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The orthogonalization process is an essential building block in Krylov spacemethods, which takes up a large portion of the computational time. Commonlyused methods, like the Gram-Schmidt method, consider the projection andnormalization separately and store the orthogonal base explicitly. We considerthe problem of orthogonalization and normalization as a QR decompositionproblem on which we apply known algorithms, namely CholeskyQR and TSQR. Thisleads to methods that solve the orthogonlization problem with reducedcommunication costs, while maintaining stability and stores the orthogonal basein a locally orthogonal representation. Furthermore, we discuss the novelmethod as a framework which allows us to combine different orthogonalizationalgorithms and use the best algorithm for each part of the hardware. After theformulation of the methods, we show their advantageous performance propertiesbased on a performance model that takes data transfers within compute nodes aswell as message passing between compute nodes into account. The theoreticresults are validated by numerical experiments.

A Hardware-aware and Stable Orthogonalization Framework