Improving the Numerical Behavior of Communication-Avoiding Krylov Subspace Methods
Speaker:
Erin Carson (Charles University, Prague)
When & Where:
1 September 2022, 10:00
SR 7, Währinger Str. 29, 1090 Vienna
Abstract:
Communication-avoiding (s-step) Krylov subspace methods have the potential to reduce the communication cost of standard Krylov methods by
an asymptotic factor of s. However, though mathematically equivalent, s-step Krylov subspace methods may be numerically less stable compared to their classical counterparts in finite precision, exhibiting slower convergence and decreased attainable accuracy. This can limit the use of s-step Krylov subspace methods in practice.
After an overview of s-step Krylov subspace methods, we present two techniques which can be used to improve the numerical behavior of -step CG while maintaining its communication-avoiding properties. First, we improve convergence behavior through the use of higher precision at critical parts of the s-step iteration and second, we integrate a residual replacement strategy into the resulting mixed precision s-step CG to improve attainable accuracy. We present performance results on the Summit Supercomputer that demonstrate that when the higher precision is implemented in hardware, these techniques have virtually no overhead on the iteration time while improving both the convergence rate and the attainable accuracy of s-step CG.
