Fetching the paper…

Reading the bibliography…

2016

Optimal weighted least-squares methods

Cohen, Albert, Migliorati, Giovanni

Original
PDF
Post on X

Understand

We consider the problem of reconstructing an unknown bounded function $u$ defined on a domain $X\subset \mathbb{R}^d$ from noiseless or noisy samples of $u$ at $n$ points $(x^i)_{i=1,\dots,n}$.

We measure the reconstruction error in a norm $L^2(X,d\rho)$ for some given probability measure $d\rho$.
Given a linear space $V_m$ with ${\rm dim}(V_m)=m\leq n$, we study in general terms the weighted least-squares approximations from the spaces $V_m$ based on independent random samples.
The contribution of the present paper is twofold.

Optimal weighted least-squares methods · Around

Built on

P. Nevai, Géza Freud, orthogonal polynomials and Christoffel Functions. A case study
1986
Earlier work this paper cites.
A. Máté, P. Nevai, and V. Totik, Szegö’s extremum problem on the unit circle
1991
Earlier work this paper cites.
E.B. Saff, and V. Totik, Logarithmic Potentials with External Fields
1997
Earlier work this paper cites.
J. Tropp, User friendly tail bounds for sums of random matrices
2012
Earlier work this paper cites.

Similar

A. Cohen , M.A. Davenport, and D. Leviatan, On the stability and accuracy of least squares approximations
2013
Cited alongside, same era.
G. Chardon, A. Cohen, and L. Daudet, Sampling and reconstruction of solutions to the Helmholtz equation
2014
Cited alongside, same era.
G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, Analysis of discrete L 2 L^{2} projection on polynomial spaces with random evaluations
2014
Cited alongside, same era.
A. Chkifa, A. Cohen, G. Migliorati, F. Nobile, and R. Tempone, Discrete least squares polynomial approximation with random evaluations - application to parametric and stochastic elliptic PDEs
2015
Cited alongside, same era.
J.D. Jakeman, A. Narayan, and T. Zhou, A Christoffel function weighted least squares algorithm for collocation approximations
Cited in the paper.

Then

A. Doostan and J. Hampton, Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression
2015
Later among the works it cites.
G. Migliorati, Multivariate Markov-type and Nikolskii-type inequalities for polynomials associated with downward closed multi-index sets
2015
Later among the works it cites.
G. Migliorati, F. Nobile, and R. Tempone, Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points
2015
Later among the works it cites.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…