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NASA NTRS1987

Implementation of an ADI method on parallel computers

The implementation of an ADI method for solving the diffusion equation on three parallel/vector computers is discussed. The computers were chosen so as to encompass a variety of architectures. They are: the MPP, an SIMD machine with 16K bit serial processors; FLEX/32, an MIMD machine with 20 processors; and CRAY/2, an MIMD machine with four vector processors. The Gaussian elimination algorithm is used to solve a set of tridiagonal systems on the FLEX/32 and CRAY/2 while the cyclic elimination algorithm is used to solve these systems on the MPP. The implementation of the method is discussed in relation to these architectures and measures of the performance on each machine are given. Simple performance models are used to describe the performance. These models highlight the bottlenecks and limiting factors for this algorithm on these architectures. Finally, conclusions are presented.

Fatoohi, Raad A.↗

NASA NTRS2001

Development of an Unstructured Mesh Code for Flows About Complete Vehicles

This report describes the research work undertaken at the Massachusetts Institute of Technology, under NASA Research Grant NAG4-157. The aim of this research is to identify effective algorithms and methodologies for the efficient and routine solution of flow simulations about complete vehicle configurations. For over ten years we have received support from NASA to develop unstructured mesh methods for Computational Fluid Dynamics. As a result of this effort a methodology based on the use of unstructured adapted meshes of tetrahedra and finite volume flow solvers has been developed. A number of gridding algorithms, flow solvers, and adaptive strategies have been proposed. The most successful algorithms developed from the basis of the unstructured mesh system FELISA. The FELISA system has been extensively for the analysis of transonic and hypersonic flows about complete vehicle configurations. The system is highly automatic and allows for the routine aerodynamic analysis of complex configurations starting from CAD data. The code has been parallelized and utilizes efficient solution algorithms. For hypersonic flows, a version of the code which incorporates real gas effects, has been produced. The FELISA system is also a component of the STARS aeroservoelastic system developed at NASA Dryden. One of the latest developments before the start of this grant was to extend the system to include viscous effects. This required the development of viscous generators, capable of generating the anisotropic grids required to represent boundary layers, and viscous flow solvers. We show some sample hypersonic viscous computations using the developed viscous generators and solvers. Although this initial results were encouraging it became apparent that in order to develop a fully functional capability for viscous flows, several advances in solution accuracy, robustness and efficiency were required. In this grant we set out to investigate some novel methodologies that could lead to the required improvements. In particular we focused on two fronts: (1) finite element methods and (2) iterative algebraic multigrid solution techniques.

Peraire, Jaime↗

NASA NTRS1989

Maximum likelihood estimation for distributed parameter models of flexible spacecraft

A distributed-parameter model of the NASA Solar Array Flight Experiment spacecraft structure is constructed on the basis of measurement data and analyzed to generate a priori estimates of modal frequencies and mode shapes. A Newton-Raphson maximum-likelihood algorithm is applied to determine the unknown parameters, using a truncated model for the estimation and the full model for the computation of the higher modes. Numerical results are presented in a series of graphs and briefly discussed, and the significant improvement in computation speed obtained by parallel implementation of the method on a supercomputer is noted.

Taylor, L. W., Jr.↗

NASA NTRS2007

Algorithm-Based Fault Tolerance for Numerical Subroutines

A software library implements a new methodology of detecting faults in numerical subroutines, thus enabling application programs that contain the subroutines to recover transparently from single-event upsets. The software library in question is fault-detecting middleware that is wrapped around the numericalsubroutines. Conventional serial versions (based on LAPACK and FFTW) and a parallel version (based on ScaLAPACK) exist. The source code of the application program that contains the numerical subroutines is not modified, and the middleware is transparent to the user. The methodology used is a type of algorithm- based fault tolerance (ABFT). In ABFT, a checksum is computed before a computation and compared with the checksum of the computational result; an error is declared if the difference between the checksums exceeds some threshold. Novel normalization methods are used in the checksum comparison to ensure correct fault detections independent of algorithm inputs. In tests of this software reported in the peer-reviewed literature, this library was shown to enable detection of 99.9 percent of significant faults while generating no false alarms.

Tumon, Michael↗

NASA NTRS1996

Satellite Angular Rate Estimation From Vector Measurements

This paper presents an algorithm for estimating the angular rate vector of a satellite which is based on the time derivatives of vector measurements expressed in a reference and body coordinate. The computed derivatives are fed into a spacial Kalman filter which yields an estimate of the spacecraft angular velocity. The filter, named Extended Interlaced Kalman Filter (EIKF), is an extension of the Kalman filter which, although being linear, estimates the state of a nonlinear dynamic system. It consists of two or three parallel Kalman filters whose individual estimates are fed to one another and are considered as known inputs by the other parallel filter(s). The nonlinear dynamics stem from the nonlinear differential equation that describes the rotation of a three dimensional body. Initial results, using simulated data, and real Rossi X ray Timing Explorer (RXTE) data indicate that the algorithm is efficient and robust.

