CATCH: Caratheodory-Tchakaloff Subsampling
A discrete version of Tchakaloff theorem on the existence
of positive algebraic cubature formulas, that can be proved by the
well-known Caratheodory theorem on conical
combinations of finite-dimensional 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
"Caratheodory-Tchakaloff
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 near-optimal 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 Caratheodory-Tchakaloff Near-Optimal
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 cubature on scattered data by
adaptive interpolation
arXiv:2307.07203 - R. Cavoretto, A. De Rossi, F. Dell'Accio,
F. Di Tommaso, N. Siar, A. Sommariva and M. Vianello
-
Qsurf: compressed QMC integration
on parametric surfaces
arXiv:2304.08090 - G. Elefante, A. Sommariva and M. Vianello
-
Compressed
QMC volume and surface integration on
union of balls
arXiv: 2303.01460 - G. Elefante, A. Sommariva and M. Vianello
-
TetraFreeQ: tetrahedra-free quadrature on polyhedral elements
preprint- A. Sommariva and M. Vianello
Appl. Numer. Math., published online 11 July 2023
-
Low-cardinality Positive Interior cubature on NURBS-shaped domains
preprint - A. Sommariva and M. Vianello
BIT Numer. Math. 63 (2023)
-
Cubature rules with positive weights on union of disks
preprint - A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 15 (2022) (Special Issue for the 60th
of S. De Marchi, 73--81)
-
CQMC: an improved code for low-dimensional Compressed
Quasi-MonteCarlo cubature
preprint - G. Elefante, A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 15 (2022)
(Special Issue "Software for Approximation 2022", 92--100)
-
Numerical hyperinterpolation over spherical triangles
preprint - A. Sommariva and M. Vianello
Math. Comput. Simul. 190 (2021), 15--22
-
Near-algebraic Tchakaloff-like quadrature on spherical triangles
preprint
- A. Sommariva and M. Vianello
Appl. Math. Lett. 120 (2021)
-
Computing Tchakaloff-like cubature rules on spline curvilinear polygons
preprint - A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 14 (2021), 1--11
-
dCATCH: a numerical package for d-variate near G-optimal Tchakaloff regression
via fast NNLS
preprint - M. Dessole, F. Marcuzzi and M. Vianello
MDPI-Mathematics 8(7) (2020), Special Issue "Numerical Methods"
-
Quadrature-based polynomial optimization
preprint - A. Martinez, F. Piazzon, A. Sommariva and M. Vianello
Optim. Lett. 14 (2020), 1027--1036
-
Accelerating the Lawson-Hanson NNLS solver for large-scale
Tchakaloff regression designs
preprint - M. Dessole, F. Marcuzzi and M. Vianello
Dolomites Res. Notes Approx. DRNA 13 (2020), 20--29
-
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), 2057--2066
-
Near G-optimal Tchakaloff designs
preprint - L. Bos, F. Piazzon and M. Vianello
Comput. Statistics 35 (2020), 803--819
-
CaTchDes: Matlab codes for Caratheodory-Tchakaloff Near-Optimal
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), 221--231
-
Near optimal Tchakaloff meshes for compact sets
with Markov exponent 2
preprint - M. Vianello
Dolomites Res. Notes Approx. DRNA 11 (2018), 92--96 (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), 95--101
-
Caratheodory-Tchakaloff
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), B571-B587
-
Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on
the sphere
preprint - P. Leopardi, A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 10 (2017), 18--24
-
Caratheodory-Tchakaloff
Subsampling
F. Piazzon, A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 10 (2017), 5--14
-
A new quasi-Monte Carlo technique based on nonnegative least-squares and
approximate Fekete points
preprint - L. Bittante, S. De Marchi and G. Elefante
Numer. Math. Theory Methods Appl. 9 (2016), 609--632
- Compressed
sampling
inequalities by Tchakaloff's theorem
preprint - M. Vianello
Math. Inequal. Appl. 19 (2016), 395--400
- Polynomial
Meshes: Computation and Approximation
preprint - S. De Marchi, F. Piazzon, A. Sommariva and M. Vianello
Proceedings of CMMSE 2015, 414--425, ISBN 978-84-617-2230-3, ISSN
2312-0177
-
Compression of multivariate discrete measures and applications
preprint - A. Sommariva and M. Vianello
Numer. Funct. Anal. Optim. 36 (2015), 1198--1223