Portable Mixed Precision Algebraic Multigrid on High Performance GPUs

Multigrid methods are algorithms for solving partial differential equations (PDE) by generating a hierarchy of successively coarser discretizations and recursively using the solution on a coarser grid to update the solution on a finer grid. The methods are attractive as they avoid the need for an expensive solver with quadratic cost on the fine discretization of the problem. Algebraic Multigrid (AMG) generalizes this concept to problems that do not originate as the discretization of a PDE but build up the hierarchy on the system matrix of the linear system. Because of its robustness and efficiency, AMG has become a central component of many scientific computing applications in academia and industry. With modern supercomputers increasingly incorporating GPU accelerators and low precision support, its popularity raises demand for redesigning to leverage the fine-grain parallelism of GPUs and employ mixed precision strategies to reduce the runtime and memory footprint.

In this dissertation, we design and implement the first open-source high-performance AMG implementation that allows users to choose the precision format used in the distinct grid levels individually while providing platform portability across GPUs from AMD, Intel, and NVIDIA. ... mehrWe note that the development of this algorithm is heavily intertwined with the development of the GINKGO open-source software library. For this reason, the dissertation also includes significant detail on how the mixed precision AMG algorithm influenced and extended the design and capabilities of the GINKGO library. We explain how we extended the scope of GINKGO from supporting NVIDIA GPUs to supporting GPUs from AMD and Intel. We discuss how the sparse matrix vector product (SpMV) is the backbone of many sparse applications and demonstrate that optimizing this kernel improves AMG performance immediately. We show that the developed high-performance AMG embraces flexibility in terms of AMG options and portability in terms of supporting different hardware platforms while remaining competitive with vendor libraries like NVIDIA's AmgX implementation. Finally, we introduce the idea of using lower precision formats for subsequent matrices to enhance the performance and memory footprint. We use experiments on real-world applications to showcase the numerical challenges that can arise and discuss problem-specific algorithmic strategies to overcome these. In performance experiments, we demonstrate that using low precision or a hierarchy of lower precision formats can reduce the overall execution time when using AMG in production for real-world problems.