Azor, Ruth↗

NASA NTRS2002

Effects of Ordering Strategies and Programming Paradigms on Sparse Matrix Computations

The Conjugate Gradient (CG) algorithm is perhaps the best-known iterative technique to solve sparse linear systems that are symmetric and positive definite. For systems that are ill-conditioned, it is often necessary to use a preconditioning technique. In this paper, we investigate the effects of various ordering and partitioning strategies on the performance of parallel CG and ILU(O) preconditioned CG (PCG) using different programming paradigms and architectures. Results show that for this class of applications: ordering significantly improves overall performance on both distributed and distributed shared-memory systems, that cache reuse may be more important than reducing communication, that it is possible to achieve message-passing performance using shared-memory constructs through careful data ordering and distribution, and that a hybrid MPI+OpenMP paradigm increases programming complexity with little performance gains. A implementation of CG on the Cray MTA does not require special ordering or partitioning to obtain high efficiency and scalability, giving it a distinct advantage for adaptive applications; however, it shows limited scalability for PCG due to a lack of thread level parallelism.

Oliker, Leonid↗

NASA NTRS1984

On the impact of communication complexity in the design of parallel numerical algorithms

This paper describes two models of the cost of data movement in parallel numerical algorithms. One model is a generalization of an approach due to Hockney, and is suitable for shared memory multiprocessors where each processor has vector capabilities. The other model is applicable to highly parallel nonshared memory MIMD systems. In the second model, algorithm performance is characterized in terms of the communication network design. Techniques used in VLSI complexity theory are also brought in, and algorithm independent upper bounds on system performance are derived for several problems that are important to scientific computation.

Gannon, D.↗

NASA NTRS1986

Automated matching of pairs of SIR-B images for elevation mapping

During the SIR-B mission in October 1984, a significant number of overlapping synthetic aperture radar (SAR) images of various ground areas was collected. This has offered the first opportunity to perform stereo analyses on images from space that cover large ground areas to determine elevation information. This paper presents the preliminary results of an investigation to obtain elevation data from stereo pairs of SIR-B images. First, the accuracy with which elevation information can be derived from SIR-B image pairs is evaluated theoretically. It is shown that elevation accuracy is a function of the slant range resolution, the incidence angles with which the stereo pair is obtained, the accuracies in spacecraft state estimation, and determination of corresponding pixels in the stereo pair. Next, a hierarchical method is developed to match the corresponding pixels. This method involves iterative removal of local distortions and correlations of pairs of local neighborhoods in the two images. Since it is necessary to perform the matching at every pixel in the image, it is very computationally intensive. Therefore, it has been implemented on the Massively Parallel Processor (MPP) at the Goddard Space Flight Center (GSFC). The MPP's speed permits two iterations of this technique to operate on a pair of 512 x 512 images within 7 s. Results of applying this algorithm of SIR-B images of Mount Shasta, CA, are shown. The matching algorithm performs well in regions of the image with significant features. An approximate elevation image derived from the matching process corresponds to published topographic map data, except for certain obvious discontinuities.

Ramapriyan, H. K.↗

NASA NTRS1987

A system for routing arbitrary directed graphs on SIMD architectures

There are many problems which can be described in terms of directed graphs that contain a large number of vertices where simple computations occur using data from connecting vertices. A method is given for parallelizing such problems on an SIMD machine model that is bit-serial and uses only nearest neighbor connections for communication. Each vertex of the graph will be assigned to a processor in the machine. Algorithms are given that will be used to implement movement of data along the arcs of the graph. This architecture and algorithms define a system that is relatively simple to build and can do graph processing. All arcs can be transversed in parallel in time O(T), where T is empirically proportional to the diameter of the interconnection network times the average degree of the graph. Modifying or adding a new arc takes the same time as parallel traversal.

Tomboulian, Sherryl↗

NASA NTRS1987

Galerkin/Runge-Kutta discretizations for parabolic equations with time dependent coefficients

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.

Keeling, Stephen L.↗

NASA NTRS1989

Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients

Keeling, Stephen L.↗

NASA NTRS1994

Empirical study of parallel LRU simulation algorithms

This paper reports on the performance of five parallel algorithms for simulating a fully associative cache operating under the LRU (Least-Recently-Used) replacement policy. Three of the algorithms are SIMD, and are implemented on the MasPar MP-2 architecture. Two other algorithms are parallelizations of an efficient serial algorithm on the Intel Paragon. One SIMD algorithm is quite simple, but its cost is linear in the cache size. The two other SIMD algorithm are more complex, but have costs that are independent on the cache size. Both the second and third SIMD algorithms compute all stack distances; the second SIMD algorithm is completely general, whereas the third SIMD algorithm presumes and takes advantage of bounds on the range of reference tags. Both MIMD algorithm implemented on the Paragon are general and compute all stack distances; they differ in one step that may affect their respective scalability. We assess the strengths and weaknesses of these algorithms as a function of problem size and characteristics, and compare their performance on traces derived from execution of three SPEC benchmark programs.

