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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 nearoptimal regression designs (geospatial analysis),
in optical system design (ray tracing method), and
in numerical cubature on
polygonal/polyhedral & curved 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

CaTchDes (Matlab codes for CaratheodoryTchakaloff NearOptimal
Regression Designs  v1.0  from SoftwareX on GitHub)
see also the corresponding
Compute Capsule in Code Ocean
by L. Bos and M. Vianello
Project
Papers

Numerical hyperinterpolation over spherical triangles
preprint  A. Sommariva and M. Vianello
Math. Comput. Simul. 190 (2021), 1522

Nearalgebraic Tchakalofflike quadrature on spherical triangles
preprint
 A. Sommariva and M. Vianello
Appl. Math. Lett. 120 (2021)

Computing Tchakalofflike cubature rules on spline curvilinear polygons
preprint  A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 14 (2021), 111

dCATCH: a numerical package for dvariate near Goptimal Tchakaloff regression
via fast NNLS
preprint  M. Dessole, F. Marcuzzi and M. Vianello
MDPIMathematics 8(7) (2020), Special Issue "Numerical Methods"

Quadraturebased polynomial optimization
preprint  A. Martinez, F. Piazzon, A. Sommariva and M. Vianello
Optim. Lett. 14 (2020), 10271036

Accelerating the LawsonHanson NNLS solver for largescale
Tchakaloff regression designs
preprint  M. Dessole, F. Marcuzzi and M. Vianello
Dolomites Res. Notes Approx. DRNA 13 (2020), 2029

Compressed cubature over polygons with applications to optical
design
preprint  B. Bauman (LLNL, USA), A. Sommariva and M. Vianello
J. Comput. Appl. Math. 370 (2020)

Algebraic cubature on polygonal elements with a circular edge
preprint  E. Artioli, A. Sommariva and M. Vianello
Comput. Math. Appl. 79 (2020), 20572066

Near Goptimal Tchakaloff designs
preprint  L. Bos, F. Piazzon and M. Vianello
Comput. Statistics 35 (2020), 803819

CaTchDes: Matlab codes for CaratheodoryTchakaloff NearOptimal
Regression Designs
Len Bos and M. Vianello
SoftwareX 10 (2019)

Near optimal polynomial regression on polynomial meshes
preprint  L. Bos, F. Piazzon and M. Vianello
Sampling Theory and Applications 2019, IEEE Xplore Digital Library,
DOI: 10.1109/SampTA45681.2019.9030910

Tchakaloff polynomial meshes
preprint  L. Bos and M. Vianello
Ann. Polon. Math. 122 (2019), 221231

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