Parallel GMRES with Polynomial Preconditioning

Welcome to mpi-gmres-analysis! This repository contains an implementation of the Generalized Minimal Residual (GMRES) method with a polynomial preconditioner. The GMRES algorithm is parallelized using the Message Passing Interface (MPI).

Introduction

The Generalized Minimal Residual (GMRES) method is an iterative technique widely employed for solving large sparse linear systems. We have utilized it to address a 3D convection-diffusion problem $$-\Delta u + \frac{\partial u}{\partial x}=f,\quad (x,y,z)\in[0,1]^3,$$ $$u(x,y,0)=g(x,y), \quad (x,y)\in[0,1]^2,$$ $$u(x,y,z)=0, \quad \text{elsewhere on boundary},$$ where $$f(x,y,z)=z\sin(2\pi x)\sin(\pi y)+8z^3,$$ $$g(x,y)=x(1-x)y(1-y).$$ We have employed a finite difference method stencil (i.e. we implement the matrix-vector multiplication in matrix-free) and a parallel GMRES solver using MPI. To increase the convergence speed, a polynomial preconditioner in conjunction with diagonal scaling was used. The truncated Neumann series $P_n(A)$ can be used as a polynomial preconditioner which is defined as $$P_n(A)=\sum_{k=0}^n(I-A)^k \approx A^{-1}.$$

Building the Project

Prerequisites

To build and run the parallel GMRES implementation, you need to have the following dependencies installed:

1. MPI Library
2. The compiler: mpic++17

Building Steps

1. Clone the repository to your local machine:

   git clone https://github.com/eliasstenhede/mpi-gmres-analysis.git

2. Navigate to the project directory:

   cd mpi-gmres-analysis

3. Compile the source code using the provided Makefile:

   make