Mixed-precision for Linear Solvers in Global Geophysical Flows

Jan Ackmann, Peter D. Düben, Tim N. Palmer, Piotr K. Smolarkiewicz

Semi-implicit time-stepping schemes for atmosphere and ocean models require elliptic solvers that work efficiently on modern supercomputers. This paper reports our study of the potential computational savings when using mixed precision arithmetic in the elliptic solvers. The essential components of a representative elliptic solver are run at precision levels as low as half (16 bits), and accompanied with a detailed evaluation of the impact of reduced precision on the solver convergence and the solution quality. A detailed inquiry into reduced precision requires a model configuration that is meaningful but cheaper to run and easier to evaluate than full atmosphere/ocean models. This study is therefore conducted in the context of a novel semi-implicit shallow-water model on the sphere, purposely designed to mimic numerical intricacies of modern all-scale weather and climate (W&C) models with the numerical stability independent on celerity of all wave motions. The governing algorithm of the shallow-water model is based on the non-oscillatory MPDATA methods for geophysical flows, whereas the resulting elliptic problem employs a strongly preconditioned non-symmetric Krylov-subspace solver GCR, proven in advanced atmospheric applications. The classical longitude/latitude grid is deliberately chosen to retain the stiffness of global W&C models posed in thin spherical shells as well as to better understand the performance of reduced-precision arithmetic in the vicinity of grid singularities. Precision reduction is done on a software level, using an emulator. The reduced-precision experiments are conducted for established dynamical-core test-cases, like the Rossby-Haurwitz wave number 4 and a zonal orographic flow. The study shows that selected key components of the elliptic solver, most prominently the preconditioning, can be performed at the level of half precision.

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