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On the Dirac–Frenkel Variational Principle on Tensor Banach Spaces

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Abstract

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.

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Notes

  1. The condition of an open covering is not needed, see [25].

  2. It suffices to have in (3.2) the terms \(n=0\) and \(n=N.\) The derivatives are to be understood as weak derivatives.

  3. \(\mathbf {v}_{0}\) can be chosen as the best approximation of \(\mathbf { u}_{0}\) in \(\mathfrak {M}_{\mathfrak {r}}(\mathbf {V}_D)\) because a best approximation exists [11].

  4. Observe that the derivative at t of a map \(\mathbf {v}:I \rightarrow \mathfrak {M}_{(1,\ldots ,1)}(\mathbf {V})\) considered as a morphism between manifolds is given by a linear map \(\mathrm {T}_t \mathbf {v}:\mathbb {R} \rightarrow \mathbb {T}_{\mathbf {v}(t)}(\mathfrak {M}_{(1,\ldots ,1)}(\mathbf {V}))\) which is characterised by the fact that \(\mathrm {T}_t \mathbf {v}(\dot{\mu }) = \dot{\mu } \mathrm {T}_t \mathbf {v}(1)\) holds for all \(\dot{\mu } \in \mathbb {R}.\) It allows us to identify the linear map \(\mathrm {T}_t \mathbf {v}\) with the vector \(\mathrm {T}_t \mathbf {v}(1),\) that represents the derivative of the curve \(\mathbf {v}(t)\) by using local coordinates which is usually written as \(\dot{\mathbf {v}}(t)\) by abuse of notation.

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Authors and Affiliations

  1. ESI International Chair@CEU-UCH, Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad Cardenal Herrera-CEU, CEU Universities, San Bartolomé 55, 46115, Alfara del Patriarca, Valencia, Spain

    Antonio Falcó

  2. Max-Planck-Institut Mathematik in den Naturwissenschaften, Inselstr. 22, 04103, Leipzig, Germany

    Wolfgang Hackbusch

  3. Centrale Nantes, LMJL UMR CNRS 6629, 1 rue de la Noë, 44321, Nantes Cedex 3, France

    Anthony Nouy

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  1. Antonio Falcó
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  2. Wolfgang Hackbusch
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  3. Anthony Nouy
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Correspondence to Antonio Falcó.

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Communicated by Joseph M. Landsberg.

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Falcó, A., Hackbusch, W. & Nouy, A. On the Dirac–Frenkel Variational Principle on Tensor Banach Spaces. Found Comput Math 19, 159–204 (2019). https://doi.org/10.1007/s10208-018-9381-4

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  • DOI: https://doi.org/10.1007/s10208-018-9381-4

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