Introduction to Multigrid and its Parallelization

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1 Introduction to Multigrid and its Parallelization! Thomas D. Economon Lecture 14a May 28, 2014

2 Announcements 2 HW 1 & 2 have been returned. Any questions? Final projects are due June 11, 5 pm. If you are not doing a research project, please complete the assigned project. Poll: would you like to have 10 min. project presentations during the exam period (starting at 9 am on Wednesday, June 11)? This would be for folks completing a research project. Tom s office hours are now Fridays from 1:00-2:45 pm in the conference room inside of Durand 464.

3 Outline 3 Motivation for Multigrid Geometric Multigrid (linear) Parallelization Nonlinear Multigrid (SUmb)

4 Multigrid Introduction 4 Multigrid is a very powerful and elegant algorithm for convergence acceleration. Often used for solving elliptic PDEs, hyperbolic PDEs (computational fluid dynamics!), purely algebraic problems. We ll focus on geometric multigrid for problems with physical grids. At first, the Poisson equation will be used as a model problem (your project). Main ideas behind multigrid: Due to local stencils, typical iterative methods damp local (high frequency) errors effectively, but global (low frequency) errors are reduced more slowly. Multigrid generates effective convergence at all length scales by using a sequence of grids at various resolutions.

5 Multigrid References 5 Some very useful references (that also list many others): Briggs, A Multigrid Tutorial, SIAM, Brandt, 1984 Guide to Multigrid Development, with applications to computational fluid dynamics. Chow et al., A Survey of Parallelization Techniques for Multigrid Solvers, on class website. Mavriplis, D., On Convergence Acceleration Techniques for Unstructured Meshes, NASA/CR (ICASE Report No ).

6 Poisson Example 6 After applying 2nd-order finite differencing on a 2D, uniform grid to the Poisson }h {i, j+1} } h {i 1,j} {i, j} {i+1,j} ar 2 = f in = g {i, j 1} a h 2 ( i,j 1 + i 1,j 4 i,j + i+1,j + i,j+1 )=f i,j in initialize i,j = g i,j and do not update

7 Poisson Example 7 Test Problem with Analytic Solution: =[0, 1] [0, 1] a =1 f(x, y) = 5.0 exp(x)exp( 2.0 y) (x, y) = exp(x)exp( 2.0 y) Set known solution values on boundaries, set initial guess to zero, and solve using Gauss-Seidel smoothing: n+1 i,j = 1 4 n+1 i,j 1 + n+1 i 1,j + n i+1,j + n i,j+1 + h2 a f i,j

8 Poisson Example 8 We might reduce cell size to improve accuracy or to include more physics.

9 Poisson Example 9 But decreasing the cell size makes the problem more stiff.

10 Geometric Multigrid 10 As we decrease the cell size on our mesh, the low frequency errors are difficult to eliminate and cause slow solver convergence. These are global errors spanning the larger solution domain that are not well treated by our stencil s local influence. Key idea: use a coarse mesh representation to provide a better initial solution guess or to correct the solution on the fine mesh. Low frequency (smooth) errors on the fine mesh appear as high frequency (oscillatory) errors on the coarse mesh. Cells on coarsest level may span the majority of the solution domain. Smoothing iterations are cheaper on the smaller coarse mesh. Apply these ideas recursively to cover all length scales (frequencies).

11 Geometric Multigrid 11 Linear multigrid for our model Poisson problem (1/2). Starting with our linear system (discrete Poisson) A = f, (1) define the error in the solution to be the di erence between the solution the approximation to the solution, or and e =, (2) where e is the error vector (one value per node in the computational mesh). We can also define a residual vector r, which is a measure of how well the discretized governing equations are being satisfied by our numerical solution procedure, as r = f A. (3)

12 Geometric Multigrid 12 Linear multigrid for our model Poisson problem (2/2). Introducing the error into our original system gives A + e = f, (1) and by then introducing our definition of the residual, we recover the following expression: A e = r, (2) which relates the error in the solution to the residual. We can now compute a measure of the error on coarser mesh levels after transferring the values of the residual from the fine mesh level onto the coarse level. After calculating e on a coarse level, we can form a correction to the solution on the fine mesh as = + e, (3)

13 Geometric Multigrid 13 So then, how do we coarsen the mesh and move information up and down between levels? Coarsening with structured meshes is straightforward: remove every other node in each direction of the grid with each level (vertex-based) or combine 2 cells in each direction (cell-centered). Some definitions: Smoother: solution relaxation method that is applied on each grid level for a number of iterations (or sweeps). Gauss-Seidel in our Poisson example. Restriction operator: an operator that transfers information from a fine grid down to a coarse grid. Prolongation operator: an operator that transfers information from a coarse grid up to a fine grid.

