An Efficient Deflation Technique for the Communication- Avoiding Conjugate Gradient Method

Communication-avoiding Krylov subspace methods (CA-KSMs) fuse s loop iterations in order to asymptotically reduce sequential and parallel communication costs by a factor of O(s). However, the actual savings depend on the nonzero structure of the system matrix A, and these savings typically diminish as A fills, as is common when preconditioning. Recent efforts target incorporating preconditioning in a communication-avoiding manner; in this paper, we explore deflation as a type of preconditioning for CA-KSMs. We derive a deflated version of communication-avoiding conjugate gradient (CA-CG), which is mathematically equivalent to deflated CG of Saad et al. [SIAM J. Sci. Comput., 21 (2000), pp.1909–1926]. Numerical experiments on a model problem demonstrate that the communication-avoiding formulations can converge at comparable rates to the classical formulations, even for large values of s. Performance modeling illustrates that O(s) speedups are possible when performance is communication bound. These results motivate deflation as a promising preconditioner for CA-KSMs.