1. MultiGrid

2. Project Goal
Parallelize and make Multigrid faster than before (with similar level of accuracy)
SharedArray, 4 procs

3. Motivation – Gauss Seidel
Solving PDE with Finite Difference discretization
Gauss-Seidel iteration – uses nearby points to update
Solving Ax = b, using
It kills high frequency errors really fast, but really slow afterwards

4. 1D Error
FAST (killing high-freq error)
SLOW (killing low-freq error)

5. Multigrid
Hierarchy of Grids

6. Multigrid – Going Down (To coarsest grid)
Sending Residual(Estimated error) recursively to coarser grid
Residual r = b – Ax = Au - Ax, where u is true solution for Au = b
Solve Residual equation, Obtain Error correction term solve Ax’ = r and then u = x+x’ (x’ corrects x when going up)
Restriction function

7. Multigrid – Going Up (To original grid)
At bottom level(coarsest grid), we could solve the residual equation exactly (There is only one point in coarsest grid!!)
Sending Error correction term recursively up to finer grids, until we reach initial grid
Prolongation(Linear Interpolation)

8. 1D Multigrid - Initialization
FAST (killing high-freq error)

9. 1D Error : Points we’re using : Points we’re not using
FAST (1 point – exactly solve)
Level 1

10. 1D Error : Points we’re using : Points we’re not using
FAST (1 point – exactly solve)
Level 1

11. 1D Error : Points we’re using : Points we’re not using
FAST (high-freq due to center point)
Level 2

12. 1D Error : Points we’re using : Points we’re not using
FAST (high-freq due to center point)
Level 2

13. 1D Error : Points we’re using : Points we’re not using
FAST (high-freq errors)
Level 3

14. 1D Error : Points we’re using : Points we’re not using
FAST (high-freq errors)
Level 3

15. 1D Error : Points we’re using : Points we’re not using
FAST (high-freq errors)
Level 4 (Original)

16. 1D Error : Points we’re using : Points we’re not using
FAST (high-freq errors)
Level 4 (Original)

17. Multigrid O(N) computational cost, N = number of grid points
Known as one of the fastest solvers
Multigrid.jl, AMG.jl, existing julia packages for Multigrid

18. Parallelizing 1 - Gauss-Seidel
WORKER 1
WORKER 2
WORKER 3
WORKER 4

19. Parallelizing 1 - Gauss-Seidel
WORKER 1
WORKER 2
WORKER 3
WORKER 4

20. Parallelizing 1 - Gauss-Seidel
function GS_mix!

21. Parallelizing 1 - Gauss-Seidel
GS gets almost 2 times faster than before
Gets faster and faster with large N
Even more accurate!

22. Parallelizing 2 – Restriction/Prolongation
Parallelizing Restriction and Prolongation functions
Restriction stencil :
Prolongation :
(Linear interpolation)

23. Parallelizing 2 – Restriction/Prolongation
Again, divide domain into 4 regions
Sending Residuals/Interpolate separately
Parallelized version beats normal version after level 12 grid (~ points)

24. V-Cycles and W-Cycles

25. Parallelizing - Around 1.5 times faster
Normal
V-cycle Level 10
2.4sec
Error rate 0.016
3.6sec
Error rate 0.026
W-cycle Level 12
192sec
Error rate 6.9E-5
306sec
Error rate
Error rate = norm(error)/norm(solution)