M2P 2023

Enabling Algebraic Multigrid for HPC industrial applications

  • Metsch, Bram (Fraunhofer SCAI)

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Parallel and high-performance computing (HPC) plays a significant role throughout industrial research and development. Large-scale HPC systems allow for a both fast and accurate numerical simulation of many physical processes relevant to engineering, like computational fluid dynamics (CFD), structural engineering, and many more. Most of these simulations involve the solution of large, sparse, linear systems of equations, often repeatedly for thousends (or even more) timesteps. As the computational models get larger and more detailed, the usage of classical methods like direct solvers is no longer feasible as they do not scale with the problem size. Algebraic multigrid methods (AMG) provide scalable (both in terms of time and memory) linear solvers for a wide class of linear systems that typically arise from finite element, finite volume or finite difference based discretization schemes. Their key idea is to build a multi-level hierarchy adapted to the linear system (setup phase), which is then used during the multigrid cycle (solution phase). While the setup phase for scalar Poisson-like problems is rather straightforward and only requires the linear system’s matrix (black-box AMG), for more challenging problems (like linear elasticity, systems with constraints etc.) the created multigrid hierarchy may be far from optimal, or the resulting linear solver may even fail. We introduce how to incorporate additional information into the AMG method. This may include physical properties, information from a non-linear iteration, or a time-stepping scheme. Hence, we do not pass just the linear system in a black-box manner to the AMG method, but rather consider the interplay between the linear solution step and the outer solver (grey-box AMG) . Especially in the industrial context, where the linear systems often do not fulfill the requirements of “textbook AMG”, this step is inevitable to obtain an overall fast and robust solver. We present various industrial applications that show the advantage of using this additional information. In addition, we show how to employ efficient parallelization techniques to obtain a fast and robust simulation process on HPC systems.