14 Geometric Multigrid 14 Typical restriction operators Injection: copy information at coincident nodes from fine to coarse mesh. Weighted: surrounding & coincident fine mesh nodes contribute information to the coarse mesh node. Typical prolongation operators Bilinear/Trilinear interpolation: information from coincident nodes is transferred coarse to fine mesh and remaining fine nodes are interpolated. Weighted: surrounding & coincident coarse mesh nodes contribute information to the fine mesh node.

15 Geometric Multigrid 15 Downstroke and upstroke of a multigrid V-cycle. SMOOTH RESTRICT SMOOTH RESTRICT SMOOTH RESTRICT SMOOTH RESTRICT SMOOTH SMOOTH PROLONGATE SMOOTH PROLONGATE SMOOTH PROLONGATE SMOOTH PROLONGATE SMOOTH

16 Geometric Multigrid 16

17 Geometric Multigrid 17 With full coarsening, solutions on any evenly divisible grid (n = 2 m +1) can be converged to machine precision in O(10) multigrid cycles.

18 Geometric Multigrid 18

19 Geometric Multigrid 19 Often use variable number of smoothing iterations on the different levels (coarser levels are cheaper). Other cycles are possible, another typical choice is a multigrid W-cycle (see SUmb slides). Full Multigrid: use the multigrid V-cycle to converge the solution on each mesh level, starting from the coarsest moving to the finest.

20 Geometric Multigrid 20 What about unstructured meshes? Could create your own sequence of coarse meshes or use an agglomeration approach for automatic generation. From Mavriplis D., On Convergence Acceleration Techniques for Unstructured Meshes

21 Parallel Multigrid 21 For mesh-based PDE problems, a common approach to parallelization of multigrid is similar to the single grid procedure: 1. Partition fine mesh using a typical decomposition approach while first ignoring coarse levels (one partition per processor). For example, divide into blocks for simple structured meshes or use METIS for unstructured grids. Construct appropriate halo layers and communication schedules. 2. Each processor constructs their own coarse mesh levels. 3. Traverse the multigrid cycle in the same manner, one level at a time.

22 Parallel Multigrid 22 Two issues in particular to consider in parallel: coarsening and smoothers. Coarsening Can inherit coarse mesh partitions directly from fine mesh partitions, i.e., nodes/cells on partition boundaries remain on boundaries on lower levels. Usually straightforward for structured meshes. See Chow et al. for more discussion on unstructured grids. Communications required after restriction/prolongation. Smoothers Classical iterative smoothers are sequential, except for Jacobi. Recall multi-coloring concepts from previous lectures on iterative solvers. A very popular choice is red-black Gauss-Seidel (SOR) for 5-point stencils. Communications required after updating each color for the smoother.

23 Parallel Multigrid 23 {i, j+1} {i 1,j} {i, j} {i+1,j} {i, j 1} Red-black ordering for Gauss-Seidel allows for the partitioning of the domain. Alternate colors and communicate between.

24 Parallel Multigrid 24 Other issues to consider when parallelizing multigrid: Load-balancing on all levels relatively easy on structured meshes, but what about unstructured? Communication costs vs. compute time on coarse levels. On coarsest level, operator may be small and communication cost high. Operator may be dense on coarsest level. Use a direct solve? Based on this, the time spent on coarse levels in parallel MG solvers is often minimized W-cycles and full MG are often avoided. Voluntarily communicate less on the coarse/coarsest mesh levels?

Multigrid Pattern. I. Problem. II. Driving Forces. III. Solution

Multigrid Pattern. I. Problem. II. Driving Forces. III. Solution Multigrid Pattern I. Problem Problem domain is decomposed into a set of geometric grids, where each element participates in a local computation followed by data exchanges with adjacent neighbors. The grids