A Cross-Entropy Approach to the Estimation of Generalized Linear Multilevel Models

In this article, we use the cross-entropy method for noisy optimization for fitting generalized linear multilevel models through maximum likelihood. We propose specifications of the instrumental distributions for positive and bounded parameters that improve the computational performance. We also introduce a new stopping criterion, which has the advantage of being problem-independent. In a second step we find, by means of extensive Monte Carlo experiments, the most suitable values of the input parameters of the algorithm. Finally, we compare the method to the benchmark estimation technique based on numerical integration. The cross-entropy approach turns out to be preferable from both the statistical and the computational point of view. In the last part of the article, the method is used to model the probability of firm exits in the healthcare industry in Italy. Supplemental materials are available online.     

A note on one-point quadrature formula for Daubechies scale function with partial support

This note is concerned with an efficient computation of integrals of products of a smooth function and Daubechies scale function with partial support by using a one-point quadrature rule. The error estimate is obtained. The rule is illustrated by considering an example from the literature.     

On Domains in Which Harmonic Functions Satisfy Generalized Mean Value Properties

Potential Analysis - Assume that a bounded domain Ω??N (N ≥ 2) has the property that there exists a signed measure µ with compact support in Ω such that, for every...     

关于一般Guass-Birlhoff求积公式的注(英文)

本文证明了对于关联矩阵E＝（E1，E2），其中E1满足Polya条件，在其内行不含奇序列，E2是正则的，一般Guass-Birkhoff求积公式的存在性。     

Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

正则长波方程的三次配点法

本文讨论了正则长波方程(RLW方程)的三次配点法,得到半离散格式的最优阶误差估计,同时给出基于向后Euler法的全离散格式,并证明了相应的误差估计.     

Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T_L,n(x) with x ∈ (0, L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev–Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.     

Quadrature formulas for the generalized Riemann-Stieltjes integral

In this paper we propose a technique of approximation for the generalized Riemann-Stieltjes integral and we found an analogue for Newton-Cotes formulas in the case n = 2 and n = 3. *Beneficiary of a Socrates fellowship at the Department of Mathematics, University of Study of Cagliari, Via Ospedale, n. 72, Cagliari, 09124, Italy, in the period February – July 2002.     

The Iyengar Type Inequalities with Exact Estimations and the Chebyshev Central Algorithms of Integrals

In this paper, both low order and high order extensions of the Iyengar type inequality are obtained. Such extensions are the best possible in the same sense as that of the Iyengar inequality. hzrthermore, the Chebyshev central algorithms of integrals for some function classes and some related problems are also considered and investigated.     

Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

S. A. Sauter Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland ^{Many important physical applications are governed by the waveequation. The formulation as time domain boundary integral equationsinvolves retarded potentials. For the numerical solution ofthis problem, we employ the convolution quadrature method forthe discretization in time and the Galerkin boundary elementmethod for the space discretization. We introduce a simple apriori cut-off strategy where small entries of the system matricesare replaced by zero. The threshold for the cut-off is determinedby an a priori analysis which will be developed in this paper.This analysis will also allow to estimate the effect of additionalperturbations such as panel clustering and numerical integrationon the overall discretization error. This method reduces thestorage complexity for time domain integral equations from O(M2N)to O(M2N logM), where N denotes the number of time steps andM is the dimension of the boundary element space.}