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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Analysis of a parallel multigrid algorithm

This paper considers the parallel multigrid algorithm of Frederickson and McBryan (1987). This algorithm uses multiple coarse-grid problems (instead of one) in the hope of accelerating convergence, and is found to have a close relationship to traditional multigrid methods. Specifically, the parallel coarse-grid correction operator is identical to a traditional multigrid coarse-grid correction operator, except that the mixing of high and low frequencies caused by aliasing error is removed. Appropriate relaxation operators can be chosen to take advantage of this property. Comparisons between the standard multigrid and the new method are made.

Chan, Tony F.↗
NASA NTRS1993

A fast algorithm for parallel computation of multibody dynamics on MIMD parallel architectures

In this paper the implementation of a parallel O(LogN) algorithm for computation of rigid multibody dynamics on a Hypercube MIMD parallel architecture is presented. To our knowledge, this is the first algorithm that achieves the time lower bound of O(LogN) by using an optimal number of O(N) processors. However, in addition to its theoretical significance, the algorithm is also highly efficient for practical implementation on commercially available MIMD parallel architectures due to its highly coarse grain size and simple communication and synchronization requirements. We present a multilevel parallel computation strategy for implementation of the algorithm on a Hypercube. This strategy allows the exploitation of parallelism at several computational levels as well as maximum overlapping of computation and communication to increase the performance of parallel computation.

Fijany, Amir↗
NASA NTRS1989

Totally parallel multilevel algorithms for sparse elliptic systems

The fastest known algorithms for the solution of a large elliptic boundary value problem on a massively parallel hypercube all require O(log(n)) floating point operations and O(log(n)) distance-1 communications, if massively parallel is defined to mean a number of processors proportional to the size n of the problem. The Totally Parallel Multilevel Algorithm (TPMA) that has, as special cases, four of these fast algorithms is described. These four algorithms are Parallel Superconvergent Multigrid (PSMG), Robust Multigrid, the Fast Fourier Transformation (FFT) based Spectral Algorithm, and Parallel Cyclic Reduction. The algorithm TPMA, when described recursively, has four steps: (1) project to a collection of interlaced, coarser problems at the next lower level; (2) apply TPMA, recursively, to each of these lower level problems, solving directly at the lowest level; (3) interpolate these approximate solutions to the finer grid, and to verage them to form an approximate solution on this grid; and (4) refine this approximate solution with a defect-correction step, using a local approximate inverse. Choice of the projection operator (P), the interpolation operator (Q), and the smoother (S) determines the class of problems on which TPMA is most effective. There are special cases in which the first three steps produce an exact solution, and the smoother is not needed (e.g., constant coefficient operators).

Frederickson, Paul O.↗
NASA NTRS1987

A survey of parallel multigrid algorithms

A typical multigrid algorithm applied to well-behaved linear-elliptic partial-differential equations (PDEs) is described. Criteria for designing and evaluating parallel algorithms are presented. Before evaluating the performance of some parallel multigrid algorithms, consideration is given to some theoretical complexity results for solving PDEs in parallel and for executing the multigrid algorithm. The effect of mapping and load imbalance on the partial efficiency of the algorithm is studied.

Chan, Tony F.↗
NASA NTRS1993

Applications and accuracy of the parallel diagonal dominant algorithm

The Parallel Diagonal Dominant (PDD) algorithm is a highly efficient, ideally scalable tridiagonal solver. In this paper, a detailed study of the PDD algorithm is given. First the PDD algorithm is introduced. Then the algorithm is extended to solve periodic tridiagonal systems. A variant, the reduced PDD algorithm, is also proposed. Accuracy analysis is provided for a class of tridiagonal systems, the symmetric, and anti-symmetric Toeplitz tridiagonal systems. Implementation results show that the analysis gives a good bound on the relative error, and the algorithm is a good candidate for the emerging massively parallel machines.

Sun, Xian-He↗
NASA NTRS1991

Parallel optimization algorithms and their implementation in VLSI design

Two new parallel optimization algorithms based on the simplex method are described. They may be executed by a SIMD parallel processor architecture and be implemented in VLSI design. Several VLSI design implementations are introduced. An application example is reported to demonstrate that the algorithms are effective.

