Introduction to Scientific Computing II Multigrid Miriam Mehl, Michael Bader

Multigrid – Algorithm iterate (GS) on the fine grid restrict residual to the coarse grid solve coarse grid equation for the error interpolate error to the fine grid correct fine grid solution

Multigrid – Things to Choose smoother relation step sizes coarse – fine grid transfer operators restriction interpolation processing order of grid levels

Multigrid – Convergence two grid analysis h-independent convergence for ‚good‘ components

Two Grid – Multigrid Example: 2D Poisson 5-point-stencil h two-grid analysis V-cycle 1/32 0.042 1/64 0.044 1/128 1/256 0.043 1/512 1/1024 1/2048

Multigrid – Some Rules smoother optimal smoothing not(!) optimal convergence small number of smoothing iterations!

Multigrid – Some Rules grid coarsening standard: doubling of h exceptions: anisotropic operators adaptively refined grids unstructured grids/general SLEs

Multigrid – Some Rules restriction/interpolation order of restriction + order of interpolation > order of discretisation

Multigrid – Some Rules V-cycle faster W-cycle more robust

Multigrid – Parallelisation parallel smoothing parallel restriction and interpolation parallel stopping criteria

Multigrid – Parallel Smoothing Gauss Seidel  Jacobi-like operation!!! processor 1 processor 2

Multigrid – Parallel Smoothing Gauss Seidel processor 1 processor 2

Multigrid – Parallel Smoothing Gauss Seidel processor 1 processor 2

Multigrid – Parallel Smoothing Gauss Seidel  Jacobi-like operation!!! processor 1 processor 2

Multigrid – Parallel Smoothing Gauss Seidel  different result than sequential GS!!! processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: Red-Black Gauss-Seidel processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: Red-Black Gauss-Seidel processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: Red-Black Gauss-Seidel processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: Red-Black Gauss-Seidel processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: Red-Black Gauss-Seidel processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: Red-Black Gauss-Seidel

Multigrid – Parallel Smoothing Alternatives: damped Jacobi processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: damped Jacobi processor 1 processor 2

Multigrid – Parallel Smoothing Alternatives: damped Jacobi processor 1 processor 2

Multigrid – Parallel Smoothing red-black GS: robust and fast smoothing further reading: Irad Yavneh, Multigrid smoothing factors for red-black Gauss-Seidel relaxation applied to a class of elliptic operators, SIAM Journal on Numerical Analysis, 32 (4), 1995 Jun Zhang, Acceleration of five-point red-black Gauss-Seidel in multigrid for Poisson equation, Applied Mathematics and Computation, 80(1), 1996 damped Jacobi: good smoothing

Ferienakademie, Sarntal, Sep 23 – Oct 5, 2012 Universität Erlangen-Nürnberg Technische Universität München Universität Stuttgart Ferienakademie, Sarntal, Sep 23 – Oct 5, 2012 Course 4: Scales and Scalability as Challenges for CSE (Bader, Schweitzer, Wellein) Deadline: Yesterday