SuperLU: Sparse Direct Solver

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Ordering for <strong>Sparse</strong> Cholesky (1/2) Local greedy: Minimum degree (upper bound on fill-in) [Tinney/Walker `67, George/Liu `79, Liu `85, Amestoy/Davis/Duff `94, Ashcraft `95, Duff/Reid `95, et al.] i j k i j k 1 ⎛× × × × ⎞ ⎜ ⎟ 1 ⎛× × × × ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ i × ⎜ ⎜ ⎟ i × • • • ⎟ ⎜ ⎟ ⎜ ⎟ Eliminate 1 ⎜ ⎟ ⎜ ⎟ j ⎜ × ⎟ j ⎜ ⎟ ⎜ × • • • ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ k ⎝× ⎠ k ⎝× • • • ⎠ 1 i j Eliminate 1 i j k k X. Li 12
Ordering for <strong>Sparse</strong> Cholesky (2/2) Model problem: discretized system Ax = b from certain PDEs, e.g., 5-point stencil on n x n grid, N = n^2 Factorization cost: O(n^3) Nested dissection ordering gave optimal complexity in exact arithmetic [George ’73, Hoffman/Martin/Ross] X. Li 13
Page 1 and 2: SuperLU: Sparse Direct Solver X. Sh
Page 3 and 4: What is SuperLU Solve general spars
Page 5 and 6: Content of SuperLU Library LAPACK-s
Page 7 and 8: Review of Gaussian Elimination (GE)
Page 9 and 10: Supernode Exploit dense submatrices
Page 11: Numerical Pivoting Goal of pivotin
Page 15 and 16: Ordering Based on Nested Dissection
Page 17 and 18: Ordering Interface in SuperLU Libra
Page 19 and 20: Symbolic Factorization Cholesky [Ge
Page 21 and 22: How to distribute the matrices Mat
Page 23 and 24: Distributed Compressed Row Storage
Page 25 and 26: Process grid and MPI communicator E
Page 27 and 28: Scalability of SuperLU_DIST Poisso
Page 29 and 30: SuperLU_DIST Example Program Super
Page 31 and 32: EXAMPLE/pddrive.c (cont.) /* Initia
Page 33 and 34: SuperLU_DIST_2.0/FORTRAN/f_5x5.f90
Page 35 and 36: f_5x5.f90 (cont.) ! Create the dist
Page 37 and 38: Other Examples in EXAMPLE/ Pddrive1
Page 39 and 40: Accelerator (cont.) Finite element
Page 41 and 42: DDS47, Linear Elements Total eigens
Page 43: Summary Efficient implementations o

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