Aggressive Coarsening through Energy Minimization for Large Scale CFD Simulations
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Due to its high effectiveness in solving elliptic PDEs, Algebraic Multigrid (AMG) is widely used as the default preconditioner in many Computational Fluid Dynamics (CFD) simulation codes. However, its strong performance comes at the price of relatively high memory consumption and significant setup and application overheads. Reducing these costs without degrading its rapid convergence can lead to substantial speed-ups. One way to achieve this is by reducing grid and operator complexities applying a more aggressive coarsening [1]. Chronos realizes aggressive coarsening by symbolically computing the k-th power of the adjacency matrix, T , associated with the graph of strongly connected unknowns. Then, it computes a maximal independent set on the resulting matrix, T k, ensuring that coarse nodes are spaced at least at distance k. Larger k leads to more aggressive coarsening and fewer multigrid levels, but at the potential cost of reduced accuracy and slower convergence. Maintaining AMG’s excellent convergence rates under aggressive coarsening requires accurate interpolation able to incorporate the contribution of long-distance nodes [2]. To interpolate nodes at long distance, Chronos implements the dynamic-pattern least squares (DPLS) interpolation [3] which leads to very low operator complexity. However, DPLS was originally designed to handle multiple test vectors and its performance degrades on CFD problems characterized by a one-dimensional near-null space. Combining DPLS with energy minimization, however, significantly enhances its quality while preserving low operator complexity, enabling AMG to converge robustly even under aggressive coarsening. Numerical results showcasing our novel aggressive coarsening GPU implementation will be presented at the conference.