Carr, Eric↗

NASA NTRS1997

Geopotential Error Analysis from Satellite Gradiometer and Global Positioning System Observables on Parallel Architecture

The recovery of a high resolution geopotential from satellite gradiometer observations motivates the examination of high performance computational techniques. The primary subject matter addresses specifically the use of satellite gradiometer and GPS observations to form and invert the normal matrix associated with a large degree and order geopotential solution. Memory resident and out-of-core parallel linear algebra techniques along with data parallel batch algorithms form the foundation of the least squares application structure. A secondary topic includes the adoption of object oriented programming techniques to enhance modularity and reusability of code. Applications implementing the parallel and object oriented methods successfully calculate the degree variance for a degree and order 110 geopotential solution on 32 processors of the Cray T3E. The memory resident gradiometer application exhibits an overall application performance of 5.4 Gflops, and the out-of-core linear solver exhibits an overall performance of 2.4 Gflops. The combination solution derived from a sun synchronous gradiometer orbit produce average geoid height variances of 17 millimeters.

Schutz, Bob E.↗

NASA NTRS1978

A single user efficiency measure for evaluation of parallel or pipeline computer architectures

A precise statement of the relationship between sequential computation at one rate, parallel or pipeline computation at a much higher rate, the data movement rate between levels of memory, the fraction of inherently sequential operations or data that must be processed sequentially, the fraction of data to be moved that cannot be overlapped with computation, and the relative computational complexity of the algorithms for the two processes, scalar and vector, was developed. The relationship should be applied to the multirate processes that obtain in the employment of various new or proposed computer architectures for computational aerodynamics. The relationship, an efficiency measure that the single user of the computer system perceives, argues strongly in favor of separating scalar and vector processes, sometimes referred to as loosely coupled processes, to achieve optimum use of hardware.

Jones, W. P.↗

NASA NTRS1984

On the impact of communication complexity on the design of parallel numerical algorithms

This paper describes two models of the cost of data movement in parallel numerical alorithms. One model is a generalization of an approach due to Hockney, and is suitable for shared memory multiprocessors where each processor has vector capabilities. The other model is applicable to highly parallel nonshared memory MIMD systems. In this second model, algorithm performance is characterized in terms of the communication network design. Techniques used in VLSI complexity theory are also brought in, and algorithm-independent upper bounds on system performance are derived for several problems that are important to scientific computation.

Gannon, D. B.↗

NASA NTRS1987

Implementation of an ADI method on parallel computers

In this paper the implementation of an ADI method for solving the diffusion equation on three parallel/vector computers is discussed. The computers were chosen so as to encompass a variety of architectures. They are the MPP, an SIMD machine with 16-Kbit serial processors; Flex/32, an MIMD machine with 20 processors; and Cray/2, an MIMD machine with four vector processors. The Gaussian elimination algorithm is used to solve a set of tridiagonal systems on the Flex/32 and Cray/2 while the cyclic elimination algorithm is used to solve these systems on the MPP. The implementation of the method is discussed in relation to these architectures and measures of the performance on each machine are given. Simple performance models are used to describe the performance. These models highlight the bottlenecks and limiting factors for this algorithm on these architectures. Finally conclusions are presented.

Fatoohi, Raad A.↗

NASA NTRS1991

Particle simulation in a multiprocessor environment

A parallel implementation of a particle simulation method that is portable between a wide class of multiprocessor computers is presented. A fine grain spatial decomposition is utilized where several subdomains having a regular structure are computed at each processing node. This leads directly to an efficient and straightforward load balancing scheme if the number of subdomains at each processor is permitted to vary in an appropriate manner. Three dimensional simulations incorporating full thermochemical nonequilibrium are possible using the resulting code. Vectorizable algorithms are retained from earlier work allowing efficient use of deeply pipelined node processors where available. Performance results are presented from three different machine architectures demonstrating the portability of the code. On a 128-node Intel iPSC/860, performance is twice that of a single Cray-Y/MP CPU running a highly vectorized simulation code. Speedup is linear over the full range of number of processors on all target machines, indicating scalability of the method to higher degrees of parallelism.

Mcdonald, Jeffrey D.↗

NASA NTRS1989

Systematic generation of multibody equations of motion suitable for recursive and parallel manipulation

The formulation of a method known as the joint coordinate method for automatic generation of the equations of motion for multibody systems is summarized. For systems containing open or closed kinematic loops, the equations of motion can be reduced systematically to a minimum number of second order differential equations. The application of recursive and nonrecursive algorithms to this formulation, computational considerations and the feasibility of implementing this formulation on multiprocessor computers are discussed.

Nikravesh, Parviz E.↗