Lee, G.↗
NASA NTRS1991

Parallel-Processing Algorithms For Dynamics Of Manipulators

Class of parallel and parallel/pipeline algorithms presented for more efficient computation of manipulator inertia matrix. Essential for implementing advanced dynamic control schemes as well as dynamic simulation of manipulator motion.

Fijany, Amir↗
DOE OSTI2024

PANDORA: A Parallel Dendrogram Construction Algorithm for Single Linkage Clustering on GPU

This paper introduces Pandora, a parallel algorithm for computing dendrograms, the hierarchical cluster trees for single linkage clustering (SLC). Current parallel approaches construct dendrograms by partitioning a minimum spanning tree and removing edges. However, they struggle with skewed, hard-to-parallelize real-world dendrograms. Consequently, computing dendrograms is the sequential bottleneck in HDBSCAN*[21], a popular SLC variant. Pandora uses recursive tree contraction to address this limitation. Pandora contracts nodes to construct progressively smaller trees. It computes the smallest contracted dendrogram and expands it by inserting contracted edges. This recursive strategy is highly parallel, skew-independent, work-optimal, and well-suited for GPUs and multicores. We develop a performance portable implementation of Pandora in Kokkos[31] and evaluate its performance on multicore CPUs and multi-vendor GPUs (e.g., Nvidia, AMD) for dendrogram construction in HDBSCAN*. Multithreaded Pandora is 2.2x faster than the current best-multithreaded implementation. Our GPU version achieves 6-20x speedup on AMD GPUs and 10-37x on NVIDIA GPUs over multithreaded Pandora. Pandora removes HDBSCAN*’s sequential bottleneck, greatly boosting efficiency, particularly with GPUs.

Sao, Piyush↗
NASA NTRS2000

A Parallel Genetic Algorithm for Automated Electronic Circuit Design

We describe a parallel genetic algorithm (GA) that automatically generates circuit designs using evolutionary search. A circuit-construction programming language is introduced and we show how evolution can generate practical analog circuit designs. Our system allows circuit size (number of devices), circuit topology, and device values to be evolved. We present experimental results as applied to analog filter and amplifier design tasks.

Lohn, Jason D.↗
NASA NTRS1990

Line-drawing algorithms for parallel machines

The fact that conventional line-drawing algorithms, when applied directly on parallel machines, can lead to very inefficient codes is addressed. It is suggested that instead of modifying an existing algorithm for a parallel machine, a more efficient implementation can be produced by going back to the invariants in the definition. Popular line-drawing algorithms are compared with two alternatives; distance to a line (a point is on the line if sufficiently close to it) and intersection with a line (a point on the line if an intersection point). For massively parallel single-instruction-multiple-data (SIMD) machines (with thousands of processors and up), the alternatives provide viable line-drawing algorithms. Because of the pixel-per-processor mapping, their performance is independent of the line length and orientation.

Pang, Alex T.↗
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 NTRS1987

A parallel solution for the symmetric Eigenproblem

A completely parallel algorithm for the symmetric eigenproblem AX = Lambda BX is outlined. The algorithm is parallel in the sense that the numerical operations do not occur in a fixed sequence. Therefore, a large number of operations can be programmed to be performed concurrently on a computer with multiple central processing units. The standard symmetric eigenvalue problem AX = Lambda X has the property that the n eigenvalues of the principal submatrix of A of order n are separated by the (n-1) eignvalues of the principal submatrix of order (n-1). The separation property delineated n intervals containing one eigenvalue. Each eigenvalue and corresponding eigenvector can be computed independently. The n eigenproblem calculations can be divided among multiple processing units.

Thurston, Gaylen A.↗
NASA NTRS1996

A Fast parallel tridiagonal algorithm for a class of CFD applications

The parallel diagonal dominant (PDD) algorithm is an efficient tridiagonal solver. This paper presents for study a variation of the PDD algorithm, the reduced PDD algorithm. The new algorithm maintains the minimum communication provided by the PDD algorithm, but has a reduced operation count. The PDD algorithm also has a smaller operation count than the conventional sequential algorithm for many applications. Accuracy analysis is provided for the reduced PDD algorithm for symmetric Toeplitz tridiagonal (STT) systems. Implementation results on Langley's Intel Paragon and IBM SP2 show that both the PDD and reduced PDD algorithms are efficient and scalable.

