Preprint No. MPIMD/11-12

Title: Towards an ADI iteration for Tensor Structured Equations

Author(s): Thomas Mach, Jens Saak


Date: 2011-12-19

Revised: 2014-07-18


We present a generalization of the alternating directions implicit (ADI) iteration to higher dimensional problems. We solve equations of the form
( I ⊗ ... ⊗ I ⊗ A1 + I ⊗ ... ⊗ I ⊗ A2 ⊗ I + ... + Ad ⊗ 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.


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{}},

Download MPIMD/11-12

Download old Version (2011-12-19)