ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS

The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra C2

Non-intrusive uncertainty quantification using reduced cubature rules

For the purpose of uncertainty quantification with collocation, a method is proposed for generating families of one-dimensional nested quadrature rules with positive weights and symmetric nodes. This is achieved through a reduction procedure: we start with a high-degree quadrature rule with positive weights and remove nodes while preserving symmetry and positivity. This is shown to be always possible, by a lemma depending primarily on Carathéodory's theorem. The resulting one-dimensional rules can be used within a Smolyak procedure to produce sparse multi-dimensional rules, but weight positivity is lost then. As a remedy, the reduction procedure is directly applied to multi-dimensional tensor-product cubature rules. This allows to produce a family of sparse cubature rules with positive weights, competitive with Smolyak rules. Finally the positivity constraint is relaxed to allow more flexibility in the removal of nodes. This gives a second family of sparse cubature rules, in which iteratively as many nodes as possible are removed. The new quadrature and cubature rules are applied to test problems from mathematics and fluid dynamics. Their performance is compared with that of the tensor-product and standard Clenshaw–Curtis Smolyak cubature rule

Product integration rules by the constrained mock-Chebyshev least squares operator

In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of "pathological" behavior, e.g. "nearly" singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates

Nonlinear Quantizer Design Based on Clenshaw-Curtis Quadrature

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2019.Esta tese visa propor um novo método para projeto de quantizadores não lineares conservadores de momentos estatísticos, baseado na quadratura de Clenshaw-Curtis. Os conceitos básicos de Conversores Analógico Digital são definidos para contextualização do problema discutido e para servir de base para o entendimento dos parâmetros de quantizadores. Então, uma definição formal da Transformada da Incerteza - Unscented Transform (UT) - é proposta para o contexto deste trabalho, e os conceitos básicos de quadratura são aplicados como uma ferramenta matemática para cálculo da UT. Finalmente, a metodologia de projeto do quantizador é detalhada, apresentando a relação entre os nós e pesos de uma quadratura com os parâmetros de quantizadores. O projeto é então aplicado a uma simulação de estudo de caso para verificação dos cálculos teóricos.This thesis aims to provide a novel method for designing nonlinear moment preserving quantizers based on the Clenshaw-Curtis quadrature. The basic concepts of Analog-to-Digital Converters (ADCs) are defined for contextualization of the discussed problem and to serve as a basis for understanding quantizers parameters. Then, a formal definition of the Unscented Transform (UT) is proposed for this work’s context, and the key concepts of quadrature are applied to it as a mathematical tool for UT calculation. Finally, the design method is detailed, presenting the relationship between quadrature’s nodes and weights and the quantizers parameters. This design is applied to a case study simulation, for validation of theoretical calculations

Approximation in the extended functional tensor train format

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, our approach often reduces the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)]

Marriage with Labor Supply

We propose a search-matching model of the marriage market that extends Shimer and Smith (2000) to allow for labor supply. We characterize the steady-state equilibrium when exogenous divorce is the only source of risk. The estimated matching probabilities that can be derived from the steady-state flow conditions are strongly increasing in both male and female wages. We estimate that the share of marriage surplus appropriated by the man increases with his wage and that the share appropriated by the woman decreases with her wage. We find that leisure is an inferior good for men and a normal good for women.Marriage search model, collective labor supply, structural estimation.