O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 1 Parallel Solution of the 3-D Laplace Equation Using a Symmetric-Galerkin Boundary Integral Approximation Talisha Haywood Physics major w/ Emphasis in Computational Science Wofford College, Spartanburg, SC Research Alliance for Minorities Mentor: Leonard J. Gray Computer Science and Mathematics Division

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 2 Problem How do we develop a parallel solution of the 3-D Laplace Equation using a Symmetric-Galerkin Boundary Integral Approximation?

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 3 Background  Boundary element method: fundamental technique used for solving partial differential equations  Discretizes boundary integral equation to find system of equations from which the boundary values can be found  After a problem has been solved, normal flux and potential are known everywhere in domain

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 4 Steps  Investigate Laplace Equation  Model and edit the single processor algorithm  Investigate the code  ScaLapack and its procedure  Develop a parallel algorithm  Test the algorithm

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 5 Laplace Partial Differential Equation  Integral equation on domain boundary (e.g. surface of a cube)  Either potential or flux is specified everywhere on the boundary

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 6 Single Processor Algorithm  Two Integral Equations  Discretize equations then simplify to matrix form: (H[potential]=G[flux])  Rearrange H and G columns  Move known values to the right-hand side  Move unknown values to the left-hand side  Simplify rearranged matrix to (Ax=b) form  Call the routine that solves the finite system of linear equations

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 7 Code Behavior  Based on a cube divided into triangular elements  Boundary value is given, 0 or 1  Input: number of elements, number of nodes  Output: coordinates of elements and nodes  Eight subroutines - bem( ) -hyp( ) -bem_c( ) -hyp_c( ) -bem_ae( ) -hyp_ae -bem_av( ) -hyp_av( )]

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 8 Breakdown of subroutine bem( ) [all non-touching elements]  H and G is a sum of integrals (E P and E Q )  Integrals are done for each pair of elements (E P, E Q )  E P integral generates 3 elements  P1  P2  P3  E Q integral generates 3 elements  Q1  Q2  Q3  After the loop a test should be made: Do I have to do this integral?

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 9 ScaLapack  Scalable Linear Algebra Package  Parallel libraries that support PVM and MPI  Designed to solve linear algebra problems on distributed memory parallel computers

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 10 ScaLapack Procedure  ScaLapack routines help in setting up number of processes  Each process constructs its own matrix  Each processor receives all of the code  Each processor has to check whether it has to do certain integrals  Sum up matrix elements and solve the linear equation

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 11 Parallel Algorithm  Execute linear solve routine in parallel  Call PDGESV  4 basic steps to call ScaLapack routine 1.Initialize process grid 2.Data distribution 3.Call the routine 4.Release process grid

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 12 Results  Techniques developed will work for boundary integral equations other than Laplace Equation  Applications  Electrochemistry  Thermal Analysis

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 13 Summary  Investigated the Laplace Equation  Modeled single processor algorithm  Edited single processor algorithm to develop a parallel algorithm  Applied ScaLapack

O AK R IDGE N ATIONAL L ABORATORY U.S. D EPARTMENT OF E NERGY 14 Acknowledgements  This research was performed under the Research Alliance for Minorities Program administered through the Computer Science and Mathematics Division, Oak ridge National Laboratory. This program is sponsored by the Mathematical, Information, and Computational Sciences Division; Office of Advanced Scientific Computing Research; U. S. Department of Energy. Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725  I would like to thank my mentor Leonard J. Gray and Bill Shelton for introducing me to another level of mathematics and computer science