Hybrid Geometric-Algebraic Matrix-Free Multigrid on Spacetrees

Weinzierl, Marion

Abstract

Linear solvers are the motor for many computer simulations that are based on partial differential equations (PDEs). For a wide range of problems, multigrid solvers belong to the most efficient ones. Their convergence rate is mostly independent of the mesh size of the underlying problem discretisation, and thus they have optimal complexity.

During the last two decades, there has been a strong focus on algebraic multigrid, which can easily employ accurate unstructured grids and is very robust. But increased accuracy requirements, complex models, and huge amounts of data from engineering, scientific or, e.g., medical applications require the use of supercomputers. Their architecture forces researchers to rethink their algorithms and data handling. For example, algebraic multigrid suffers from a serious performance decrease on parallel architectures, due to high setup costs and communication overhead, unstructured data access, indirect addressing, and a large memory footprint. Therefore, the less robust geometric multigrid is now reconsidered. For the data, spacetrees have turned out to not only provide fast data access but also an efficient structure for performing the computations.

This work combines the advantages of geometric and algebraic multigrid. It defines the solver on a geometrically coarsened structured grid, but instead of geometric multigrid operations the much more robust BoxMG by Dendy using operator dependent intergrid transfer operators and Petrov-Galerkin coarse-grid operators is used. The solver is implemented on a spacetree as underlying data- and computational structure, ensuring efficient data handling, data locality, and low communication overhead. It is integrated into and parallelised in the PDE solver framework Peano, which has a small memory footprint and is very memory-efficient. This results in a robust solver that is tailored to high performance computers

Citation

Weinzierl, M. Hybrid Geometric-Algebraic Matrix-Free Multigrid on Spacetrees. (Thesis). Technische Universität München Technische Universität München. https://durham-repository.worktribe.com/output/1618042

• Thesis Type: Thesis
• Deposit Date: Dec 15, 2014
• Public URL: https://durham-repository.worktribe.com/output/1618042
• External URL: http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:91-diss-20130730-1138173-0-3
• Award Date: 2013