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Number representations, processing architectures and algorithms, optical linear algebra processor fabrication and test results, case study descriptions, and future system plans are covered.
An explicit flow solver, applicable to the hierarchy of model equations ranging from Euler to full Navier-Stokes, is combined with several techniques designed to reduce computational expense. The computational domain consists of local grid refinements embedded in a global coarse mesh, where the locations of these refinements are defined by the physics of the flow. Flow characteristics are also used to determine which set of model equations is appropriate for solution in each region, thereby reducing not only the number of grid points at which the solution must be obtained, but also the computational effort required to get that solution. Acceleration to steady-state is achieved by applying multigrid on each of the subgrids, regardless of the particular model equations being solved. Since each of these components is explicit, advantage can readily be taken of the vector- and parallel-processing capabilities of machines such as the Cray X-MP and Cray-2.
An explicit flow solver, applicable to the hierarchy of model equations ranging from Euler to full Navier-Stokes, is combined with several techniques designed to reduce computational expense. The computational domain consists of local grid refinements embedded in a global coarse mesh, where the locations of these refinements are defined by the physics of the flow. Flow characteristics are also used to determine which set of model equations is appropriate for solution in each region, thereby reducing not only the number of grid points at which the solution must be obtained, but also the computational effort required to get that solution. Acceleration to steady-state is achieved by applying multigrid on each of the subgrids, regardless of the particular model equations being solved. Since each of these components is explicit, advantage can readily be taken of the vector- and parallel-processing capabilities of machines such as the Cray X-MP and Cray-2.
A reflector antenna computer program based upon a simple discreet approximation of the radiation integral has proven to be extremely easy to adapt to the parallel computing architecture of the modest number of large-gain computing elements such as are used in the Intel iPSC and Touchstone Delta parallel machines.
In this paper, the first domain decomposition parallelization of the Fast Marching Method for level sets has been presented. Parallel speedup has been demonstrated in both the optimal and non-optimal domain decomposition case. The parallel performance of the proposed method is strongly dependent on load balancing separately the number of nodes on each side of the interface. A load imbalance of nodes on either side of the domain leads to an increase in communication and rollback operations. Furthermore, the amount of inter-domain communication can be reduced by aligning the inter-domain boundaries with the interface normal vectors. In the case of optimal load balancing and aligned inter-domain boundaries, the proposed parallel FMM algorithm is highly efficient, reaching efficiency factors of up to 0.98. Future work will focus on the extension of the proposed parallel algorithm to higher order accuracy. Also, to further enhance parallel performance, the coupling of the domain decomposition parallelization to the G(sub 0)-based parallelization will be investigated.
Adaptive local grid refinement and coarsening results in unequal distribution of workload among the processors of a parallel system. A novel method for balancing the load in cases of dynamically changing tetrahedral grids is developed. The approach employs local exchange of cells among processors in order to redistribute the load equally. An important part of the load balancing algorithm is the method employed by a processor to determine which cells within its subdomain are to be exchanged. Two such methods are presented and compared. The strategy for load balancing is based on the Divide-and-Conquer approach which leads to an efficient parallel algorithm. This method is implemented on a distributed-memory MIMD system.
In this research an efficient parallel algorithm for 3-D directionally split problems is developed. The proposed algorithm is based on a reformulated version of the pipelined Thomas algorithm that starts the backward step computations immediately after the completion of the forward step computations for the first portion of lines This algorithm has data available for other computational tasks while processors are idle from the Thomas algorithm. The proposed 3-D directionally split solver is based on the static scheduling of processors where local and non-local, data-dependent and data-independent computations are scheduled while processors are idle. A theoretical model of parallelization efficiency is used to define optimal parameters of the algorithm, to show an asymptotic parallelization penalty and to obtain an optimal cover of a global domain with subdomains. It is shown by computational experiments and by the theoretical model that the proposed algorithm reduces the parallelization penalty about two times over the basic algorithm for the range of the number of processors (subdomains) considered and the number of grid nodes per subdomain.
