In this paper our aim is to estimate integrals of the form [,"if} = J~... f(ei9)dl-£(8) where 1-£ is, in general a complex measure. We consider quadrature formulas like [n if} = L:7=1 Aj,nf(xj,n) with n distinct nodes Xj,n on the unit circle and so that [,"if} = [n{f} for any f E 'Rn (a certain subspace of Laurent polynomials with dimension n). Under appropriate assumptions on the function f we...

In many applications of ageand size-structured population models, there is an interest is obtaining good approximations of total population numbers rather than of their densities. Therefore, it is reasonable in such cases to solve numerically not the PDE model equations themselves, but rather their integral equivalents. For this purpose quadrature formulae are used in place of the integrals. Si...

Interval quadrature formulae of Gaussian type on R and R+ for exponential weight functions of the form w(x) = exp(−Q(x)), where Q is a continuous function on its domain and such that all algebraic polynomials are integrable with respect to w, are considered. For a given set of nonoverlapping intervals and an arbitrary n, the uniqueness of the n-point interval Gaussian rule is proved. The result...

An error analysis is given for convolution quadratures based on strongly A-stable RungeKutta methods, for the non-sectorial case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation depends on the classical order and stage order of the Runge-Kutta method and on the growth exponent of the Laplace transform. Numerical experiment...

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as a possible competitor to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadra-ture formulas consists in computing their L 2-discrepancy. Especially for larger dimensions, such computations are...

The expectation-maximization algorithm has been advocated recently by a number of authors for fitting generalized linear mixed models. Since the E-step typically involves analytically intractable integrals, one approach is to approximate them by Monte Carlo methods. However, in practice, the Monte Carlo sample sizes required for convergence are often prohibitive. In this paper we show how rando...

[1] Because of transverse pore-scale dispersion, a mixing layer develops along the interface between a lighter fluid (fresh water) that flows toward a well or sink above a heavier fluid (seawater) in a steady axisymmetric configuration. The Peclet number characterizing mixing, Pe = L/aT, where L is the aquifer thickness and aT is transverse dispersivity, is generally much larger than unity. The...

One of the benefits of shading with linear light sources is also one of its major challenges: generating soft shadows. The primary difficulty in this task is determining the discontinuities in the linear light source integrals that are caused by occluding objects. We demonstrate in this paper that the computed location of each discontinuity only needs to be moderately accurate, provided that th...