computational complexity. Alternatively, we use a few auxiliary variables in LE-AUX and DPD-AUX models to replace the non-Markovian dynamics with a Markovian dynamics in a higher dimensional space, leading to a much reduced memory footprint and computational cost. In our numerical benchmarks, the GLE and non-Markovian DPD models are constructed from molecular dynamics (MD) simulations of star-polymer melts. Results show that a Markovian dynamics with auxiliary variables successfully generates equivalent non-Markovian dynamics consistent with the reference MD system, while maintaining a tractable computational cost. Also, transient subdiffusion of the star-polymers observed in the MD system can be reproduced by the coarse-grained models. The non-interacting particle models, LE-NM/AUX, are computationally much cheaper than the interacting particle models, DPD-NM/AUX. Yet, the pairwise models with momentum conservation are more appropriate for correctly reproducing the long-time hydrodynamics characterised by an algebraic decay in the velocity autocorrelation function.
Here, we introduce an energy functional for ground-state electronic structure calculations. Its variables are the natural spin-orbitals of singlet many-body wave functions and their joint occupation probabilities deriving from controlled approximations to the two-particle density matrix that yield algebraic scaling in general, and Hartree–Fock scaling in its seniority-zero version. Results from the latter version for small molecular systems are compared with those of highly accurate quantum-chemical computations. The energies lie above full configuration interaction calculations, close to doubly occupied configuration interaction calculations. Their accuracy is considerably greater than that obtained from current density-functional theory approximations and from current functionals of the oneparticle density matrix.
During my visit to LLNL during the summer of 2013, I worked on algebraic multilevel solvers for large sparse systems of linear equations arising from discretizations of partial differential equations.
The need for fault-tolerant computing is addressed from the viewpoints of (1) why it is needed, (2) how to apply it in the current state of technology, and (3) what it means in the context of the Phoenix computer system and other related systems. To this end, the value of concurrent error detection and correction is described. User protection, program retry, and repair are among the factors considered. The technology of algebraic codes to protect memory systems and arithmetic codes to protect memory systems and arithmetic codes to protect arithmetic operations is discussed.
A set of partial differential equations, describing the two-dimensional supersonic chemically-reacting flow of the hydrogen-air system, is formulated such that the equations are parabolic in the streamwise direction. A fully-implicit fully-coupled finite-difference algorithm is used to develop a computer code which solves the governing equations by marching in the streamwise direction. The combustion process is modeled by a two-step finite-rate chemistry whereas turbulence is simulated by an algebraic turbulence model. Results of two calculations of internal supersonic reacting flow show fairly good agreement with the results obtained by the more costly full Navier-Stokes procedure.
There is a practical need to model high speed flows that exist in jet engine inlets. The boundary layers that form in these inlets may be turbulent or laminar and either separated or attached. Also, unsteady supersonic inlets may be subject to frequent changes in operating conditions. Some changes in the operating conditions of the inlets may include varying the inlet geometry, bleeds and bypasses, and rotating or translating the centerbody. In addition, the inlet may be either started or unstarted. Therefore, a CFD code, used to model these inlets, may have to run for several different cases. Also, since the flow conditions through an unsteady inlet may be continually fluctuating, the CFD code which models these flows may have to be run over many time steps. Therefore, it would be beneficial that the code run quickly. Many turbulence models, however, are cumbersome to implement and require a lot of computer time to run, since they add to the number of differential equations to be solved to model a flow. The Baldwin-Lomax turbulence model is a popular model. It is an algebraic, eddy viscosity model. The Baldwin-Lomax model is used in many CFD codes because it is quick and easy to implement. In this paper, we will discuss implementing the Baldwin-Lomax turbulence model for both steady and unsteady compressible flows. In addition, these flows may be either separated or attached. In order to apply this turbulence model to flows which may be subjected to these conditions, certain modifications should be made to the original Baldwin-Lomax model. We will discuss these modifications and determine whether the Baldwin-Lomax model is a viable turbulence model that produces reasonably accurate results for high speed flows that can be found in engine inlets.
Assume-guarantee reasoning is a popular and expressive paradigm for modular and compositional specification of programs. It is becoming a fundamental concept in some computer-aided design tools for embedded system design. In this paper, we elaborate foundations for contract-based embedded system design by proposing a general-purpose module language based on a Boolean algebra allowing to define contracts. In this framework, contracts are used to negotiate the correctness of assumptions made on the definition of a component at the point where it is used and provides guarantees to its environment. We illustrate this presentation with the specification of a simplified 4-stroke engine model.
Power and energy consumption are becoming an increasing concern in high performance computing. Compared to multi-core CPUs, GPUs have a much better performance per watt. In this paper we discuss efforts to redesign the most computation intensive parts of BLAST, an application that solves the equations for compressible hydrodynamics with high order finite elements, using GPUs BLAST, Dobrev. In order to exploit the hardware parallelism of GPUs and achieve high performance, we implemented custom linear algebra kernels. We intensively optimized our CUDA kernels by exploiting the memory hierarchy, which exceed the vendor's library routines substantially in performance. We proposed an auto tuning technique to adapt our CUDA kernels to the orders of the finite element method. Compared to a previous base implementation, our redesign and optimization lowered the energy consumption of the GPU in two aspects: 60% less time to solution and 10% less power required. Compared to the CPU-only solution, our GPU accelerated BLAST obtained a 2.5× overall speedup and 1.42× energy efficiency (green up) using 4th order (Q_4) finite elements, and a 1.9× speedup and 1.27× green up using 2nd order (Q2) finite elements.