Algorithms sought to minimize time and cost of computation. Report describes research on scheduling of computations tasks in system of multiple identical data processors operating in parallel. Computational intractability requires use of suboptimal heuristic algorithms. First algorithm called "list heuristic", variation of classical list scheduling. Second algorithm called "cluster heuristic" applied to tightly coupled tasks and consists of four phases. Third algorithm called "exchange heuristic", iterative-improvement algorithm beginning with initial feasible assignment of tasks to processors and periods of time. Fourth algorithm is iterative one for optimal assignment of tasks and based on concept called "simulated annealing" because of mathematical resemblance to aspects of physical annealing processes.
Computational algorithms for structural analysis on parallel-vector supercomputers are reviewed. These parallel algorithms, developed by the authors, are for the assembly of structural equations, 'out-of-core' strategies for linear equation solution, massively distributed-memory equation solution, unsymmetric equation solution, general eigensolution, geometrically nonlinear finite element analysis, design sensitivity analysis for structural dynamics, optimization search analysis and domain decomposition. The source code for many of these algorithms is available.
Computational algorithms for structural analysis on parallel-vector supercomputers are reviewed. These parallel algorithms, developed by the authors, are for the assembly of structural equations, 'out-of-core' strategies for linear equation solution, massively distributed-memory equation solution, unsymmetric equation solution, general eigen-solution, geometrically nonlinear finite element analysis, design sensitivity analysis for structural dynamics, optimization algorithm and domain decomposition. The source code for many of these algorithms is available from NASA Langley.
Space missions have a growing interest in putting multi-core processors onboard spacecraft. For many missions processing power significantly slows operations. We investigate how continual planning and scheduling algorithms can exploit multi-core processing and outline different potential design decisions for a parallelized planning architecture. This organization of choices and challenges helps us with an initial design for parallelizing the CASPER planning system for a mesh multi-core processor. This work extends that presented at another workshop with some preliminary results.
In part 1 architecture of NETRA is presented. A performance evaluation of NETRA using several common vision algorithms is also presented. Performance of algorithms when they are mapped on one cluster is described. It is shown that SIMD, MIMD, and systolic algorithms can be easily mapped onto processor clusters, and almost linear speedups are possible. For some algorithms, analytical performance results are compared with implementation performance results. It is observed that the analysis is very accurate. Performance analysis of parallel algorithms when mapped across clusters is presented. Mappings across clusters illustrate the importance and use of shared as well as distributed memory in achieving high performance. The parameters for evaluation are derived from the characteristics of the parallel algorithms, and these parameters are used to evaluate the alternative communication strategies in NETRA. Furthermore, the effect of communication interference from other processors in the system on the execution of an algorithm is studied. Using the analysis, performance of many algorithms with different characteristics is presented. It is observed that if communication speeds are matched with the computation speeds, good speedups are possible when algorithms are mapped across clusters.
In geophysics, we are confronted with many under-determined inverse problems. For example, all of our observations of earthquakes are made at the Earth’s surface. So, when we try to infer how slip during an earthquake evolves in space and time, we find that there are many potential slip histories that are consistent with our limited observations and our understanding of earthquake physics. One way to approach these problems is with Bayesian analysis which allows us to infer the ensemble of all potential slip models that satisfy the observations and our prior knowledge of earthquake physics. In Bayesian analysis, our prior knowledge is known as the prior probability density function or prior PDF, the fit to the data is known as the data likelihood, and the target PDF that satisfies both the prior PDF and data likelihood is known as the posterior PDF. However, simulating the posterior PDF typically requires using Markov Chain Monte Carlo (MCMC) to draw tens of billions of random realizations of earthquake slip models, which may not be computationally feasible. To make this and similar geophysical inversions computationally tractable, we developed the Cascading Adaptive Transitional Metropolis In Parallel (CATMIP) algorithm. CATMIP is an efficient parallel Markov Chain Monte Carlo (MCMC) sampler that is used for model fitting and uncertainty quantification in geophysics. Example use cases are earthquake rupture modeling, determining mineral composition on Mars, reconstructing the history of ocean salinity, and historical earthquake relocation. CATMIP employs many parallel instances of the Metropolis algorithm for sampling in a transitioning framework. Transitioning is a process in which a set of random samples at equilibrium with a known probability density function (PDF) are used as seeds for the Markov chains to sample successive target PDFs that incrementally move the distribution from the starting seeds to the final desired PDF that describes the relative plausibility of potential values for the model parameters. The algorithm is implemented as a Master-Worker model employing MPI for communication. The worker processes are loosely coupled with global parameters periodically optimized by the master process. This provides a very high amount of parallelism with little communication between updates. During the presentation we will discuss the history of the algorithm and elaborate the earthquake rupture modeling use case for the CATMIP package. Our first step toward GPU optimization was to optimize the code for the CPU. CPU profiling revealed that most of the compute time is spent in calls to level 2 BLAS routines and calls to GSL random number generators. We revised the algorithm to employ level 3 BLAS routines instead. In our presentation we will describe how this was accomplished. Adding GPU support to CATMIP consisted mostly of replacing the calls to GSL with calls to GPU vendor-provided library routines. A small number of loops were directly implemented in CUDA. In the presentation will provide implementation details. Finally, we will discuss methods for profiling and opportunities for further optimizing GPU execution. By creating a code with the flexibility to run on either a CPU or GPU architecture, CATMIP can be used on systems ranging from large CPU-based HPC environments to single servers with GPU acceleration and everything in between.
