numerical evaluation of highly oscillatory bessel transforms abstract in this paper we mainly focus on numerical evaluation of highly oscillatory bessel transforms based on the multiple integral the schemes for computing this class of the transform are presented by means of the bounds of special functions we give the corresponding errors of the schemes to demonstrate the efficiency and accuracy of the proposed methods some test examples are shown

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

note on the homotopy perturbation method for multivariate vector value oscillatory integrals this paper considers a homotopy perturbation method for approximating multivariate vector value highly oscillatory integrals the asymptotic formulae of the integrals and the asymptotic order of the asymptotic method are presented numerical examples show the efficiency of the approximation method

on evaluation of bessel transforms with oscillatory and algebraic singular integrands in this paper we study efficient methods for computing the integrals of the form 0 1x a 1 x bf x j v wx dx where a b v w are the given constants and w 1 j v is the bessel function of the first kind and of order v f is a sufficiently smooth function on 0 1 firstly we express the moments in a closed form with the aid of special functions secondly we induce the filon type method based on the taylor interpolation polynomial at two endpoints and the hermite interpolation polynomial at clenshaw curtis points on evaluating the highly oscillatory bessel integrals with algebraic singularity theoretical results and numerical experiments perform that the methods are very efficient in obtaining very high precision approximations if w is sufficiently large

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

asymptotic expansions of bessel anger and weber transformations in this paper we explore higher order numerical quadrature for the integrations of systems containing bessel anger and weber functions the method is constructed by finding the approximate solution of the differential equation and truncating the asymptotic series numerical examples based on theoretical results are presented to demonstrate the efficiency of the method

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

computation of integrals with oscillatory singular factors of algebraic and logarithmic type in this paper we present the clenshaw curtis filon methods and the higher order methods for computing many classes of oscillatory integrals with algebraic or logarithmic singularities at the two endpoints of the interval of integration the methods first require an interpolant of the nonoscillatory and nonsingular parts of the integrands in n 1 clenshaw curtis points then the required modified moments can be accurately and efficiently computed by constructing some recurrence relations moreover for these quadrature rules their absolute errors in inverse powers of the frequency are given the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

on set valued kalman filtering and its application to event based state estimation motivated by challenges in state estimation with event based measurement updates the properties of the exact and approximate set valued kalman filters with multiple sensor measurements for linear time invariant systems are investigated in this work first we show that the exact and the proposed approximate set valued filters are independent of the fusion sequence at each time instant second the boundedness of the size of the set of estimation means is proved for the exact set valued filter for the approximate set valued filter if the closed loop matrix is contractive then the set of estimation means has a bounded size asymptotically otherwise a nonsingular linear transform is constructed such that the size of the set of estimation means for the transformed states is asymptotically bounded third the effect of set valued measurements on the size of the set of estimation means is analyzed and conditions for performance improvement in terms of smaller size of the set of estimation means are proposed finally the results are applied to event based estimation which allow the event triggering conditions to be designed by considering requirements on performance and communication rates the efficiency of the proposed results are illustrated by simulation examples and comparison with the approximate event based mmse estimator and the kalman filter with intermittent observations

on van der corput type lemmas for bessel and airy transforms and applications abstract this paper presents van der corput type lemmas for bessel and airy transforms with shifted jacobi weight functions these results on the asymptotic orders of the highly oscillatory integrals on the frequency are optimal furthermore from these estimates the convergence rates on filon type methods are easily derived

large deviations for expanding transformations with additive white noise large deviations estimates for the autocorrelations of order kof the random process zn xn n n 0 are obtained the processes xn n 0and n n 0are independent n n 0 are i i d bounded random variables xn tn x0 n mathbb n t m mis expanding leaving invariant a gibbs measure on a compact set m and m mathbb r is a continuous function a possible application of this result is the case where mis the unit circle and the gibbs measure is the one absolutely continuous with respect to the lebesgue measure on the circle the case when tis a uniquely ergodic map was studied in carmona et al 1998 in the present paper tis an expanding map however it is possible to derive large deviations properties for the autocorrelations samples 1 n n 1j 0zjzj k but the deviation function is quite different from the uniquely ergodic case because it is necessary to take into account the entropy of invariant measures for tas an important information the method employed here is a combination of the variational principle of the thermodynamic formalism with donsker and varadhan s large deviations approach

