  The orthogonalization process is an essential building block in Krylov space
methods, which takes up a large portion of the computational time. Commonly
used methods, like the Gram-Schmidt method, consider the projection and
normalization separately and store the orthogonal base explicitly. We consider
the problem of orthogonalization and normalization as a QR decomposition
problem on which we apply known algorithms, namely CholeskyQR and TSQR. This
leads to methods that solve the orthogonlization problem with reduced
communication costs, while maintaining stability and stores the orthogonal base
in a locally orthogonal representation. Furthermore, we discuss the novel
method as a framework which allows us to combine different orthogonalization
algorithms and use the best algorithm for each part of the hardware. After the
formulation of the methods, we show their advantageous performance properties
based on a performance model that takes data transfers within compute nodes as
well as message passing between compute nodes into account. The theoretic
results are validated by numerical experiments.
