Lec15-Parallel Multifrontal method for sparse systems of equations

# Lec15-Parallel Multifrontal method for sparse systems of equations

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Parallel Multifrontal Methods Information Sciences Institute, Computational Sciences Division 8 November 2006 Bob Lucas [email protected]

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Collaborators Over the Years Stanford Bob Dutton (advisor) & Tom Blank (now Microsoft) Jerry Tiemann (GE CRD) & Arthur Raefsky (now SGI) IDA John Conroy, Steve Kratzer & Aaron Naiman (now JCT) LSTC Roger Grimes & Cleve Ashcraft Many discussions with others Golub, Duff, Simon, Amestoy, etc.
Import of Multifrontal Method The linear solver is a major computational bottleneck in Mechanical Computer Aided Engineering (MCAE) Multifrontal method is used in: NASTRAN – Vibration ANSYS – Linear Analysis ABAQUS – Implicit Non-Linear LS-DYNA – Explicit Non-Linear

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Outline Review of Multifrontal Method Distributed Memory Shared Memory SIMD Petascale?
Definition of Sparsity A matrix is sparse if it has a non-zero structure that can be exploited to reduce storage and/or operations Iain Duff’s definition of sparsity

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Sparsity and Fill-in Example 9 XXXXXXXXX 1 XX******* 2 X*X****** 3 X**X***** 4 X***X**** 5 X****X*** 6 X*****X** 7 X******X* 8 X*******X O(N) factor operations 1 X X 2 X X 3 X X 4 X X 5 X X 6 X X 7 X X 8 XX 9 XXXXXXXXX O(N^3) factor operations
Toy Problem 1 2 3 7 8 9 4 5 6 1 X X X 3 XX X 2 XXX *X* 7 X XX 9 XX X 8 XXX*X* 4 X *X *XX* 5 X XXXX 6 X* X**XX do 4 k = 1, 9 do 1 i = k + 1, 9 a(i, k) = a(i,k) / a(k,k) 1 continue do 3 j = k + 1, 9 do 2 i = k + 1, 9 a(i,j) = a(i,j) – 1 a(i,k) * 2 a(k,j) 2 continue 3 continue 4 continue

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Multifrontal View of Toy 8 4 5 6 2 4 5 6 7 4 8 9 6 8 3 2 6 1 2 4 Duff and Reid, ACM TOMS 1983 1 2 3 7 8 9 4 5 6
Multifrontal Attributes Exploits fill-reducing orderings (i.e., METIS, Multisection, or MMD) Dense arithmetic kernels Matrix-matrix operations High performance for large frontal matrices Easily adapts for pivoting Dynamically form frontal matrices Allows symmetric indefinite problems Relatively small working set Good for hierarchical memories Out-of-core

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Toy Problem (with stack) Post-order traversal of Elimination Tree Minimizes working storage (Liu ’85) Limits concurrency time 1 1 3 2 7 7 2 9 Stack 2 1 3 2 7 9 4-8 8 4 5 6 2 4 5 6 7 4 8 9 6 8 3 2 6 1 2 4
Notional Control Flow do 1 sn = 1, nsn call Assemble () call Factor () call Stack () 1 continue 8 4 5 6 2 4 5 6 7 4 8 9 6 8 3 2 6 1 2 4

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In-Core Memory Trace Total Memory Factor, working Updates, stacked Updates, working Factor, complete time 1 1 1 3 3 1 1 3 9 1 3 2 1 3 2 7 1 3 2 7 9 2 2 7 2 7 7 9 2 1 3 2 7 9 8,4,5,6 1 3 2 7 9 8,4,5,6 Stack
A Real Problem : “Hood” Automotive Hood Inner Panel Springback using LS-DYNA

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“Hood” Elimination Tree Each frontal matrix’s triangle scaled by operations required to factor it.
Hood Storage Trace with Stack

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Outline Review of Multifrontal Method Distributed Memory Shared Memory SIMD Petascale?
Design Constraints Practical problems run on O(10) CPUs ANSYS runs on up to 8 Toyota runs LS-DYNA on 32 Target platform is the ubiquitous Beowulf implies MPI-1 want to minimize communication Integrate with LS-DYNA code suggests f77

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Two Sources of Concurrency Concurrency within frontal matrices Small P => column wrap
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## This note was uploaded on 02/20/2008 for the course CSE 260 taught by Professor Baden during the Fall '06 term at UCSD.

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Lec15-Parallel Multifrontal method for sparse systems of equations

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