Adaptively restarted block Krylov subspace methods with low-synchronization skeletons

Abstract With the recent realization of exascale performance by Oak Ridge National Laboratory’s Frontier supercomputer, reducing communication in kernels like QR factorization has become even more imperative. Low-synchronization Gram-Schmidt methods, first introduced Świrydowicz et al. ( Numer. Lin. Alg. Appl. 28(2):e2343, 2020), have been shown to improve scalability Arnoldi method high-performance distributed computing. Block versions low-synchronization show further potential for speeding up algorithms, as column-batching allows maximizing cache usage with matrix-matrix operations. In this work, block variants from Carson Linear Algebra 638:150–195, 2022) are transformed into use full orthogonalization methods (BFOM) and generalized minimal residual (BGMRES). An adaptive restarting heuristic is developed handle instabilities that arise increasing condition number Krylov basis. The performance, accuracy, stability these assessed via a flexible benchmarking tool written MATLAB. modularity additionally permits inner products, global product.