Abstract
Multigrid algorithms are among the most efficient solvers for elliptic partial differential equations. However, we have to invest into an expensive matrix setup phase before we kick off the actual solve. This assembly effort is non-negligible; particularly if the fine grid stencil integration is laborious. Our manuscript proposes to start multigrid solves with very inaccurate, geometric fine grid stencils which are then updated and improved in parallel to the actual solve. This update can be realised greedily and adaptively. We furthermore propose that any operator update propagates at most one level at a time, which ensures that multiscale information propagation does not hold back the actual solve. The increased asynchronicity, i.e. the laziness improves the runtime without a loss of stability if we make the grid update sequence take into account that multiscale operator information propagates at finite speed.
The work was funded by an EPSRC DTA PhD scholarship (award no. 1764342). It made use of the facilities of the Hamilton HPC Service of Durham University. The underlying project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 671698 (ExaHyPE).
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Murray, C.D., Weinzierl, T. (2020). Lazy Stencil Integration in Multigrid Algorithms. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2019. Lecture Notes in Computer Science(), vol 12043. Springer, Cham. https://doi.org/10.1007/978-3-030-43229-4_3
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