On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces
Antonio Falcó, Wolfgang Hackbusch, and Anthony Nouy
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Submission date: 30. Dec. 2017
published in: Foundations of computational mathematics, 19 (2019) 1, p. 159-204
DOI number (of the published article): 10.1007/s10208-018-9381-4
MSC-Numbers: 15A69, 46B28, 46A32
Keywords and phrases: tensor spaces, Banach manifolds, Tensor formats, Tensor rank
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The main goal of this paper is to extend the so-called Dirac-Frenkel Variational Principle in the framework of tensor Banach spaces. To this end we observe that a tensor product of normed spaces can be described as a union of disjoint connected components. Then we show that each of these connected components, composed by tensors in Tucker format with a fixed rank, is a Banach manifold modelled in a particular Banach space, for which we provide local charts. The description of the local charts of these manifolds is crucial for an algorithmic treatment of high-dimensional partial differential equations and minimisation problems. In order to describe the relationship between these manifolds and the natural ambient space we prove under natural conditions that each connected component can be immersed in a particular ambient Banach space. This fact allows us to finally extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.