Moitra, Stuti↗
NASA NTRS1993

Mapping robust parallel multigrid algorithms to scalable memory architectures

The convergence rate of standard multigrid algorithms degenerates on problems with stretched grids or anisotropic operators. The usual cure for this is the use of line or plane relaxation. However, multigrid algorithms based on line and plane relaxation have limited and awkward parallelism and are quite difficult to map effectively to highly parallel architectures. Newer multigrid algorithms that overcome anisotropy through the use of multiple coarse grids rather than relaxation are better suited to massively parallel architectures because they require only simple point-relaxation smoothers. In this paper, we look at the parallel implementation of a V-cycle multiple semicoarsened grid (MSG) algorithm on distributed-memory architectures such as the Intel iPSC/860 and Paragon computers. The MSG algorithms provide two levels of parallelism: parallelism within the relaxation or interpolation on each grid and across the grids on each multigrid level. Both levels of parallelism must be exploited to map these algorithms effectively to parallel architectures. This paper describes a mapping of an MSG algorithm to distributed-memory architectures that demonstrates how both levels of parallelism can be exploited. The result is a robust and effective multigrid algorithm for distributed-memory machines.

Overman, Andrea↗
NASA NTRS1993

Mapping robust parallel multigrid algorithms to scalable memory architectures

The convergence rate of standard multigrid algorithms degenerates on problems with stretched grids or anisotropic operators. The usual cure for this is the use of line or plane relaxation. However, multigrid algorithms based on line and plane relaxation have limited and awkward parallelism and are quite difficult to map effectively to highly parallel architectures. Newer multigrid algorithms that overcome anisotropy through the use of multiple coarse grids rather than line relaxation are better suited to massively parallel architectures because they require only simple point-relaxation smoothers. The parallel implementation of a V-cycle multiple semi-coarsened grid (MSG) algorithm or distributed-memory architectures such as the Intel iPSC/860 and Paragon computers is addressed. The MSG algorithms provide two levels of parallelism: parallelism within the relaxation or interpolation on each grid and across the grids on each multigrid level. Both levels of parallelism must be exploited to map these algorithms effectively to parallel architectures. A mapping of an MSG algorithm to distributed-memory architectures that demonstrate how both levels of parallelism can be exploited is described. The results is a robust and effective multigrid algorithm for distributed-memory machines.

Overman, Andrea↗
NASA NTRS1995

Parallel CFD Algorithms for Aerodynamical Flow Solvers on Unstructured Meshes

The Advisory Group for Aerospace Research and Development (AGARD) has requested my participation in the lecture series entitled Parallel Computing in Computational Fluid Dynamics to be held at the von Karman Institute in Brussels, Belgium on May 15-19, 1995. In addition, a request has been made from the US Coordinator for AGARD at the Pentagon for NASA Ames to hold a repetition of the lecture series on October 16-20, 1995. I have been asked to be a local coordinator for the Ames event. All AGARD lecture series events have attendance limited to NATO allied countries. A brief of the lecture series is provided in the attached enclosure. Specifically, I have been asked to give two lectures of approximately 75 minutes each on the subject of parallel solution techniques for the fluid flow equations on unstructured meshes. The title of my lectures is "Parallel CFD Algorithms for Aerodynamical Flow Solvers on Unstructured Meshes" (Parts I-II). The contents of these lectures will be largely review in nature and will draw upon previously published work in this area. Topics of my lectures will include: (1) Mesh partitioning algorithms. Recursive techniques based on coordinate bisection, Cuthill-McKee level structures, and spectral bisection. (2) Newton's method for large scale CFD problems. Size and complexity estimates for Newton's method, modifications for insuring global convergence. (3) Techniques for constructing the Jacobian matrix. Analytic and numerical techniques for Jacobian matrix-vector products, constructing the transposed matrix, extensions to optimization and homotopy theories. (4) Iterative solution algorithms. Practical experience with GIVIRES and BICG-STAB matrix solvers. (5) Parallel matrix preconditioning. Incomplete Lower-Upper (ILU) factorization, domain-decomposed ILU, approximate Schur complement strategies.

Barth, Timothy J.↗
NASA NTRS1984

Acoustooptic linear algebra processors - Architectures, algorithms, and applications

Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.

Casasent, D.↗