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CATCH: CaratheodoryTchakaloff Subsampling
A discrete version of Tchakaloff theorem on the existence
of positive algebraic cubature formulas, that can be proved by the
wellknown Caratheodory theorem on conical
combinations of finitedimensional vectors, entails that the information
required for multivariate polynomial approximation can be suitably
compressed. The framework here is approximating a discrete
measure by another one, with the same polynomial moments up to a
certain degree, and a (much) smaller support. Extracting such
"CaratheodoryTchakaloff
points" from the support
of discrete measures by Linear or Quadratic Programming, we are able to
compress algebraic quadrature, QMC
integration, Least
Squares approximation and polynomial meshes on multivariate compact
sets and manifolds. Applications arise, for example, in the construction
of efficient sensor networks (geospatial analysis), and
in the optimization of 3D finite element and finite volume methods for CFD
simulations.
Poster
Software

comprexcub (Matlab code for the compression of bivariate
cubature formulas)
version 2.0, by F. Piazzon, A. Sommariva and M. Vianello
Papers

On the use of compressed polyhedral quadrature formulas in embedded
interface
methods
draft  Y. Sudhakar, A. Sommariva, M. Vianello and W.A. Wall
SIAM J. Sci. Comput., to appear

CaratheodoryTchakaloff
Least Squares
preprint  F. Piazzon, A. Sommariva and
M. Vianello
Extended Abstract accepted at SampTA 2017, IEEE Xplore Digital Library, in
press

Optimal polynomial meshes and CaratheodoryTchakaloff submeshes on
the sphere
P. Leopardi, A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 10 (2017), 1824

CaratheodoryTchakaloff
Subsampling
F. Piazzon, A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 10 (2017), 514

A new quasiMonte Carlo technique based on nonnegative leastsquares and
approximate Fekete points
preprint  L. Bittante, S. De Marchi and G. Elefante
Numer. Math. Theory Methods Appl. 9 (2016), 609632
 Compressed
sampling
inequalities by Tchakaloff's theorem
preprint  M. Vianello
Math. Inequal. Appl. 19 (2016), 395400
 Polynomial
Meshes: Computation and Approximation
preprint  S. De Marchi, F. Piazzon, A. Sommariva and M. Vianello
Proceedings of CMMSE 2015, 414425, ISBN 9788461722303, ISSN
23120177

Compression of multivariate discrete measures and applications
preprint  A. Sommariva and M. Vianello
Numer. Funct. Anal. Optim. 36 (2015), 11981223