Avoiding Communication in Nonsymmetric Lanczos-Based Krylov Subspace Methods

Krylov subspace methods are iterative methods for solving large, sparse linear systems and eigenvalue problems in a variety of scientific domains. On modern computer architectures, communication, or movement of data, takes much longer than the equivalent amount of computation. Classical formulations of Krylov subspace methods require data movement in each iteration, creating a performance bottleneck, and thus increasing runtime. This motivated $s$-step, or communication-avoiding, Krylov subspace methods, which only perform data movement every $O(s)$ iterations. We present new communication-avoiding Krylov subspace methods, CA-BICG and CA-BICGSTAB. We are the first to provide derivations of these methods. For both sequential and parallel implementations, our methods reduce data movement by a factor of $O(s)$ versus the classical algorithms. We implement various polynomial bases and perform convergence experiments to enable comparison with the classical algorithm. We discuss recent results in improving both numerical behavior and performance in communication-avoiding Krylov subspace methods.