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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
52/2009

Variational Calculus with Sums of Elementary Tensors of Fixed Rank

Mike Espig, Wolfgang Hackbusch, Thorsten Rohwedder and Reinhold Schneider

Abstract

In this article we introduce a calculus of variations for sums of elementary tensors and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the solution of a linear system in structured tensor format. Moreover, we discuss the solution of an eigenvalue problem with sums of elementary tensors. This example can be viewed as a prototype of a constrained minimization problem. For the numerical treatment, we suggest a method which has the same order of complexity as the popular alternating least square algorithm and demonstrate the rate of convergence in numerical tests.

Received:
28.08.09
Published:
28.08.09

Related publications

inJournal
2012 Repository Open Access
Mike Espig, Wolfgang Hackbusch, Thorsten Rohwedder and Reinhold Schneider

Variational calculus with sums of elementary tensors of fixed rank

In: Numerische Mathematik, 122 (2012) 3, pp. 469-488