on quadrature of highly oscillatory integrals with logarithmic singularities in this paper a quadrature rule is discussed for highly oscillatory integrals with logarithmic singularities at the same time its error depends on the frequency and the computation of its moments are given the new rule is implemented by interpolating f at chebyshev nodes and singular point where the interpolation polynomial satisfies some conditions numerical experiments conform the efficiency for obtaining the approximations

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

a sparse discretization for integral equation formulations of high frequency scattering problems we consider two dimensional scattering problems formulated as an integral equation defined on the boundary of the scattering obstacle the oscillatory nature of high frequency scattering problems necessitates a large number of unknowns in classical boundary element methods in addition the corresponding discretization matrix of the integral equation is dense we formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies the method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns but retains accuracy for lower frequencies new in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals as a result we obtain a sparse discretization matrix for the oscillatory problem moreover numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency the sparse discretization applies to problems where the phase of the solution can be predicted a priori for example in the case of smooth and convex scatterers

a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels based on the filon clenshaw curtis method for highly oscillatory integrals and together with the sommariva s result sommariva 2013 for clenshaw curtis quadrature rule we present a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels whose unknown function is assumed to be less oscillatory than the kernel in the proposed method the filon clenshaw curtis method is used to compute the involved oscillatory integrals which makes the proposed method very precise by solving only a small system of linear equations we can obtain a very satisfactory numerical solution the performance of the presented method is illustrated by several numerical examples compared with the method proposed by li et al 2012 this method enjoys a lower computational complexity furthermore numerical examples show that the presented method has a competitive advantage on the accuracy compared with the method in li et al 2012

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

the superconvergence of newton cotes rules for the hadamard finite part integral on an interval we study the general composite newton cotes rules for the computation of hadamard finite part integral with the second order singularity and focus on their pointwise superconvergence phenomenon i e when the singular point coincides with some a priori known point the convergence rate is higher than what is globally possible we show that the superconvergence rate of the composite newton cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points several numerical examples are provided to validate the theoretical analysis

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

efficient filon type methods for int_a bf x rm e rm i omega g x rm d x based on the transformation y g x some new efficient filon type methods for integration of highly oscillatory function int a bf x rm e rm i omega g x rm d x with an irregular oscillator are presented one is a moment free filon type method for the case that g x has no stationary points in a b the others are based on the filon type method or the asymptotic method together with filon type method for the case that g x has stationary points the effectiveness and accuracy are tested by numerical examples

numerical evaluation of a class of highly oscillatory integrals involving airy functions in this paper we consider the implementation of the clenshaw curtis filon type method for a class of highly oscillatory integrals 0 1 x 1 x f x ai x dx where ai x is an airy function and 1 1 by replacing f by its chebyshev interpolation polynomial at the clenshaw curtis points so that the modified moments can be computed by recursive formula based on special functions an efficient and stable method for this integral is presented error analysis for the presented method is given moreover the method shares the property that the larger the the higher the precision theoretical results and numerical examples show that the method is very efficient in obtaining very high precision approximations if is sufficiently large

letter to the editor on quadrature of bessel transformations a method for integral transformations of highly oscillatory functions bessel functions is presented it is based on the filon type method and the decay of the error can be increased as increases the effectiveness and accuracy of the quadrature is tested for both large arguments and higher orders of bessel functions in the case where the orders are nonnegative integers

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

on the numerical approximation for fourier type highly oscillatory integrals with gauss type quadrature rules in this paper we present an improved numerical steepest descent method for the approximation of fourier type highly oscillatory integrals based on the previous numerical steepest descent method the new method used the integrand information at endpoints and stationary points the asymptotic order is given that is improved both for the case of stationary points and stationary points free several numerical examples are presented which show the high efficiency of the proposed method numerical results support our theoretical analyses