On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators

In this paper we set out to understand Filon-type quadratureof highly-oscillating integrals of the form ₀¹ f(x) e^ig(x) dx,where g is a real-valued function and >> 1. Employingad hoc analysis, as well as perturbation theory, we demonstratethat for most functions g of interest the moments behave asymptoticallyaccording to a specific model that allows for an optimal choiceof quadrature nodes. Filon-type methods that employ such quadraturenodes exhibit significantly faster decay of the error for highfrequencies . Perhaps counterintuitively, as long as optimalquadrature nodes are used, rapid oscillation leads to significantlymore precise and more affordable quadrature.     

Tensor product Gauss-Lobatto points are Fekete points for the cube

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the -dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral methods based on Gauss-Lobatto points to non-tensor-product domains, since Fekete points are known to exist and have been computed in the triangle and tetrahedron. In one dimension this result was proved by Fejér in 1932, but the extension to higher dimensions in non-trivial.

     

Multi-Dimensional Quadrature of Singular and Discontinuous Functions

In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.     

Analysis of a class of nonconforming finite elements for crystalline microstructures

An analysis is given for a class of nonconforming Lagrange-type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure. These elements were first proposed and analyzed in 1992 by Rannacher and Turek for the Stokes equation. Our analysis highlights the features of these elements which make them effective for the computation of microstructure. New results for superconvergence and numerical quadrature are also given.

     

Numerical solution of modified Black–Scholes equation pricing stock options with discrete dividend

This paper deals with the numerical solution of the modified Black–Scholes equation modelling the valuation of stock options with discrete dividend payments. By using a delta-defining sequence of the involved generalized Dirac delta function and applying the Mellin transform, an integral formula for the solution is obtained. Then, numerical quadrature approximations and illustrative examples are given.     

Gaussian quadrature and acceleration of convergence

The aim of this paper is to take up again the study done in previous papers, to the case where the integrand possesses an algebraic singularity within the interval of integration. The singularities or poles close to the interval of integration considered in this paper are only real or purely imaginary. This revised version was published online in August 2006 with corrections to the Cover Date.     

Error expansions for multidimensional trapezoidal rules with Sidi transformations

In 1993, Sidi introduced a set of trigonometric transformations x = ψ(t) that improve the effectiveness of the one-dimensional trapezoidal quadrature rule for a finite interval. In this paper, we extend Sidi's approach to product multidimensional quadrature over [0,1] ^N . We establish the Euler–Maclaurin expansion for this rule, both in the case of a regular integrand function f(x) and in the cases when f(x) has homogeneous singularities confined to vertices. This revised version was published online in August 2006 with corrections to the Cover Date.     

Sard-optimal prefilters for the Fast Wavelet Transform

A direct input of function samples into the Fast Wavelet Transform often gives inaccurate results. We use the refinement equation for the construction of prefiltering quadrature formulas which are optimal in Sard's sense, i.e.,in the standard class of functions with ‖ f ^(p)‖₂ < ∞. A detailed analysis of the error and several applications are given. This revised version was published online in August 2006 with corrections to the Cover Date.     

CMOS多相位滤波器的优化与合成

开发了一个新颖的考虑布局因素的用于射频CMOS多相位滤波器的优化和合成工具——PPFOPTIMA.在优化引擎中使用了遗传算法用来避免局部的优化解.实验结果证明PPFOPTIMA是射频CMOS多相位滤波器设计和优化中有效的计算机辅助设计工具     

On the stability of Volterra integral equations with a lagging argument

Stability properties of numerical methods for Volterra integral equations with lagging argument are considered. Some suitable definitions for the stability of the numerical methods are included, and plots of the stability regions for particular cases are included.