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dc.contributor.otherUCH. Departamento de Matemáticas, Física y Ciencias Tecnológicas-
dc.creatorFalcó Montesinos, Antonio.es
dc.creatorNouy, Anthony.es
dc.creatorHackbusch, Wolfgang.es
dc.identifier.citationFalcó, A., Hackbusch, W. & Nouy, A. (2015). Geometric structures in tensor representations : final release.-
dc.descriptionEste documento de trabajo se encuentra disponible en http://arxiv.org/pdf/1505.03027v2.pdf-
dc.description.abstractThe main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components where each of them is a manifold of tensors in the tree-based format with a fixed tree-based rank. The local chart representation of these manifolds is often crucial for an algorithmic treatment of high-dimensional PDEs and minimization problems. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the bodywork of topological tensor spaces.-
dc.subjectAnálisis numérico - Documentos de trabajo.es
dc.subjectCálculo tensorial - Documentos de trabajo.es
dc.subjectAnálisis funcional - Documentos de trabajo.es
dc.subjectÁlgebra de tensores - Documentos de trabajo.es
dc.subjectTensor algebra - Working papers.es
dc.subjectEspacios generalizados - Documentos de trabajo.es
dc.subjectFunctional analysis - Working papers.es
dc.subjectFunction spaces - Working papers.es
dc.subjectCalculus of tensors - Working papers.es
dc.subjectGeneralized spaces - Working papers.es
dc.subjectEspacios funcionales - Documentos de trabajo.es
dc.subjectBanach, Espacios de - Documentos de trabajo.es
dc.subjectBanach spaces - Working papers.es
dc.subjectGeometría diferencial - Documentos de trabajo.es
dc.subjectGeometry, Differential - Working papers.es
dc.subjectNumerical analysis - Working papers.es
dc.titleGeometric structures in tensor representations : final release / Antonio Falcó, Wolfgang Hackbusch and Anthony Nouy.es
dc.typeDocumento de trabajoes
europeana.dataProviderUNIVERSIDAD SAN PABLO CEU-
europeana.providerUNIVERSIDAD SAN PABLO CEU-
Aparece en las colecciones: Dpto. Matemáticas, Física y Ciencias Tecnológicas

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