Quasi-Monte Carlo integration on manifolds with mapped low-discrepancy points and greedy minimal Riesz s-energy points

In this paper we consider two sets of points for Quasi-Monte Carlo integration on two- dimensional manifolds. The first is the set of mapped low-discrepancy sequence by a measure preserving map, from a rectangle U⊂R2 to the manifold. The second is the greedy minimal Riesz s-energy points extracted from a suitable discretization of the manifold. Thanks to the Poppy-seed Bagel Theorem we know that the classes of points with minimal Riesz s-energy, under suitable assumptions, are asymptotically uniformly distributed with respect to the normalized Hausdorff measure. They can then be considered as quadrature points on manifolds via the Quasi-Monte Carlo (QMC) method. On the other hand, we do not know if the greedy minimal Riesz s-energy points are a good choice to integrate functions with the QMC method on manifolds. Through theoretical considerations, by showing some properties of these points and by numerical experiments, we attempt to answer to these questions

Austrian High-Performance-Computing meeting (AHPC2020)

This booklet is a collection of abstracts presented at the AHPC conference

A new quasi-monte carlo technique based on nonnegative least squares and approximate Fekete points

The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be ”easily” addressed by means of the quasi- Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains

More properties of (β,γ)(\beta,\gamma)-Chebyshev functions and points

Recently, $(\beta,\gamma)$-Chebyshev functions, as well as the corresponding zeros, have been introduced as a generalization of classical Chebyshev polynomials of the first kind and related roots. They consist of a family of orthogonal functions on a subset of $[-1,1]$, which indeed satisfies a three-term recurrence formula. In this paper we present further properties, which are proven to comply with various results about classical orthogonal polynomials. In addition, we prove a conjecture concerning the Lebesgue constant's behavior related to the roots of $(\beta,\gamma)$-Chebyshev functions in the corresponding orthogonality interval

Polynomial mapped bases: theory and applications

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects

On (β,γ)(\beta,\gamma)-Chebyshev functions and points of the interval

In this paper, we introduce the class of $(\beta,\gamma)$-Chebyshev functions and corresponding points, which can be seen as a family of {\it generalized} Chebyshev polynomials and points. For the $(\beta,\gamma)$-Chebyshev functions, we prove that they are orthogonal in certain subintervals of $[-1,1]$ with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev-Lobatto points are instances of $(\beta,\gamma)$-Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters $\beta$ and $\gamma$

La Reine d\u2019\uc9cosse di Montchrestien, pp. 15-21; Lettres di Guez de Balzac, pp. 53-59

Presentazione in cinque capoversi dell'autore, della sua produzione, della sua poetica, dell'opera in questione e dell'estratto selezionato e corredato di note linguistiche e critiche