Lect11 - Slide 1 / 32 Sparse Matrices and Optimized...

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Unformatted text preview: Slide 1 / 32 Sparse Matrices and Optimized Parallel Implementations ________________________________________________ CS 594 Lecture Notes 03/25/2009 Stan Tomov Innovative Computing Laboratory Computer Science Department The University of Tennessee March 25, 2009 Slide 2 / 32 Topics Projection in Scientific Computing PDEs, Numerical solution, Tools, etc. Sparse matrices, parallel implementations Iterative Methods Slide 3 / 32 Outline • Part I – Discussion • Part II – Sparse matrix computations • Part III – Reordering algorithms and parallelization Slide 4 / 32 Part I Discussion Slide 5 / 32 Orthogonalization • We can orthonormalize non-orthogonal basis. How? Other approaches : QR using Householder transformation (as in LAPACK), Cholesky, or/and SVD on normal equations ( as in homeworks 7 and 8 ) 1 2 4 8 16 32 64 128 256 512 0.5 1 1.5 2 2.5 MGS QR_it_svd QR_it_Chol/svd Vector size x 1,000 Gflop/s 1 2 4 8 16 32 64 128 256 512 20 40 60 80 100 120 140 QR_it_svd QR_it_Chol/svd Vector size x 1,000 Gflop/s Hybrid CPU-GPU (NVIDIA Quadro FX 5600) computation as in Homework #9 CPU computation AMD Opteron (tm), Processor 265 (1.8 Ghz, 1 GB cache) 128 vectors Slide 6 / 32 What if the basis is not orthonormal? • If we do not want to orthonormalize: u P u = c 1 x 1 + c 2 x 2 + . . . + c m x m (u , x 1 ) = c 1 (x 1 , x 1 )+ c 2 (x 2 , x 1 )+ . . . + c m (x m , x 1 ) . . . (u, x m ) = c 1 (x 1 , x m )+ c 2 (x 2 , x m )+ . . . + c m (x m , x m ) • These are the so called Petrov-Galerkin conditions • We saw examples of their use in * optimization, and * PDE discretization, e.g. FEM / 'Multiply' by x 1 , ..., x m to get Slide 7 / 32 What if the basis is not orthonormal? • If we do not want to orthonormalize, e.g. in FEM u P u = c 1 1 + c 2 2 + . . . + c 7 7 a( c 1 1 + c 2 2 + . . . + c 7 7 , i ) = F( i ) for i = 1, ... , 7 / 'Multiply' by 1 , ..., 7 to get a 7x7 system (Image taken from http://www.urel.feec.vutbr.cz/~raida) Two examples of basis functions i The more i overlap, the denser the resulting matrix Spectral element methods (high-order FEM) Slide 8 / 32 Stencil Computations • K. Datta, S. Kamil, S. Williams, L. Oliker, J. Shaft, K. Yelick, K....
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Lect11 - Slide 1 / 32 Sparse Matrices and Optimized...

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