Mapping Dense LU Factorization on Multicore Supercomputer Nodes
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1 Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander, Phil Miller, Ramprasad Venkataraman, nshu rya, erry Jones, Laxmikant Kale niversity Oak of Illinois rbana-champaign Ridge National Laboratory May 22, 2012
2 N N L Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 2/42
3 Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 3/42
4 Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 4/42
5 ctive panel block block railing submatrix block Previously factored block Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 5/42
6 ctive panel block block railing submatrix block Previously factored block Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 6/42
7 ctive panel block block railing submatrix block Previously factored block Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 7/42
8 ctive panel block block railing submatrix block Previously factored block Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 8/42
9 Process grid with 16 cores Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 9/42
10 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 10/42
11 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 11/42
12 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 12/42
13 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 13/42
14 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 14/42
15 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 15/42
16 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 16/42
17 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 17/42
18 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 18/42
19 N N Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 19/42
20 Overview Striding: vary the amount of locality in the active panel Rotation: vary the degree of parallelism in the row/column estbed implemented in Charm++ Cray X5: 67% of peak for 8k cores utilizing 75% of memory for the matrix Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 20/42
21 estbed llows easy experimentation with various mappings int map(const int coor[2]) { int gridx = coor[0] % P; int gridy = coor[1] % Q; int proc = gridy P + gridx; return proc; } // N/b x N/b array of blocks, b is block size CkrrayOptions opts(n/b, N/b); // set block to processor mapping opts.setmap(map); // create parallel array luproxy = CProxy_LBlock::ckNew(opts); Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 21/42
22 Striding Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 22/42
23 node node node node 3 Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 23/42
24 Rank-1 pdate imes (ms) Microarchitecture Cores/socket performing updates Efficiency Intel Nehalem-EP % MD Istanbul % IBM Blue Gene/P % Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 24/42
25 Weak-Scaled Single-Node Microbenchmark Number of L3 Cache Misses (x1000) Expected Compulsory Misses Measured Misses Number of Cores Performing Rank-1 pdates Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 25/42
26 Striding: a range of values for u Block-cylic: Block-cylic with stride (s = 2): Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 26/42
27 Formulæ: augmenting block-cyclic with a stride s Block-cylic: m 1 = x mod P (x in grid) n 1 = y mod Q (y in grid) f r(x, y, P, Q) = m 1 Q + n 1 (1) f c(x, y, P, Q) = n 1 P + m 1 (2) Block-cylic with striding: y p = Q m 3 =x mod P (grid y index) (x in grid) n 3 =y mod s (y in subgrid) y mod Q q = (subgrid y) s f s(x, y, P, Q, s) =m 3 s + n 3 + P sq (3) where 1 s Q and s is a factor of Q. Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 27/42
28 cores 528 cores 2112 cores 6 GFlop/s/core row major ctive panel cores/node column major Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 28/42
29 Rotation Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 29/42
30 Process grid with 24 cores Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 30/42
31 Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 31/42
32 7 6 Wide (8 x 32) Process Grid Square (16 x 16) Process Grid all (32 x 8) Process Grid 5 ime (seconds) ctive Panel Step Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 32/42
33 cores 528 cores 2112 cores 6 GFlop/s/core tall process grid spect ratio wide process grid Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 33/42
34 Rotation: increasing row parallelism Block-cylic: Block-cylic with rotation (r = 2): Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 34/42
35 Formulæ: augmenting block-cyclic with a rotation r Block-cylic: m 1 = x mod P (x in grid) n 1 = y mod Q (y in grid) f r(x, y, P, Q) = m 1 Q + n 1 (4) f c(x, y, P, Q) = n 1 P + m 1 (5) Block-cylic with rotation: y p = Q m 2 = (x + pr) mod P n 2 = y mod Q (grid y index) (x in grid) (y in grid) f rotrow (x, y, P, Q, r) = m 2 Q + n 2 (6) f rotcol (x, y, P, Q, r) = n 2 P + m 2 (7) Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 35/42
36 cores 528 cores 2112 cores 6 GFlop/s/core less rotation more rotation Cores performing triangular solves Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 36/42
37 cores 528 cores 2112 cores 6 GFlop/s/core Cores performing triangular solves / sqrt(#cores) Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 37/42
38 Combining striding and rotation Block-cylic: Block-cylic with striding and rotation: Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 38/42
39 Formulæ: augmenting parameters r and s Block-cylic: m 1 = x mod P (x in grid) n 1 = y mod Q (y in grid) f r(x, y, P, Q) = m 1 Q + n 1 (8) f c(x, y, P, Q) = n 1 P + m 1 (9) Block-cylic with striding and rotation: y p = Q m 4 =(x + pr) mod P n 4 =y mod s y mod Q q = s (grid y index) (x in grid) (y in grid) (subgrid y) f sr(x, y, P, Q, r, s) =m 4 s + n 4 + P sq (10) Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 39/42
40 Data Movement Overhead Number of Cores Factorization ime (seconds) Data Movement ime (seconds) otal ime (seconds) Data Movement Percentage of otal 1.9% 1.0% Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 40/42
41 Conclusion Performance compared to best square grid Striding: 8% performance increase Rotation: 3% performance increase Combined: 11% performance increase, corresponding to a 7% increase in peak performance With memory usage constant at about 75% on Cray X5: Future work bout 67% of peak from 120 cores to 8064 cores Recursive panel factorization Communication-avoiding Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 41/42
42 100 heoretical peak on X5 Weak scaling on X5 heoretical peak on BG/P Strong scaling on BG/P 10 otal Flop/s Number of Cores Mapping Dense L Factorization on Multicore Supercomputer Nodes Jonathan Lifflander (IC) 42/42
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