Towards an ADI iteration for tensor structured equations / Thomas Mach and Jens Saak

Abstract

Abstract: 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.