Issues in the Design of Scalable Out-of-Core Dense Symmetric Indefinite Factorization Algorithms

Strazdins, Peter

doi:10.1007/3-540-44863-2_70

Cited by 3 publications

(4 citation statements)

References 10 publications

(28 reference statements)

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“…Most of the previous out‐of‐core symmetric indefinite factorization algorithms rely on the Bunch‐Kaufman algorithm. For instance, the challenges of designing an out‐of‐core implementation of the algorithm were studies on a distributed‐memory CPU computer . There, to reduce the communication overheads and get scalable performance, diagonal pivots were selected within or near the current elimination block.…”

Section: Related Workmentioning

confidence: 99%

“…For instance, the challenges of designing an out-of-core implementation of the algorithm were studies on a distributed-memory CPU computer. 10 There, to reduce the communication overheads and get scalable performance, diagonal pivots were selected within or near the current elimination block. Similarly, out-of-core factorization of a sparse symmetric indefinite matrix based on the Bunch-Kaufman algorithm was considered.…”

Section: Introductionmentioning

confidence: 99%

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Non‐GPU‐resident symmetric indefinite factorization

Yamazaki

Tomov

Dongarra

2016

Concurrency and Computation

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Summary We study various algorithms to factorize a symmetric indefinite matrix that does not fit in the core memory of a computer. There are two sources of the data movement into the memory: one needed for selecting and applying pivots and the other needed to update each column of the matrix for the factorization. It is a challenge to obtain high performance of such an algorithm when the pivoting is required to ensure the numerical stability of the factorization. For example, when factorizing each column of the matrix, a diagonal entry, which ensures the stability, may need to be selected as a pivot among the remaining diagonals, and moved to the leading diagonal by swapping both the corresponding rows and columns of the matrix. If the pivot is not in the core memory, then it must be loaded into the core memory. For updating the matrix, the data locality may be improved by partitioning the matrix. For example, a right‐looking partitioned algorithm first factorizes the leading columns, called panel, and then uses the factorized panel to update the trailing submatrix. This algorithm only accesses the trailing submatrix after each panel factorization (instead of after each column factorization) and performs most of its floating‐point operations (flops) using BLAS‐3, which can take advantage of the memory hierarchy. However, because the pivots cannot be predetermined, the whole trailing submatrix must be updated before the next panel factorization can start. When the whole submatrix does not fit in the core memory all at once, loading the block columns into the memory can become the performance bottleneck. Similarly, the left‐looking variant of the algorithm would require to update each panel with all of the previously factorized columns. This makes it a much greater challenge to implement an efficient out‐of‐core symmetric indefinite factorization compared with an out‐of‐core nonsymmetric LU factorization with partial pivoting, which only requires to swap the rows of the matrix and accesses the trailing submatrix after each in‐core factorization (instead of after each panel factorization by the symmetric factorization). To reduce the amount of the data transfer, in this paper we uses the recently proposed left‐looking communication‐avoiding variant of the symmetric factorization algorithm to factorize the columns in the core memory, and then perform the partitioned right‐looking out‐of‐core trailing submatrix updates. This combination may still require to load the pivots into the core memory, but it only updates the trailing submatrix after each in‐core factorization, while the previous algorithm updates it after each panel factorization.Although these in‐core and out‐of‐core algorithms can be applied at any level of the memory hierarchy, we apply our designs to the GPU and CPU memory, respectively. We call this specific implementation of the algorithm a non–GPU‐resident implementation. Our performance results on the current hybrid CPU/GPU architecture demonstrate that when the matrix is much larger than the GPU mem...

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non‐GPU‐resident symmetric indefinite factorization

Yamazaki

Tomov

Dongarra

2016

Concurrency and Computation

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“…If pivots were searched only inside tile A ii , the factorization would depend only and exclusively on A kk . However, for most pivoting techniques, pivots are searched throughout columns, which make the design of efficient parallel algorithm very difficult [18].…”

Section: Tile-wise Pivotingmentioning

confidence: 99%

“…The search for pivots outside a given tile curtails memory locality and data dependence between tiles (or tasks). The former has a direct impact on the performance of serial kernels and the latter on parallel performance (by increasing data dependence among tiles, granularity is decreased and therefore scalability) [18].…”

Section: Introductionmentioning

confidence: 99%

Reducing the Amount of Pivoting in Symmetric Indefinite Systems

Becker

Baboulin

Dongarra

2012

Parallel Processing and Applied Mathematics

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Abstract. This paper illustrates how the communication due to pivoting in the solution of symmetric indefinite linear systems can be reduced by considering innovative approaches that are different from pivoting strategies implemented in current linear algebra libraries. First a tiled algorithm where pivoting is performed within a tile is described and then an alternative to pivoting is proposed. The latter considers a symmetric randomization of the original matrix using the so-called recursive butterfly matrices. In numerical experiments, the accuracy of tile-wise pivoting and of the randomization approach is compared with the accuracy of the Bunch-Kaufman algorithm.

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On the performance of parallel factorization of out-of-core matrices

Caron

Utard

2004

Parallel Computing

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Issues in the Design of Scalable Out-of-Core Dense Symmetric Indefinite Factorization Algorithms

Cited by 3 publications

References 10 publications

Non‐GPU‐resident symmetric indefinite factorization

Non‐GPU‐resident symmetric indefinite factorization

Reducing the Amount of Pivoting in Symmetric Indefinite Systems

On the performance of parallel factorization of out-of-core matrices

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