In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank k-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of the CURdecomposition are most suitable.For d-mode tensors T ∈ ⊗ d i=1 R n i , with d > 2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r1, . . . , r d )-approximation, which maximizes the 2 norm of the projection of T on ⊗ d i=1 Ui, where Ui is an ri-dimensional subspace R n i . One of the most common methods is the alternating maximization method (AMM). It is obtained by maximizing on one subspace Ui, while keeping all other fixed, and alternating the procedure repeatedly for i = 1, . . . , d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CUR-approximation method for tensors.The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best (r1, . . . , r d )-approximation. We compare numerically different approximation methods.2000 Mathematics Subject Classification. 14M15, 15A18, 15A69, 65H10, 65K10.

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The accurate simulation of linear electromagnetic scattering by diffraction gratings is crucial in many technologies of scientific and engineering interest. In this contribution we describe a High-Order Perturbation of Surfaces (HOPS) algorithm built upon a class of Integral Equations due to the analysis of Fokas and collaborators, now widely known as the Unified Transform Method. The unknowns in this formalism are boundary quantities (the electric field and current at the layer interface) which are an order of magnitude fewer than standard volumetric approaches such as Finite Differences and Finite Elements. With detailed numerical experiments we show the efficiency, fidelity, and high-order accuracy one can achieve with an implementation of this algorithm.