Incorrect computer hardware behavior may corrupt intermediate computations in numerical algorithms, possibly resulting in incorrect answers. Prior work models misbehaving hardware by randomly flipping bits in memory. We start by accepting this premise, and present an analytic model for the error introduced by a bit flip in an IEEE 754 floating-point number. We then relate this finding to the linear algebra concepts of normalization and matrix equilibration. In particular, we present a case study illustrating that normalizing both vector inputs of a dot product minimizes the probability of a single bit flip causing a large error in the dot product's result. Moreover, the absolute error is either less than one or very large, which allows detection of large errors. Then, we apply this to the GMRES iterative solver. We count all possible errors that can be introduced through faults in arithmetic in the computationally intensive orthogonalization phase of GMRES, and show that when the matrix is equilibrated, the absolute error is bounded above by one.
The objective of this research is to develop turbulence models to predict the flow and heat transfer fields dominated by the curvature effect such as those encountered in turbine cascades and turn-around ducts. A Navier-Stokes code has been developed using an explicit Runge-Kutta method with a two layer k-epsilon/ARSM (Algebraic Reynolds Stress Model), Chien's Low Reynolds Number (LRN) k-epsilon model and Coakley's LRN q-omega model. The near wall pressure strain correlation term was included in the ARSM. The formulation is applied to Favre-averaged N-S equations and no thin-layer approximations are made in either the mean flow or turbulence transport equations. Anisotropic scaling of artificial dissipation terms was used. Locally variable timestep was also used to improve convergence. Detailed comparisons were made between computations and data measured in a turbine cascade by Arts et al. at Von Karman Institute. The surface pressure distributions and wake profiles were predicted well by all the models. The blade heat transfer is predicted well by k-epsilon/ARSM model, as well as the k-epsilon model. It's found that the onset of boundary layer transition on both surfaces is highly dependent upon the level of local freestream turbulence intensity, which is strongly influenced by the streamline curvature. Detailed computation of the flow in the turn around duct has been carried out and validated against the data by Monson as well as Sandborn. The computed results at various streamwise locations both on the concave and convex sides are compared with flow and turbulence data including the separation zone on the inner well. The k-epsilon/ARSM model yielded relatively better results than the two-equation turbulence models. A detailed assessment of the turbulence models has been made with regard to their applicability to curved flows.
The goal of the ExaWind project is to enable predictive simulations of wind farms composed of many MW-scale turbines situated in complex terrain. Predictive simulations will require computational fluid dynamics (CFD) simulations for which the mesh resolves the geometry of the turbines, and captures the rotation and large deflections of blades. Whereas such simulations for a single turbine are arguably petascale class, multi-turbine wind farm simulations will require exascale-class resources. The primary code in the ExaWind project is Nalu, which is an unstructured-grid solver for the acoustically-incompressible Navier-Stokes equations, and mass continuity is maintained through pressure projection. The model consists of the mass-continuity Poisson-type equation for pressure and a momentum equation for the velocity. For such modeling approaches, simulation times are dominated by linear-system setup and solution for the continuity and momentum systems. For the ExaWind challenge problem, the moving meshes greatly affect overall solver costs as re-initialization of matrices and re-computation of preconditioners is required at every time step We describe in this report our efforts to decrease the setup and solution time for the mass-continuity Poisson system with respect to the benchmark timing results reported in FY18 Q1. In particular, we investigate improving and evaluating two types of algebraic multigrid (AMG) preconditioners: Classical Ruge-Stfiben AMG (C-AMG) and smoothed-aggregation AMG (SA-AMG), which are implemented in the Hypre and Trilinos/MueLu software stacks, respectively. Preconditioner performance was optimized through existing capabilities and settings.
This research brief contains the progress reports of the research staff of the Center for Modeling of Turbulence and Transition (CMOTT) from June 1992 to July 1993. It is also an annual report to the Institute for Computational Mechanics in Propulsion located at Ohio Aerospace Institute and NASA Lewis Research Center. The main objectives of the research activities at CMOTT are to develop, validate, and implement turbulence and transition models for flows of interest in propulsion systems. Currently, our research covers eddy viscosity one- and two-equation models, Reynolds-stress algebraic equation models, Reynolds-stress transport equation models, nonequilibrium multiple-scale models, bypass transition models, joint scalar probability density function models, and Renormalization Group Theory and Direct Interaction Approximation methods. Some numerical simulations (LES and DNS) have also been carried out to support the development of turbulence modeling. Last year was CMOTT's third year in operation. During this period, in addition to the above mentioned research, CMOTT has also hosted the following programs: an eighteen-hour short course on 'Turbulence--Fundamentals and Computational Modeling (Part I)' given by CMOTT at the NASA Lewis Research Center; a productive summer visitor research program that has generated many encouraging results; collaborative programs with industry customers to help improve their turbulent flow calculations for propulsion system designs; a biweekly CMOTT seminar series with speakers from within and without the NASA Lewis Research Center including foreign speakers. In addition, CMOTT members have been actively involved in the national and international turbulence research activities. The current CMOTT roster and organization are listed in Appendix A. Listed in Appendix B are the abstracts of the biweekly CMOTT seminar. Appendix C lists the papers contributed by CMOTT members.
Analytical study of ordinary and partial differential equations, illustrating initial value problem and large matrix inversion