Key words tensor, low rank, multivariate functions, linear systems, eigenvalue problems MSC (2010) 15A69, 65F10, 65F15During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors.

In tensor completion, the goal is to fill in missing entries of a partially known tensor under a low-rank constraint. We propose a new algorithm that performs Riemannian optimization techniques on the manifold of tensors of fixed multilinear rank. More specifically, a variant of the nonlinear conjugate gradient method is developed. Paying particular attention to efficient implementation, our algorithm scales linearly in the size of the tensor. Examples with synthetic data demonstrate good recovery even if the vast majority of the entries are unknown. We illustrate the use of the developed algorithm for the recovery of multidimensional images and for the approximation of multivariate functions.

We consider linear systems A(α)x(α) = b(α) depending on possibly many parameters α = (α 1 ,. .. , αp). Solving these systems simultaneously for a standard discretization of the parameter range would require a computational effort growing drastically with the number of parameters. We show that a much lower computational effort can be achieved for sufficiently smooth parameter dependencies. For this purpose, computational methods are developed that benefit from the fact that x(α) can be well approximated by a tensor of low rank. In particular, low-rank tensor variants of short-recurrence Krylov subspace methods are presented. Numerical experiments for deterministic PDEs with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach. * Supported by the SNF research module Preconditioned methods for large-scale model reduction within the SNF ProDoc Efficient Numerical Methods for Partial Differential Equations.

The numerical solution of linear systems with certain tensor product structures is considered. Such structures arise, for example, from the finite element discretization of a linear PDE on a d-dimensional hypercube. Linear systems with tensor product structure can be regarded as linear matrix equations for d = 2 and appear to be their most natural extension for d > 2. A standard Krylov subspace method applied to such a linear system suffers from the curse of dimensionality and has a computational cost that grows exponentially with d. The key to breaking the curse is to note that the solution can often be very well approximated by a vector of low tensor rank. We propose and analyse a new class of methods, so called tensor Krylov subspace methods, which exploit this fact and attain a computational cost that grows linearly with d. * Supported by the SNF research module Preconditioned methods for large-scale model reduction within the SNF ProDoc Efficient Numerical Methods for Partial Differential Equations.

Abstract-Effective information analysis generally boils down to properly identifying the structure or geometry of the data, which is often represented by a graph. In some applications, this structure may be partly determined by design constraints or predetermined sensing arrangements, like in road transportation networks for example. In general though, the data structure is not readily available and becomes pretty difficult to define. In particular, the global smoothness assumptions, that most of the existing works adopt, are often too general and unable to properly capture localized properties of data. In this paper, we go beyond this classical data model and rather propose to represent information as a sparse combination of localized functions that live on a data structure represented by a graph. Based on this model, we focus on the problem of inferring the connectivity that best explains the data samples at different vertices of a graph that is a priori unknown. We concentrate on the case where the observed data is actually the sum of heat diffusion processes, which is a quite common model for data on networks or other irregular structures. We cast a new graph learning problem and solve it with an efficient nonconvex optimization algorithm. Experiments on both synthetic and real world data finally illustrate the benefits of the proposed graph learning framework and confirm that the data structure can be efficiently learned from data observations only. We believe that our algorithm will help solving key questions in diverse application domains such as social and biological network analysis where it is crucial to unveil proper geometry for data understanding and inference.