1. A Parallel Hierarchical Solver for the Poisson Equation
Seung Lee
R.Sudarshan
18th March 2003

2. Outline Introduction Single level vs. multilevel approaches
Preconditioning and adaptivity
Galerkin formulation
Solution of equations
Scope for parallelization
Implementation issues
Road Map/Conclusions

3. Introduction Problem being considered:
Discretize using piecewise bilinear hierarchical bases
Approximate solution as

4. Single Level Vs. Multilevel Approaches+  + 

5. (Galerkin) Weak Form Formally,
Leads to a multilevel system of equations
Coarse <=> coarse interactions
Coarse <=> fine interactions
Fine <=> fine interactions

6. Solution of Equations K is not banded (but sparse nonetheless)
Entries decay away from the diagonal
Solve system using preconditioned conjugate gradient method with diagonal preconditioning
Hierarchical solution also possible (solve coarsest level, next finer level, next finer level, …. ala multigrid)

7. How to Parallelize This Problem?
Hierarchical domain decomposition
Assembly of the stiffness matrix (almost embarrassingly parallel)
Need to account for boundary vertices (shown as )
Some message passing involved
Imposing Dirichlet boundary conditions
Parallel conjugate gradient
Probably use a canned solver?

8. Implementation Issues
Parallel assembly
Allocating geometry/level information among procs
Need a suitable sparse matrix data structure
PCG Solver
Need to be able to compute A x for sparse A (BLAS?)
Computing the ||Residue|| needs some communication
Synthesis of the final solution vector
Can be done in parallel
Results can be bumped up to the root node to be consolidated

9. Parallel distribution of data Solve using PCG solver
Road Map / Conclusions
Identify the problems
Parallel distribution of data
Solve using PCG solver
Retrieve the final solution