# Preprint No. MPIMD/11-12

#### Author(s): Thomas Mach, Jens Saak

#### Email: thomas.mach@googlemail.com

#### Date: 2011-12-19

#### Revised: 2014-07-18

#### Abstract:

We present a generalization of the alternating directions implicit (ADI)
iteration to higher dimensional problems. We solve equations of the form

( I ⊗ ... ⊗ I ⊗ A_{1} + I ⊗ ...
⊗ I ⊗ A_{2} ⊗ I + ... + A_{d} ⊗ I
⊗ ... ⊗ I ) vec(X) = vec(B),

with B given in the tensor train format. The solution X is
computed in the tensor train format, too. The accuracy of X depends
exponentially on the local rank of X and on the rank of B. To prove this
we adapt a result for right hand sides of low Kronecker rank to low
tensor train rank. Further we give a convergence proof for the generalized
ADI iteration in the single shift case and show first ideas for more
sophisticated shift strategies. The conditioning of tensor-structured
equations is investigated by generalizing results for the matrix equations
case. Finally we present first numerical results.

#### BibTeX:

@TECHREPORT{MPIMD11-12,

author = {Thomas Mach and Jens Saak},

title = {Towards an ADI iteration for Tensor Structured Equations},

number = {MPIMD/11-12},

month = dec,

year = 2011,

institution = {Max Planck Institute Magdeburg},

type = {Preprint},

note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},

}