6,240 research outputs found
Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
We provide a framework for the sparse approximation of multilinear problems
and show that several problems in uncertainty quantification fit within this
framework. In these problems, the value of a multilinear map has to be
approximated using approximations of different accuracy and computational work
of the arguments of this map. We propose and analyze a generalized version of
Smolyak's algorithm, which provides sparse approximation formulas with
convergence rates that mitigate the curse of dimension that appears in
multilinear approximation problems with a large number of arguments. We apply
the general framework to response surface approximation and optimization under
uncertainty for parametric partial differential equations using kernel-based
approximation. The theoretical results are supplemented by numerical
experiments
Progress on Polynomial Identity Testing - II
We survey the area of algebraic complexity theory; with the focus being on
the problem of polynomial identity testing (PIT). We discuss the key ideas that
have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve
Some techniques on nonlinear analysis and applications
In this paper we present two different results in the context of nonlinear
analysis. The first one is essentially a nonlinear technique that, in view of
its strong generality, may be useful in different practical problems. The
second result, more technical, but also connected to the first one, is an
extension of the well-known Pietsch Domination Theorem. The last decade
witnessed the birth of different families of Pietsch Domination-type results
and some attempts of unification. Our result, that we call "full general
Pietsch Domination Theorem" is potentially a definitive Pietsch Domination
Theorem which unifies the previous versions and delimits what can be proved in
this line.The connections to the recent notion of weighted summability are
traced.Comment: 24 page
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