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Dense Linear Algebra (Data Distributions) Sathish Vadhiyar

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Gaussian Elimination - Review Version 1 for each column i zero it out below the diagonal by adding multiples of row i to later rows for i= 1 to n-1 for each row j below row i for j = i+1 to n add a multiple of row i to row j for k = i to n A(j, k) = A(j, k) – A(j, i)/A(i, i) * A(i, k) 000000000000 0000000000 i i i,i X x j k

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Gaussian Elimination - Review Version 2 – Remove A(j, i)/A(i, i) from inner loop for each column i zero it out below the diagonal by adding multiples of row i to later rows for i= 1 to n-1 for each row j below row i for j = i+1 to n m = A(j, i) / A(i, i) for k = i to n A(j, k) = A(j, k) – m* A(i, k) 000000000000 0000000000 i i i,i X x j k

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Gaussian Elimination - Review Version 3 – Dont compute what we already know for each column i zero it out below the diagonal by adding multiples of row i to later rows for i= 1 to n-1 for each row j below row i for j = i+1 to n m = A(j, i) / A(i, i) for k = i+1 to n A(j, k) = A(j, k) – m* A(i, k) 000000000000 0000000000 i i i,i X x j k

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Gaussian Elimination - Review Version 4 – Store multipliers m below diagonals for each column i zero it out below the diagonal by adding multiples of row i to later rows for i= 1 to n-1 for each row j below row i for j = i+1 to n A(j, i) = A(j, i) / A(i, i) for k = i+1 to n A(j, k) = A(j, k) – A(j, i)* A(i, k) 000000000000 0000000000 i i i,i X x j k

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GE - Runtime Divisions Multiplications / subtractions Total 1+ 2 + 3 + … (n-1) = n 2 /2 (approx.) 1 2 + 2 2 + 3 2 + 4 2 +5 2 + …. (n-1) 2 = n 3/ 3 – n 2 /2 2n 3 /3

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Parallel GE 1 st step – 1-D block partitioning along blocks of n columns by p processors 000000000000 0000000000 i i i,i X x j k

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1D block partitioning - Steps 1. Divisions 2. Broadcast 3. Multiplications and Subtractions Runtime: n 2 /2 xlog(p) + ylog(p-1) + zlog(p-3) + … log1 < n 2 logp (n-1)n/p + (n-2)n/p + …. 1x1 = n 3 /p (approx.) < n 2 /2 +n 2 logp + n 3 /p

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2-D block To speedup the divisions 000000000000 0000000000 i i i,i X x j k P Q

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2D block partitioning - Steps 1. Broadcast of (k,k) 2. Divisions 3. Broadcast of multipliers logQ n 2 /Q (approx.) xlog(P) + ylog(P-1) + zlog(P-2) + …. = n 2 /Q logP 4. Multiplications and subtractions n 3 /PQ (approx.)

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Problem with block partitioning for GE Once a block is finished, the corresponding processor remains idle for the rest of the execution Solution? -

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Onto cyclic The block partitioning algorithms waste processor cycles. No load balancing throughout the algorithm. Onto cyclic 0123023023110 cyclic 1-D block-cyclic2-D block-cyclic Load balance Load balance, block operations, but column factorization bottleneck Has everything

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Block cyclic Having blocks in a processor can lead to block-based operations (block matrix multiply etc.) Block based operations lead to high performance

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GE: Miscellaneous GE with Partial Pivoting 1D block-column partitioning: which is better? Column or row pivoting 2D block partitioning: Can restrict the pivot search to limited number of columns Column pivoting does not involve any extra steps since pivot search and exchange are done locally on each processor. O(n-i-1) The exchange information is passed to the other processes by piggybacking with the multiplier information Row pivoting Involves distributed search and exchange – O(n/P)+O(logP)

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Triangular Solve – Unit Upper triangular matrix Sequential Complexity - Complexity of parallel algorithm with 2D block partitioning (P 0.5 *P 0.5 ) O(n 2 ) O(n 2 )/P 0.5 Thus (parallel GE / parallel TS) < (sequential GE / sequential TS) Overall (GE+TS) – O(n 3 /P)

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Dense LU on GPUs

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LU for Hybrid Multicore + GPU Systems (Tomov et al., Parallel Computing, 2010) Assume the CPU host has 8 cores Assume NxN matrix; Divided into blocks of size NB; Split such that the first N-7NB columns are on GPU memory Last 7NB on the host

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Load Splitting for Hybrid LU

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Steps Current panel is downloaded to CPU; the dark blue part in the figure Panel factored on CPU and result sent to GPU to update trailing sub-matrix; the red part; GPU updates the first NB columns of the trailing submatrix Updated panel sent to CPU; Asynchronously factored on CPU while the GPU updates the rest of the trailing submatrix The rest of the 7 host cores update the last 7 NB host columns

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Dense Cholesky on GPUs

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Cholesky Factorization Symmetric positive definite matrix A A = LL T Similar to Gaussian elimination; succession of block column factorizations followed by updates of the trailing submatrix Can be parallelized using fork-join model similar to parallel GE Matrix-matrix multiplications in parallel (fork) and synchronization needed for next block column factorization (join)

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Heterogeneous Tile Algorithms (Song et al., ICS 2012) Divide a matrix into a set of small and large tiles Small tiles for CPU, and large tiles for GPUs At the top level, divide the matrix into large square tiles of size B x B Then subdivide each top level tile into a number of small rectangular tiles of size Bxb and a remaining tile

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Heterogeneous Tile Cholesky Factorization Factor the first tile to solve L(1,1) Apply L(1,1) to update its right A(1,2)

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Heterogeneous Tile Cholesky Factorization Factor the two tiles below L(1,1) Update all tiles on the right of the first tile column

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Heterogeneous Tile Cholesky Factorization At the second iteration, repeat the above steps on the trailing submatrix starting from the second tile column

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Two Level Block-Cyclic Distribution Divide a matrix A into p x (s.p) rectangular tiles discussed earlier i.e., first divided into pxp large tiles, then partition each large tile into s tiles On a hybrid CPU and GPU machine, allocate the tiles to the host and a number of P GPUs in a 1-D block-cyclic way Those columns whose indices are multiples of s are mapped to the P GPUs in a cyclic way, and remaining columns go to all CPU cores in the host

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Two Level Block-Cyclic Distribution - Example

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Tuning Tile Size Depends on relative performance of CPU and GPU cores For example, to determine the tile width, Bh for CPU:

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References E. Agullo, C. Augonnet, J. Dongarra, H. Ltaief, R. Namyst, S. Thibault, S. Tomov, and W. m. Hu. Faster, Cheaper, Better: a Hybridization Methodology to Develop Linear Algebra Software for GPUs. GPU Computing Gems, 2, 2010. F. Song, S. Tomov, and J. Dongarra. Enabling and Scaling Matrix Computations on Heterogeneous Multi-core and Multi-GPU Systems. In In the Proceedings of IEEE/ACM Supercomputing Conference (SC), pages 365{376, 2012.

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