  The paper is concerned with methods for computing the best low multilinear
rank approximation of large and sparse tensors. Krylov-type methods have been
used for this problem; here block versions are introduced. For the computation
of partial eigenvalue and singular value decompositions of matrices the
Krylov-Schur (restarted Arnoldi) method is used. We describe a generalization
of this method to tensors, for computing the best low multilinear rank
approximation of large and sparse tensors. In analogy to the matrix case, the
large tensor is only accessed in multiplications between the tensor and blocks
of vectors, thus avoiding excessive memory usage. It is proved that, if the
starting approximation is good enough, then the tensor Krylov-Schur method is
convergent. Numerical examples are given for synthetic tensors and sparse
tensors from applications, which demonstrate that for most large problems the
Krylov-Schur method converges faster and more robustly than higher order
orthogonal iteration.
