SUMMARYIn this paper, we propose a method for the numerical solution of linear systems of equations in low rank tensor format. Such systems may arise from the discretisation of PDEs in high dimensions, but our method is not limited to this type of application. We present an iterative scheme, which is based on the projection of the residual to a low dimensional subspace. The subspace is spanned by vectors in low rank tensor format which—similarly to Krylov subspace methods—stem from the subsequent (approximate) application of the given matrix to the residual. All calculations are performed in hierarchical Tucker format, which allows for applications in high dimensions. The mode size dependency is treated by a multilevel method. We present numerical examples that include high‐dimensional convection–diffusion equations.Copyright © 2012 John Wiley & Sons, Ltd.

We derive and analyse a scheme for the approximation of order d tensors A ∈ R n 1 ×···×n d in the hierarchical (H-) Tucker format, a dimension-multilevel variant of the Tucker format and strongly related to the TT (tensor train) format. For a fixed rank parameter k, the storage complexity of a tensor in H-Tucker format is O dk 3 + k d i=1 n i and we present a (heuristic) algorithm that finds an approximation to a tensor in the H-Tucker format in O dk 4 + log(d)k 2 d i=1 n i by inspection of only O dk 3 + log(d)k 2 d i=1 n i entries. Under mild assumptions, tensors in the H-Tucker format are reconstructed. For general tensors we derive error bounds that are based on the approximability of matrices (matricizations of the tensor) by few outer products of its rows and columns. The construction parallelizes with respect to the order d and we also propose an adaptive approach that aims at finding the rank parameter for a given target accuracy ε automatically.

uncertainty quantification or optimisation. In many cases, one is interested in scalar output quantities induced by the parameter-dependent solution. The output can be interpreted as a tensor living on a high-dimensional parameter space. Our aim is to adaptively construct an approximation of this tensor in a data-sparse hierarchical tensor format. Once this approximation from an offline computation is available, the evaluation of the output for any parameter value becomes a cheap online task. Moreover, the explicit tensor representation can be used to compute stochastic properties of the output in a straightforward way. The potential of this approach is illustrated by numerical examples.

In this paper we consider time-dependent electromagnetic scattering problems from conducting objects. We discretize the time-domain electric field integral equation using RungeKutta convolution quadrature in time and a Galerkin method in space. We analyze the involved operators in the Laplace domain and obtain convergence results for the fully discrete scheme. Numerical experiments indicate the sharpness of the theoretical estimates.

The hierarchical tensor format allows for the low-parametric representation of tensors even in high dimensions d. The efficiency of this representation strongly relies on an appropriate hierarchical splitting of the different directions 1, . . . , d such that the associated ranks remain sufficiently small. This splitting can be represented by a binary tree which is usually assumed to be given. In this paper, we address the question of finding an appropriate tree from a subset of tensor entries without any a priori knowledge on the tree structure. We propose an agglomerative strategy that can be combined with rank-adaptive cross approximation techniques such that tensors can be approximated in the hierarchical format in an entirely black box way. Numerical examples illustrate the potential and the limitations of our approach.