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ε-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs

dc.contributor.authorDũng, Dinh
dc.contributor.authorGriebel, Michael
dc.contributor.authorHuy, Vu Nhat
dc.contributor.authorRieger, Christian
dc.date.accessioned2024-08-13T15:02:03Z
dc.date.available2024-08-13T15:02:03Z
dc.date.issued2017
dc.identifier.urihttps://hdl.handle.net/20.500.11811/11838
dc.description.abstractIn this article, we present a cost-benefit analysis of the approximation in tensor products of Hilbert spaces of Sobolev-analytic type. The Sobolev part is defined on a finite dimensional domain, whereas the analytical space is defined on an infinite dimensional domain. As main mathematical tool, we use the ε-dimension of a subset in a Hilbert space. The ε-dimension gives the lowest number of linear information that is needed to approximate an element from the set in the norm of the Hilbert space up to an accuracy ε > 0. From a practical point of view this means that we a priori fix an accuracy and ask for the amount of information to achieve this accuracy. Such an analysis usually requires sharp estimates on the cardinality of certain index sets which are in our case infinite-dimensional hyperbolic crosses. As main result, we obtain sharp bounds of the ε-dimension of the Sobolev-analytic-type function classes which depend only on the smoothness differences in the Sobolev spaces and the dimension of the finite dimensional domain where these spaces are defined. This implies in particular that, up to constants, the costs of the infinite dimensional (analytical) approximation problem is dominated by the finite-variate Sobolev approximation problem. We demonstrate this procedure with examples of functions spaces stemming from the regularity theory of parametric partial differential equations.en
dc.format.extent25
dc.language.isoeng
dc.relation.ispartofseriesINS Preprints ; 1703
dc.rightsIn Copyright
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/
dc.subjectinfinite-dimensional hyperbolic cross approximation
dc.subjectmixed Sobolev-analytic-type smoothness
dc.subjectKolmogorov n-width
dc.subjectlinear information
dc.subjectparametric elliptic PDEs
dc.subjectcollective Galerkin approximation
dc.subjectepsilon-dimension
dc.subject.ddc510 Mathematik
dc.subject.ddc518 Numerische Analysis
dc.titleε-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs
dc.typePreprint
dc.publisher.nameInstitut für Numerische Simulation (INS)
dc.publisher.locationBonn
dc.rights.accessRightsopenAccess
dc.relation.doihttps://doi.org/10.1016/j.jco.2017.12.001
ulbbn.pubtypeZweitveröffentlichung
dcterms.bibliographicCitation.urlhttps://ins.uni-bonn.de/publication/preprints


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