Preconditioned Krylov Solvers and Methods for Runtime Loop Parallelization

Baxter, Doug; Salz, Joel; Schultz, Martin H.; Eisenstat, Stanley C.

doi:10.21236/ada206388

Cited by 4 publications

(5 citation statements)

References 7 publications

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“…In order to satisfy data dependences, processors busy-wait for rows on which they are dependent. The Wavefront algorithm, studied by , Baxter et al [1988], Sadayappan and Visvanathan [1988], and Liu [1986], avoids this excess busy-waiting by sorting the rows that can be executed in parallel into groups (wavefronts). The processors are assigned rows from the wavefronts in a round-robin fashion.…”

Section: Parallel Sparse Solversmentioning

confidence: 99%

See 1 more Smart Citation

Efficient decomposition and performance of parallel PDE, FFT, Monte Carlo simulations, simplex, and Sparse solvers

Cvetanovic

Freedman

Nofsinger³

1991

J Supercomput

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In this paper, we describe the decomposition of six algorithms: two partial differential equations (PDE) solvers (successive over-relaxation [SOR] and alternating direction implicit [ADI]), fast Fourier transform (FFT), Monte Carlo simulations, Simplex linear programming, and Sparse solvers. The algorithms were selected not only because of their importance in scientific applications, but also because they represent a variety of computational (structured to irregular) and communication (low to high) requirements. We present the performance results TM 1 of these algorithms on two shared-memory VAX/VMS multiprocessor prototypes: the VAX 6300 series with up to 8 processors and the M31 with up to 22 processors. We demonstrate that by efficient decomposition it is possible to achieve high performance for all algorithms on both prototypes. We describe the efficient decomposition techniques applied to optimize the performance of parallel algorithms. Also, we discuss the performance implications due to different cache designs on two multiprocessors.

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Section: Parallel Sparse Solversmentioning

confidence: 99%

“…We have investigated two shared-memory algorithms that solve triangular sparse matrices, the Busy-Wait algorithm and the Wavefront algorithm, which was previously studied by , Baxter et al [1988], and Anderson [1988]. Both algorithms achieved good performance (efficiency >70%) for large (> 2500 rows) sparse matrices.…”

Section: Introductionmentioning

confidence: 99%

Efficient decomposition and performance of parallel PDE, FFT, Monte Carlo simulations, simplex, and Sparse solvers

Cvetanovic

Freedman

Nofsinger³

1991

J Supercomput

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“…On the other hand, for those shared memory machines in which hardware synchronization is available and inexpensive, such as the Alliant FX-8, dynamic scheduling would have some disadvantages since it requires managing queues and generating explicitly busy waits. Both approaches have been tested and compared in [21] where it was concluded that on the Encore Multimax dynamic scheduling is usually preferable except for problems with few synchronization points and large amount of parallelism. In [54] a combination of prescheduling and dynamic scheduling was found to be the best approach on a Sequent balance 21000.…”

Section: Algorithm: Forward Eliiiiinatioii W I T H Level Schedulingmentioning

confidence: 99%

“…In [22,21] and [5,6] a number of additional experiments are presented to study the performance of level scheduling within the context of preconditioned conjugate gradient methods.…”

Section: Eiki-1 In (16)mentioning

confidence: 99%

Krylov Subspace Methods on Supercomputers

Saad¹

1989

SIAM J. Sci. and Stat. Comput.

207

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This paper presents a short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three dimensional models gain importance. A conservative approach to derive effective iterative techniques for supercomputers has been to find efficient parallel/ vector implementations of the standard algorithms. The main source of difficulty in the incomplete factorization preconditionings is in the solution of the triangular systems at each step. We describe in detail a few approaches consisting of implementing efficient forward and backward triangular solutions. Then we discuss polynomial preconditioning as an alternative to standard incomplete factorization techniques. Another efficient approach is to reorder the equations so as improve the structure of the matrix to achieve better parallelism or vectorization. We give an overview of these ideas and others and attempt to comment on their effectiveness or potential for different types of architectures.

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“…For example in the SPE5 problem, the selfexecuting solve requires 23. 4 milliseconds, the prescheduled solve (in Table 3) required 29.0 milliseconds and the doacross version of the solve took 45.0 milliseconds.…”

Section: Where Does the Time Gomentioning

confidence: 99%

Run-time parallelization and scheduling of loops

Baxter

Mirchandaney

Saltz

1989

Proceedings of the First Annual ACM Symposium on Parallel Algorithms and Architectures

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In this paper, we extend t h e class of problenis t h a t can be effcrtively m npiled by parallelizing compilers. This is accornplislictl with t h e doconsider coilstruct which would allow these compilers t o paralldizc~ many prohlcnis i n \vliit 11 siil)staiitial loop-lcvel parallelisin is available h i t cattnot bc tlctcctctl hy staiitl;irtl corn pile-t iriie analysis. \Ire describe and espcriincii till IJ analyze mcdia nisni.; II srd to parallclize t h e work required for these types of loops. In each of t hcse met Iiotls.a new loop structure is produced hy modifying t h e loop to be parallclizcd. \\'P also prcscnt t h e rules by wliicli these loop transformations may be autoiiiatcd i n oi t1c.r.that they b e iriclridccl in langiiage conipilers. Tlic i t i a i i i application a1c.n of o u r wscarcli involves problems i n scientific computations aittl engineering. The wot kIo;itl iiscd in our experirnents iitcludcs a mixture of real ~)i~)l)lcins as wc4l as synt 1101 irally gcncratetl inputs. I2rorn our estcwsivc tests oii t l i t 1 I:iicorc hliil t i i t t ; i s / : l~(~, IVV have reaclicd t h e conclusion that for t h e types of \voi liloatls \vv Iiavc. itivfsligat (VI, self-execution almost always perforins better tltitii ~~tc~-sc~liccli~li~ig. b'iirtlier. 1 IIV i r i iprovcment in performance t h a t accriics as a resiil t of global topological sori irig of indices as opposed to the less experisivc local sortiiig. is not very sigiiifirant i r i t IIP case of self-execution.

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Preconditioned Krylov Solvers and Methods for Runtime Loop Parallelization

Cited by 4 publications

References 7 publications

Efficient decomposition and performance of parallel PDE, FFT, Monte Carlo simulations, simplex, and Sparse solvers

Efficient decomposition and performance of parallel PDE, FFT, Monte Carlo simulations, simplex, and Sparse solvers

Krylov Subspace Methods on Supercomputers

Run-time parallelization and scheduling of loops

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