Parallel Solution of Triangular Systems on Distributed-Memory Multiprocessors

Heath, Michael T.; Romine, Charles H.

doi:10.1137/0909037

Cited by 131 publications

(26 citation statements)

References 9 publications

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“…Algorithm 2 below uses this blocking schedule with blocking factor b to compute the triangular solve. Our algorithm is similar to the wavefront algorithm given by Heath et al [11]. The parallel Algorithm 2 can be executed using p = n/b processors.…”

Section: Attainabilitymentioning

confidence: 99%

Tradeoffs Between Synchronization, Communication, and Work in Parallel Linear Algebra Computations

Solomonik¹,

Carson²,

Knight³

et al. 2014

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This paper derives tradeoffs between three basic costs of a parallel algorithm: synchronization, data movement, and computational cost. Our theoretical model counts the amount of work and data movement as a maximum of any execution path during the parallel computation. By considering this metric, rather than the total communication volume over the whole machine, we obtain new insight into the characteristics of parallel schedules for algorithms with non-trivial dependency structures. The tradeoffs we derive are lower bounds on the execution time of the algorithm which are independent of the number of processors, but dependent on the problem size. Therefore, these tradeoffs provide lower bounds on the parallel execution time of any algorithm computed by a system composed of any number of homogeneous components each with associated computational, communication, and synchronization payloads. We first state our results for general graphs, based on expansion parameters, then we apply the theorem to a number of specific algorithms in numerical linear algebra, namely triangular substitution, Gaussian elimination, and Krylov subspace methods. Our lower bound for LU factorization demonstrates the optimality of Tiskin's LU algorithm [24] answering an open question posed in his paper, as well as of the 2.5D LU [20] algorithm which has analogous costs. We treat the computations in a general manner by noting that the computations share a similar dependency hypergraph structure and analyzing the communication requirements of lattice hypergraph structures.

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Section: Attainabilitymentioning

confidence: 99%

Tradeoffs Between Synchronization, Communication, and Work in Parallel Linear Algebra Computations

Solomonik¹,

Carson²,

Knight³

et al. 2014

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show abstract

“…A large number of technical papers have been written in this area (Angus, et al, 1988;Fox, et al, 1988;Geist & Romine, 1988;and Heath & Romine, 1988). This chapter discusses the parallel matrix factor and solve routines that were implemented as part of this lED.…”

Section: Parallel Matrix Factor and Solvementioning

confidence: 99%

“…Several algorithms have been developed. Two of the most popular are the wavefront algorithm and the cyclic algorithm (Heath & Romine, 1988). These algorithms assume the matrix is column-wrap mapped onto the processors.…”

Section: Parallel Matrix Solvementioning

confidence: 99%

Parallel Processing Applied to Computational Electromagnetics

Russell¹,

Rockway²

1991

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MA SUMMARY OBJECTIVETo apply parallel processing techniques to a computational electromagnetic code and show that high-performance computing can improve the utility and efficiency of techniques used in ship electromagnetic designs. RESULTS" Efficient parallel matrix filling, factoring, and solving routines were developed, implemented, and evaluated. " The performances of the parallel matrix factoring and solving routines were independent of the structure being analyzed, whereas the performance of the parallel matrix filling routine was found to be very dependent on the structure being analyzed. " Several different techniques for mapping the matrix columns onto the processors were implemented and evaluated. * Efficiencies for all parallel algorithms increased as the size of the matrix being solved increased. " Performance comparisons were made between a MicroVax, the four transputer array, and the Convex Model C-220 mini-supercomputer. PAYOFFSSeveral payoffs have resulted, or will result, from the work done on this lED project: " A computer platform now exists for the efficient transition of computational electromagnetics to high-performance computing. The transputer array can be used to initiate parallelizing an existing computer code.

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“…2 distributed among the diagonal processors in each of these quadrants. The stategy for merging is to move the even numbered blocks b 6 , bs, bl 0 , b 12 down, to the diagonal processors of the lower (southern) quadrants, and the odd numbered blocks bs, b 7 , bg, b 11 up (north). The cost is 2(-SL + 4s -1) communication steps for the bidirectional move.…”

Section: Lmentioning

confidence: 99%

Some Aspects of Parallel Implementation of the Finite Element Method on Message Passing Architectures

Babuska¹,

Elman²

1988

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Parallel Solution of Triangular Systems on Distributed-Memory Multiprocessors

Cited by 131 publications

References 9 publications

Tradeoffs Between Synchronization, Communication, and Work in Parallel Linear Algebra Computations

Tradeoffs Between Synchronization, Communication, and Work in Parallel Linear Algebra Computations

Parallel Processing Applied to Computational Electromagnetics

Some Aspects of Parallel Implementation of the Finite Element Method on Message Passing Architectures

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