GeorgiaTechCSE12mar2010

GeorgiaTechCSE12mar2010 - Challenges in Combinatorial...

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1   Challenges in Combinatorial Scientific Computing John R. Gilbert University of California, Santa Barbara Georgia Tech CSE Colloquium March 12, 2010 Support: NSF, DARPA, SGI
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2 Combinatorial Scientific Computing “I observed that most of the  coefficients in our matrices were  zero; i.e., the nonzeros were  ‘sparse’ in the matrix, and that  typically the triangular matrices  associated with the forward and  back solution provided by Gaussian  elimination would remain sparse if  pivot elements were chosen with  care” - Harry Markowitz, describing the 1950s work on portfolio theory that won the 1990 Nobel Prize for Economics
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3 Graphs and Sparse Matrices : Cholesky factorization 10 1 3 2 4 5 6 7 8 9 10 1 3 2 4 5 6 7 8 9 G(A) G + (A) [chordal] Symmetric Gaussian elimination: for j = 1 to n     add edges between j’s     higher-numbered neighbors Fill new nonzeros in factor
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4 Large graphs are everywhere… WWW snapshot, courtesy Y. Hyun Yeast protein interaction network, courtesy H. Jeong Internet structure Social interactions Scientific datasets: biological, chemical, cosmological, ecological, …
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5 An analogy? As the “middleware” of scientific computing, linear algebra has supplied or enabled: Mathematical tools “Impedance match” to computer operations High-level primitives High-quality software libraries Ways to extract performance from computer architecture Interactive environments Computers Continuous physical modeling Linear algebra
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6 An analogy? Computers Continuous physical modeling Linear algebra Discrete structure analysis Graph theory Computers
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7 An analogy? Well, we’re not there yet …. Discrete structure analysis Graph theory Computers   Mathematical tools ? “Impedance match” to computer operations ?   High-level primitives High-quality software libs Ways to extract performance from computer architecture Interactive environments
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8 The Case for Primitives
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9 All-Pairs Shortest Paths on a GPU [Buluc et al.] A B C D A B D C A = A*;      % recursive call B = AB;  C = CA;   D = D + CB; D = D*;      % recursive call B = BD;  C = DC; A = A + BC;  is “min”,    ×   is “add” Based on R-Kleene algorithm Well suited for GPU architecture: In-place computation => low memory bandwidth Few, large MatMul calls => low GPU dispatch overhead Recursion stack on host CPU, not on multicore GPU Careful tuning of GPU code Fast matrix-multiply kernel
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10 The Case for Primitives 480x 480x Lifting Floyd-Warshall to  GPU The right primitive! Unorthodox R- Kleene algorithm Runtime vs. Matrix Dimension,  log-log APSP: Experiments and observations
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11 The Case for Primitives 480x 480x Lifting Floyd-Warshall to  GPU The right primitive!
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This note was uploaded on 12/27/2011 for the course CMPSC 240A taught by Professor Gilbert during the Fall '09 term at UCSB.

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GeorgiaTechCSE12mar2010 - Challenges in Combinatorial...

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