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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 obtain
compression of 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),
in optical system design (ray tracing method), and
in numerical cubature on
polygonal/polyhedral elements for the discretization of PDEs.
Posters
Software

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

Near optimal polynomial regression on polynomial meshes
draft  L. Bos, F. Piazzon and M. Vianello

Near Goptimal Tchakaloff designs
draft  L. Bos, F. Piazzon and M. Vianello

Tchakaloff polynomial meshes
draft  L. Bos and M. Vianello

Quadraturebased polynomial optimization
draft  A. Martinez, F. Piazzon, A. Sommariva and M. Vianello
Optim. Lett., to appear

Near optimal Tchakaloff meshes for compact sets
with Markov exponent 2
preprint  M. Vianello
Dolomites Res. Notes Approx. DRNA 11 (2018), 9296 (Special Issue on Norm
Levenberg's 60th birthday)

Nearly optimal nested sensors location for polynomial
regression on
complex geometries
preprint  A. Sommariva and
M. Vianello
Sampl. Theory Signal Image Process. 17 (2018), 95101

CaratheodoryTchakaloff
Least Squares
preprint  F. Piazzon, A. Sommariva and
M. Vianello
Sampling Theory and Applications 2017, IEEE Xplore Digital Library, DOI:
10.1109/SAMPTA.2017.8024337

On the use of compressed polyhedral quadrature formulas in embedded
interface
methods
preprint  Y. Sudhakar, A. Sommariva, M. Vianello and W.A. Wall
SIAM J. Sci. Comput. 39 (2017), B571B587

Optimal polynomial meshes and CaratheodoryTchakaloff submeshes on
the sphere
preprint  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