A parallel-processing algorithm and a hardware architecture to implement the algorithm have been devised for timeslot synchronization in the reception of pulse-position-modulated (PPM) optical or radio signals. As in the cases of some prior algorithms and architectures for parallel, discrete-time, digital processing of signals other than PPM, an incoming broadband signal is divided into multiple parallel narrower-band signals by means of sub-sampling and filtering. The number of parallel streams is chosen so that the frequency content of the narrower-band signals is low enough to enable processing by relatively-low speed complementary metal oxide semiconductor (CMOS) electronic circuitry. The algorithm and architecture are intended to satisfy requirements for time-varying time-slot synchronization and post-detection filtering, with correction of timing errors independent of estimation of timing errors. They are also intended to afford flexibility for dynamic reconfiguration and upgrading. The architecture is implemented in a reconfigurable CMOS processor in the form of a field-programmable gate array. The algorithm and its hardware implementation incorporate three separate time-varying filter banks for three distinct functions: correction of sub-sample timing errors, post-detection filtering, and post-detection estimation of timing errors. The design of the filter bank for correction of timing errors, the method of estimating timing errors, and the design of a feedback-loop filter are governed by a host of parameters, the most critical one, with regard to processing very broadband signals with CMOS hardware, being the number of parallel streams (equivalently, the rate-reduction parameter).
The implementation and performance of a finite-difference algorithm for the compressible Navier-Stokes equations in two or three dimensions on the Connection Machine are described. This machine is a single-instruction multiple-data machine with up to 65536 physical processors. The implicit portion of the algorithm is of particular interest. Running times and megadrop rates are given for two- and three-dimensional problems. Included are comparisons with the standard codes on a Cray X-MP/48.
The method of simultaneous iteration with shift is extended to extraction of m-eigenpairs of a general eigenvalue problem of large order n in a parallel processing environment. The algorithm combines the power method and the Jacobi technique, and reduces to performing four basic operations. Parallel implementation of the algorithm is discussed in detail. The analysis accounts for computation and communication costs, and utilizes a parallel processing architecture of the ensemble type. Expressions for the computational efficiency and speedup are defined as a function of the problem and hardware parameters. Selected representative problems exhibit efficiencies ranging from 60 to 98 percent.
Master processor supervises slave processors, each with its own memory. Computer with parallel processing serves as inexpensive tool for experimentation with parallel mathematical algorithms. Speed enhancement obtained depends on both nature of problem and structure of algorithm used. In parallel-processing architecture, "bank select" and control signals determine which one, if any, of N slave processor memories accessible to master processor at any given moment. When so selected, slave memory operates as part of master computer memory. When not selected, slave memory operates independently of main memory. Slave processors communicate with each other via input/output bus.
A parallel compression algorithm for the 16,384 processor MPP machine was developed. The serial version of the algorithm can be viewed as a combination of on-line dynamic lossless test compression techniques (which employ simple learning strategies) and vector quantization. These concepts are described. How these concepts are combined to form a new strategy for performing dynamic on-line lossy compression is discussed. Finally, the implementation of this algorithm in a massively parallel fashion on the MPP is discussed.