CISM COURSE COMPUTATIONAL ACOUSTICS

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1 CISM COURSE COMPUTATIONAL ACOUSTICS Solvers Part 5: Multigrid II Ulrich Langer and Martin Neumüller Institute of Computational Mathematics Johannes Kepler University Linz Udine, May 23-27, 2016

2 Outline 1. Higher dimensions 2. Algebraic multigrid method 3. Parallelization 4. Summary CISM Course Computational Acoustics 1/8

3 Outline 1. Higher dimensions 2. Algebraic multigrid method 3. Parallelization 4. Summary CISM Course Computational Acoustics 2/8

4 Higher dimensions Changes: Smoother: all introduced smoothers possible only other optimal damping parameter Restriction/ prolongation: CISM Course Computational Acoustics 2/8

5 Outline 1. Higher dimensions 2. Algebraic multigrid method 3. Parallelization 4. Summary CISM Course Computational Acoustics 3/8

6 Algebraic multigrid method Some aspects: Construct coarse problems by using only matrix information Construction of coarse grid find coarse and fine nodes (e.g. Ruge/Stüben coarsening) Compute weights for restriction and prolongation Generate coarse system matrix Many possible algorithms and options After the construction of the coarse levels standard multigrid procedure Parallel implementation possible (e.g. software package hypre) CISM Course Computational Acoustics 3/8

7 Outline 1. Higher dimensions 2. Algebraic multigrid method 3. Parallelization 4. Summary CISM Course Computational Acoustics 4/8

8 Parallelization Some remarks: Partition mesh over the processors METIS Distributed assembling of matrices and vectors Communication over the interfaces Coarse grid solver: parallel or sequentiell on one processor? CISM Course Computational Acoustics 4/8

9 Outline 1. Higher dimensions 2. Algebraic multigrid method 3. Parallelization 4. Summary CISM Course Computational Acoustics 5/8

10 Consider: t u(t) + u(t) = f(t) t (0, T ), u(0) = u 0. Implicit Euler: (1 + τ)u k+1 = τ f(t k+1 ) + u k k = 0, 1, 2,... Big linear system: 1 + τ τ τ τ u 1 u 2. u m 1 u m τf(t 1 ) + u 0 τf(t 2 ) =.. τf(t m 1 ) τf(t m ) apply multigird idea CISM Course Computational Acoustics 5/8

11 Figure: k = 0

12 Figure: k = 1

13 Figure: k = 2

14 Figure: k = 3

15 Figure: k = 4

16 Figure: k = 5

17 Figure: k = 10

18 Figure: k = 15

19 Figure: k = 20

20 Figure: k = 25

21 Figure: k = 30

22 Figure: k = 35

23 Figure: k = 40

24 Figure: k = 45

25 Figure: k = 50

26 Figure: k = 55

27 Figure: k = 60

28 Parabolic multigrid Ω = (0, 1) 3, τ = 10 1, p t = 3, space level 4 ( elements) computed on VULCAN (IBM Blue Gene/Q), LLNL, USA Scaling results: cores time steps dof iter time fwd. sub Table: Weak scaling. cores time steps dof iter time Table: Strong scaling. CISM Course Computational Acoustics 7/8

29 Outline 1. Higher dimensions 2. Algebraic multigrid method 3. Parallelization 4. Summary CISM Course Computational Acoustics 8/8

30 Summary Multigrd method in higher dimensions Algebraic multigird method Parallelization CISM Course Computational Acoustics 8/8

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