GS 535 : Parallel tools for scientific modeling

1. Introduction

2. First steps with MPI

2.1 Preliminaries, compilers, programming

2.2 Calculating the value of pi=3.1415... in parallel

• Description of problem/solution
• Fortran source (serial)
• Fortran source (parallel)
• Notes on using mpirun across the panoramix cluster
• Notes on timing program speed and a few suggested exercises

2.3 Using send and receive

• Simple illustration of master-slave communication (Fortran)
• Description and a few suggested exercises

3. Finite difference methods for solving partial differential equations

3.1. A one-dimensional example

A few suggestions for programming exercises:

• Write a series of programs to compare the performance of Jacobi, Red-Black Gauss-Seidel and SOR for the 1D example on grids with 3, 5, 9, 17 etc. nodal points. Do you find similar behavior as that shown in the description?
• To investigate the properties of multigrid it is beneficial to consider the two level problem first. Use in this case the differential equation with homogeneous boundary conditions and right hand side f(x)=pi*pi*sin(pi*x), which yields the analytical solution u(x)=sin(pi*x).
  Develop subroutines for smoothing, computation of the residual, restriction, interpolation and addition and test the coarse grid correction algorithm for a fine grid with, say, 65 grid points. Make sure you test this thoroughly! Do you see any improvements in convergence compared to strict Red-Black Gauss Seidel?
• Develop and test a subroutine for a V-cycle. A possible way to keep track of the solution, residual and right hand side at each level is to 'layer' the solutions within 1D vectors and have a set of pointers that indicate the start of the vector at each level. In our 1D example, coarsest level has 3 nodal points and the next level has 5 points. For this two level grid we could define an 8-element array in which we store the coarsest grid first. We can then have a 2 element pointer array with first entry 1 (to indicate that the information for the coarsest grid starts at element 1) and the second entry 4 (which points to the element at which the information for the next level starts). This can be easily extended for multiple levels.
  Test the V-cycle subroutine thoroughly for our simple problem.
• Write a program for full multigrid that uses the subroutines for V-cycle and interpolation.
• Write a driver for the tridag subroutine for the direct solution of a Poisson equation. Modify the subroutine for a symmetric matrix. How does the performance of this algorithm compare to that of the iterative methods?