Multifrontal sparse QR factorization on the GPU

Transcription

1 Multifontal spase QR factoization on the GPU Tim Davis, Sanjay Ranka, Shaanyan Chetlu, Nui Yealan Univesity of Floida Feb 2012

2 GPU-based Multifontal QR factoization why spase QR? multifontal spase QR in a nutshell multi-theaded spase QR spase multifontal QR on the GPU ou stategy wok in pogess

3 Why multifontal spase QR factoization? wide applicability of QR numeically stable bette paallelism independent poblems decoupled, unlike LU o Cholesky Communication-Avoiding QR (CAQR) othogonal methods have highe flops pe memoy efeence QR assembly step is GPU-fiendly elated to othe diect methods (LU, Cholesky, LDL T )

4 Multifontal spase QR factoization in a nutshell ows can be opeated on in any ode goup togethe ows with left-most nonzeos in the same column factoize each block of ows independently each block of ows takes on the same nonzeo patten (a fontal mati) mege of fontal matices: copy, not add (unlike LU, Cholesky) epeat until the mati becomes uppe tiangula

5 X X X X X X X X X X X The dots: union of the nonzeo pattens of all ows in each block

6 Householde Spase QR Sot the ows of A by column of leftmost nonzeo, and annihilate X X X X X X X X X X X Can do the othe blocks at the same time X X X X X X X X

7 Householde Spase QR Key obsevation: each block of ows to annihilate has the same nonzeo patten So place them in a dense submati and use dense mati kenels Fo column 1: X X X X X X X X X X X use this: X X X

8 Multifontal QR factoization in a nutshell Goup ows of A with nonzeo in same leftmost column Apply Householde to educe each goup to uppe tiangula, one ow becomes a ow of R Append emainde to the goup fo the net nonzeo column net: a tee of columns (the column elimination tee) Lump adjacent columns togethe if thei ows of R have the same nonzeo patten (supenodes)

9 the mati A the QR facto R the column elimination tee 5

10 QR factoization of a leaf fontal mati h h h h c c c c c c h h h h h h h h factoized font 1 ows of A fo font 1

11 QR fo a non-leaf fontal mati child font c1 c1 c1 c1 c1 c 1 child c c2 c2 c2 c 8 2 c c c2 c2 c 2 child c3 c3 c3 c3 14 c3 c3 c3 15 c3 c

12 Non-leaf fontal mati: childen to assemble child font c1 c1 c1 c1 c1 c 1 child c c2 c2 c2 c 8 2 c c c2 c2 c 2 child c3 c3 c3 c3 14 c3 c3 c3 15 c3 c ows of A fo font Assembly: shuffle the above data into a single mati (net slide)

13 Fontal mati assembly: no ead-modify-wite child c3 c 3 c 3 c3 14 c3 c 3 c3 15 c3 c3 child font c1 c 1 c 1 c1 c 1 c assembled font c 3 c 3 c 3 c 3 c 1 c 1 c 1 c2 c 2 c 2 c2 c 2 c 2 c 2 c3 c3 c3 c1 c1 c 2 c 2 c3 c 3 c 1 c 2 ows of A fo font child c c 2 c 2 c2 c c 2 c2 c c2 2 c 2

14 Fontal mati: afte factoization factoized font h h h h h h c c c c h h h c c c h h h h c c h h h h h c h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h R facto c: contibution to paent h: Householde vectos (Q)

15 Multifontal QR: assembly step is GPU-fiendly Assembly fo multifontal LU, Cholesky equies data shuffling and addition in paallel: two thead blocks would need to synchonize the summation (ead-modify-wite) fo multifontal QR equies just data shuffling; no addition in paallel: no ead-modify-wite

16 Fontal mati QR factoization on the GPU Kenel 1: suppose one thead block can factoize one stipe with a fied maimum numbe of ows If stipe has too many columns, slice it and apply Q afte computed

17 Fontal mati QR factoization on the GPU Kenel 2: take two stipes; annihilate below diagonal Befoe: afte: numeically stable because of othogonal opeations

18 Fontal mati QR factoization on the GPU Combine with moe stipes:

19 Fontal mati QR factoization on the GPU Rinse and epeat, always woking on pais of stipes at a time, whee two pais fit in shaed memoy:

20 Fontal mati QR factoization on the GPU Kenel dependencies fo a fontal mati of 4 stipes and 6 blocks of columns:

21 Fontal mati QR factoization on the GPU Futhe pipelining fo additional paallelism

22 Algoithm outline symbolic analysis: on the CPU numeic factoization: on the GPU

23 Symbolic analysis: on the CPU Fill-educing odeing (typically O( A ) time) Symbolic analysis (nealy O( A ), without foming A T A find the column etee ow counts of R find elaed supenodes sot ows of A task assignment (subtees = paallel subtasks) total time: about O( A )

24 Numeical factoization: on the GPU numeical factoization of subtees: fontal mati assembly fontal mati factoization contibution block stacked fo paent the challenge of heteogeneous computations within a subtee some factoize while othes assemble fonts vay wildly in size fom tiny to huge tee diven by mati; not simple balanced binay tee staging: factoize one subtee while tansfeing anothe CPU GPU multi-gpu: each GPU handles independent subtees if font is huge, teat like multi-gpu dense QR factoization with blocking/stiping

25 Pefomance esults: pe-gpu method Least squaes poblem: 2 million by 110 thousand Method odeing pocs time =A\b COLMMD 1? =A\b AMD 1 11 days MA49 AMD 1 35 hous SuiteSpaseQR AMD 1 15 hous SuiteSpaseQR METIS 1 45 minutes SuiteSpaseQR METIS minutes Algoithmic speedup vs =A\b: 375 Paallel speedup: 575 on 16 coes Total: 2,155 (14 Gflops on 70 Gflops machine) Single coe: 25 Gflop peak, same as LAPACK QR

26 Gflop vs LAPACK (single coe) 4 3 n=100 n=1000 n= GFlops SuiteSpaseQR Dense QR (DGEQRF) Flop count / memoy usage in bytes

27 Multifontal QR on the GPU Tesla C2050 double-pecision fontal mati QR (65 GFlops) fonts emain on the GPU fontal mati assembly in-pogess: stip-mining scheduling nodes of tee = one fontal mati split each font into a subtee paallel assembly of some fonts while othes ae factoized

28 Multifontal QR on the GPU: stip-mining

29 Multifontal QR on the GPU: stip-mining etc 4a 4b 4c 4d A A 5th A A A 8c 4th 2 3e 8a 8b 3d A 3c 3d A A 2nd 1 3a 3b A 1st kenel launch

30 Multifontal Spase QR on the GPU: Summay Fast symbolic analysis and fill-educing odeing ( O( A )) Dense mati kenels to eploit tightly-coupled egula paallelism within the GPU Elimination tee fo loosely-coupled iegula paallelism High pefomance in pe-gpu vesion peak Gflop ate same as LAPACK ample paallel speedup Appeas as the built-in =A\b and q in MATLAB R2009a If A is all nonzeo, =spase(a)\b can be faste than =A\b GPU method in pogess dense QR fo fontal matices: Shaanyan Chetlu assembly: by the GPU Regula memoy taffic to/fom global memoy; all iegula taffic in shaed memoy within each thead block: Nui Yealan stip-mining scheduling of the epanded fontal mati tee staging subtees to handle lage poblems (> 6 GB)

31 Acknowledgements

32 Postscipt Please send me you matices!