Jan Schneider

Error estimates for two-dimensional cross approximation

In this paper we deal with the approximation of a given function f on [0, 1]2 by special bilinear forms ∑k i=1 gi ⊗ hi via the so-called cross approximation. In particular we are interested in estimating the error function f − ∑k i=1 gi ⊗ hi of the corresponding algorithm in the maximum norm. There is a large amount of publications available that successfully deal with similar matrix algorithms in applied situations, for example in connection with H-matrices (see Boerm and Grasedyck (2003) [9] or Hackbusch (2007) [16] for many references). But as they do not give satisfactory error estimates, we concentrate on the theoretical issues of the problem in the language of functions. We connect it with related results from other areas of analysis in a historical survey and give a lot of references. Our main result is the connection of the error of our algorithm with the error of best approximation by arbitrary bilinear forms. This will be compared with the different approach in Bebendorf (2008) [6]. c

Equivalence of anchored and ANOVA spaces via interpolation

We consider weighted anchored and ANOVA spaces of functions with first order mixed derivatives bounded in L p . Recently, Hefter, Ritter and Wasilkowski established conditions on the weights in the cases p = 1 and p = ∞ which ensure equivalence of the corresponding norms uniformly in the dimension or only polynomially dependent on the dimension. We extend these results to the whole range of p ? 1 , ∞ . It is shown how this can be achieved via interpolation.

Literature survey on low rank approximation of matrices

Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic algorithms for low rank approximation. But these techniques are very expensive

Generalized cross approximation for 3d-tensors

(O(n^{3})

A note on tensor chain approximation

operations are required for

On the efficient convolution with the Newton potential

n\times n

Function spaces of varying smoothness I

matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n . In this article we review low rank approximation techniques briefly and give extensive references of many techniques.Abstract Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization, Interpolative decomposition, etc. are classical deterministic algorithms for low rank approximation. But these techniques are very expensive ( operations are required for matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n. In this article we review low rank approximation techniques briefly and give extensive references of many techniques.

Characterization of a rearrangement-invariant hull of a Besov space via interpolation

In this article we present a generalized version of the Cross Approximation for 3d-tensors. The given tensor

A note on approximation in Tensor Chain format

{a\in\mathbb{R}^{n\times n\times n}}