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[null,"1.","Record Nr.",null,"UNINA9910151929403321"]
[null,null,"Autore",null,"Novak Erich"]
[null,null,"Titolo",null,"Tractability of Multivariate Problems [[electronic resource] ] : Volume II: Standard Information for Functionals / / Erich Novak, Henryk Woźniakowski"]
[null,null,"Pubbl/distr/stampa",null,"Zuerich, Switzerland, : European Mathematical Society Publishing House, 2010"]
[null,null,"ISBN",null,"3-03719-584-3"]
[null,null,"Descrizione fisica",null,"1 online resource (675 pages)"]
[null,null,"Collana",null,"EMS Tracts in Mathematics (ETM) ; 12"]
[null,null,"Classificazione",null,"65-xx68-xx"]
[null,null,"Soggetti",null,"Mathematical theory of computation"]
[{"$nulls":4},"Numerical analysis"]
[{"$nulls":4},"Computer science"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,null,"Sommario/riassunto","This three-volume set is a comprehensive study of the  tractability of multivariate problems.  The present second volume deals with algorithms using  standard information consisting of function values  for the approximation of linear and selected nonlinear  functionals. An important  example is numerical multivariate integration.            The proof techniques used in volumes I and II  are quite different.  It is especially hard to establish meaningful lower error  bounds for the approximation of functionals  by using finitely many function values.  Here, the concept  of decomposable reproducing kernels is  helpful, allowing it to find matching lower and upper error bounds  for some linear functionals.  It is then possible to conclude tractability results  from such error bounds.        Tractability results even for linear functionals are very rich in  variety. There are infinite-dimensional Hilbert spaces for which  the approximation with an arbitrarily small error  of all linear functionals requires only one function  value. There are Hilbert spaces for which all  nontrivial linear functionals suffer from the curse of dimensionality.  This holds for unweighted spaces, where the role of all variables and  groups of variables is the same. For weighted spaces one can monitor the  role of all variables and groups of variables. Necessary and  sufficient conditions on the decay of the weights are given to obtain various  notions of tractability.    The text "]
[{"$nulls":4},"contains extensive  chapters on discrepancy and integration,  decomposable kernels and lower bounds,  the Smolyak/sparse grid algorithms,  lattice rules and the  CBC (component-by-component) algorithms.  This is done in various settings.  Path integration and quantum computation are also discussed.    The book is of interest for researchers working in computational  mathematics, especially in approximation of high-dimensional  problems. It is also well suited for graduate courses and  seminars.  61 open problems..."]