The reduced basis method was introduced for the accurate online evaluation of solutions to
a parameter dependent family of elliptic PDEs. Abstractly, it can be viewed as determining a …

The present paper is dedicated to the application of the pivoted Cholesky decomposition to
compute low-rank approximations of dense, positive semi-definite matrices. The resulting …

In this article, we study the approximation of a probability measure μ on Rd by its empirical
measure ˆμN interpreted as a random quantization. As error criterion we consider an …

In 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 …

In this paper we propose and analyze stable variational formulations for convection diffusion
problems starting from concepts introduced by Sangalli. We derive efficient and reliable a …

We derive and analyse a scheme for the approximation of order d tensors [Formula: see text]
in the hierarchical (H-) Tucker format, a dimension-multilevel variant of the Tucker format …

We review two similar concepts of hierarchical rank of tensors (which extend the matrix rank
to higher order tensors): the TT-rank and the H-rank (hierarchical or H-Tucker rank). Based …

We analyze and characterize the possibility to represent or approximate tensors that stem
from a tensorization of vectors, matrices, or tensors by low (hierarchical) rank. Our main …

We discuss the calculus of variations in tensor representations with a special focus on tensor
networks and apply it to functionals of practical interest. The survey provides all necessary …

In this article, we describe an efficient approximation of the stochastic Galerkin matrix which
stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed …