Lect13 - Iterative Methods in Linear Algebra (part 1) Stan...

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Unformatted text preview: Iterative Methods in Linear Algebra (part 1) Stan Tomov Innovative Computing Laboratory Computer Science Department The University of Tennessee Wednesday April 8, 2009 CS 594, 04-08-2009 CS 594, 04-08-2009 Outline Part I Discussion Motivation for iterative methods Part II Stationary Iterative Methods Part III Nonstationary Iterative Methods CS 594, 04-08-2009 Part I Discussion CS 594, 04-08-2009 Discussion Part 1: CS 594, 04-08-2009 Discussion Part 2: CS 594, 04-08-2009 Discussion Original matrix: >> load matrix.output >> S = spconvert(matrix); >> spy(S) Reordered matrix: >> load A.data >> S = spconvert(A); >> spy(S) CS 594, 04-08-2009 Discussion Original sub-matrix: >> load matrix.output >> S = spconvert(matrix); >> spy(S(1:8660),1:8660)) Reordered sub-matrix: >> load A.data >> S = spconvert(A); >> spy(S(8602:8700,8602:8700)) CS 594, 04-08-2009 Discussion Note: Columns have to be sorted CS 594, 04-08-2009 Discussion Results on torc0: Pentium III 933 MHz, 16 KB L1 and 256 KB L2 cache Original matrix: Mflop/s = 42.4568 Reordered matrix: Mflop/s = 45.3954 BCRS Mflop/s = 72.17374 Results on woodstock: Intel Xeon 5160 Woodcrest 3.0GHz, L1 32 KB, L2 4 MB Original matrix: Mflop/s = 386.8 Reordered matrix: Mflop/s = 387.6 BCRS Mflop/s = 894.2 CS 594, 04-08-2009 Homework 9 CS 594, 04-08-2009 Questions? Homework #9 PART I PETSc Many iterative solvers We will see how is projection used in iterative methods PART II hybrid CPU-GPUs computations CS 594, 04-08-2009 Hybrid CPU-GPU computations CS 594, 04-08-2009 Iterative Methods CS 594, 04-08-2009 Motivation So far we have discussed and have seen: Many engineering and physics simulations lead to sparse matrices e.g. PDE based simulations and their discretizations based on FEM/FVM finite differences Petrov-Galerkin type of conditions, etc. How to optimize performance on sparse matrix computations Some software how to solve sparse matrix problems (e.g. PETSc) The question is: Can we solve sparse matrix problems faster than using direct sparse methods? In certain cases Yes : using iterative methods CS 594, 04-08-2009 Sparse direct methods Sparse direct methods These are based on Gaussian elimination (LU) Performed in sparse format Are there limitations of the approach? Yes, they have fill-ins which lead to more memory requirements more flops being performed Fill-ins can become prohibitively high CS 594, 04-08-2009 Sparse direct methods Consider LU for the matrix below a nonzero is represented by a * 1st step of LU factorization will introduce fill-ins marked by an F CS 594, 04-08-2009 Sparse direct methods Fill-ins can be improved by reordering Remember: we talked about it in slightly different context (for speed and parallelization)...
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Lect13 - Iterative Methods in Linear Algebra (part 1) Stan...

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