A Parallel Hierarchical Solver for the Poisson Equation Seung Lee selee@mit.edu R.Sudarshan darshan@mit.edu 18th March 2003

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

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

Single Level Vs. Multilevel Approaches +  + 

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

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)

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?

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

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