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OpenMG: A New Multigrid Implementation in Python

Tom S. Bertalan
The University of Alabama, Department of Chemical and Biological Engineering

Akand W. Islam
The University of Alabama, Department of Chemical and Biological Engineering

Roger B. Sidje
The University of Alabama, Department of Mathematics

Eric Carlson
The University of Alabama, Department of Chemical and Biological Engineering

Abstract

In many large-scale computations, systems of equations arise in the form , where is a linear operation to be performed on the unknown data , producing the known right-hand side, , which represents some constraint of known or assumed behavior of the system being modeled. Since such systems can be very large, solving them directly can be too slow. In contrast, a multigrid solver solves partially at full resolution, and then solves directly only at low resolution. This creates a correction vector, which is then interpolated to full resolution, where it corrects the partial solution. This project aims to create an open-source multigrid solver called OpenMG, written only in Python. The existing PyAMG multigrid implementation is a highly versatile, configurable, black-box solver, but is difficult to read and modify due to its C core. Our proposed OpenMG is a pure Python experimentation environment for testing multigrid concepts, not a production solver. By making the code simple and modular, we make the algorithmic details clear. We thereby create an opportunity for education and experimentation with the partial solver (Jacobi, Gauss Seidel, SOR, etc.), the restriction mechanism, the prolongation mechanism, and the direct solver, or the use of GPGPUs, multiple CPUs, MPI, or grid computing. The resulting solver is tested on an implicit pressure reservoir simulation problem with satisfactory results.

Keywords

python, multigrid, numpy, partial differential equations

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