performance comparison of numerical inversion methods for laplace and hankel integral transforms in engineering problems abstract different methods for the numerical evaluations of the inverse laplace and inverse of joint laplace hankel integral transforms are applied to solve a wide range of initial boundary value problems often arising in engineering and applied mathematics the aim of the paper is to present a performance comparison among different numerical methods when they are applied to transformed functions related to actual engineering problems found in the literature most of our selected test functions have been found in the solution of boundary value problems of applied mechanics such as those related to transient responses of isotropic and transversely isotropic half space to concentrated impulse or those related to viscoelastic wave motion in layered media these classes of test functions are frequently encountered in similar problems such as those in boundary element or boundary integral equations theoretical seismology soil structure interaction in time domain and so on therefore their behavior with different numerical inversion algorithms could make a useful guide to a precise choice of more suitable inversion method to be used in similar problems some different methods are also investigated in detail and compared for the inversion of the joint hankel laplace transforms where more sophisticated integrand functions are encountered it is shown that durbin crump d amore fixed talbot gaver whyn rho gwr and direct integration methods have excellent performance and produce good results when applied to the same problems on the contrary gaver stehfest and piessens methods furnish results not very reliable for almost all classes of transformed functions and they seem good only for simple transformed functions particularly the performance of gwr algorithm is very good even for transformed functions with infinite number of singularities where the other methods fail in addition in case of double integral transforms only the fixed talbot durbin and weeks methods are recommended

a parameter method for computing highly oscillatory integrals this paper presents a new efficient parameter method for integration of the highly oscillatory integral 0 1f x e i w g x dx with an irregular oscillator the effectiveness and accuracy are tested by means of numerical examples for the case where g x has stationary points

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

almost invariant sets and invariant manifolds connecting probabilistic and geometric descriptions of coherent structures in flows abstract we study the transport and mixing properties of flows in a variety of settings connecting the classical geometrical approach via invariant manifolds with a probabilistic approach via transfer operators for non divergent fluid like flows we demonstrate that eigenvectors of numerical transfer operators efficiently decompose the domain into invariant regions for dissipative chaotic flows such a decomposition into invariant regions does not exist instead the transfer operator approach detects almost invariant sets we demonstrate numerically that the boundaries of these almost invariant regions are predominantly comprised of segments of co dimension 1 invariant manifolds for a mixing periodically driven fluid like flow we show that while sets bounded by stable and unstable manifolds are almost invariant the transfer operator approach can identify almost invariant sets with smaller mass leakage thus the transport mechanism of lobe dynamics need not correspond to minimal transport the transfer operator approach is purely probabilistic it directly determines those regions that minimally mix with their surroundings the almost invariant regions are identified via eigenvectors of a transfer operator and are ranked by the corresponding eigenvalues in the order of the sets invariance or leakiness while we demonstrate that the almost invariant sets are often bounded by segments of invariant manifolds without such a ranking it is not at all clear which intersections of invariant manifolds form the major barriers to mixing furthermore in some cases invariant manifolds do not bound sets of minimal leakage our transfer operator constructions are very simple and fast to implement they require a sample of short trajectories followed by eigenvector calculations of a sparse matrix

characterizing the metastable balance between chaos and diffusion abstract we examine some new diagnostics for the behavior of a field evolving in an advective diffusive system one of these diagnostics is approximately the fourier second moment denoted as 2 and the other is the linear entropy or field intensity s the latter being significantly easier to compute or measure we establish that as a result of chaos the increasing structure in is accompanied by increasing exponentially rapidly in time at a rate given by dependent lyapunov exponents i and dominated by the largest one max noise or diffusive coarse graining of causes to decrease as 2 1 4 dt where d is a measure of the diffusion when both effects are present the competition between the processes leads to metastability for followed by a final diffusive tail the initial stages may be chaotic or diffusive depending upon the value of 1max2d 2 0 but the metastable value of 2 is given by 2 i i 2d irrespective since s 2d 2 similar analysis applies to s and in particular there exists a metastable decay rate for s given by s i i these arguments are verified for a simple case the arnol d cat map with added diffusive noise

on the implementation of discontinuous galerkin methods for volterra integral equations with highly oscillatory bessel kernels in this paper we consider implementation of discontinuous galerkin methods for weakly singular volterra integral equations of first and second kinds with highly oscillatory bessel kernels which cost the same operations independent of large values of frequencies preliminary numerical results show that the proposed method is efficient for verifying accuracy of approximate solutions

on the evaluation of infinite integrals involving bessel functions abstract in this paper we consider the numerical method for computing the infinite highly oscillatory bessel integrals of the form a f x c v x dx where c v x denotes bessel function j v x of the first kind y v x of the second kind h v 1 x and h v 2 x of the third kind f is a smooth function on a lim x f k x 0 k 0 1 2 is large and a 1 k with k 1 we construct the method based on approximating f by a combination of the shifted chebyshev polynomial so that the generalized moments can be evaluated efficiently by the truncated formula of whittaker w function the method is very efficient in obtaining very high precision approximations if is sufficiently large furthermore we give the error which depends on the endpoint a numerical examples are provided to confirm our results

parareal multiscale methods for highly oscillatory dynamical systems we introduce a new strategy for coupling the parallel in time parareal iterative methodology with multiscale integrators following the parareal framework the algorithm computes a low cost approximation of all slow variables in the system using an appropriate multiscale integrator which is refined using parallel fine scale integrations convergence is obtained using an alignment algorithm for fast phase like variables the method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables or to resolve all the state variables the numerical scheme does not require that the system is split into slow and fast coordinates moreover the dynamics may involve hidden slow variables for example due to resonances we propose an alignment algorithm for almost periodic solutions in which case convergence of the parareal iterations is proved the applicability of the method is demonstrated in numerical examples

a method for computing bessel function integrals abstract a method for numerical calculation of integrals containing bessel functions of integer or integer plus one half order is described the calculation involves first a one dimensional fourier sine or cosine transform followed by evaluation of the coefficient of the chebyshev series of the fourier transformed function in the case of the bessel function and evaluation of the legendre expansion coefficient in the case of the spherical bessel function a generalization of the method for the computation of an integral involving the bessel function of arbitrary real order v is presented as well

decay of passive scalars under the action of single scale smooth velocity fields in bounded two dimensional domains from non self similar probability distribution functions to self similar eigenmodes we examine the decay of passive scalars with small but nonzero diffusivity in bounded two dimensional 2d domains the velocity fields responsible for advection are smooth i e they have bounded gradients and of a single large scale moreover the scale of the velocity field is taken to be similar to the size of the entire domain the importance of the initial scale of variation of the scalar field with respect to that of the velocity field is strongly emphasized if these scales are comparable and the velocity field is time periodic we see the formation of a periodic scalar eigenmode the eigenmode is numerically realized by means of a deterministic 2d map on a lattice analytical justification for the eigenmode is available from theorems in the dynamo literature weakening the notion of an eigenmode to mean statistical stationarity we provide numerical evidence that the eigenmode solution also holds for aperiodic flows represented by random maps turning to the evolution of an initially small scale scalar field we demonstrate the transition from an evolving i e non self similar probability distribution function pdf to a stationary self similar pdf as the scale of variation of the scalar field progresses from being small to being comparable to that of the velocity field and of the domain furthermore the non self similar regime itself consists of two stages both stages are examined and the coupling between diffusion and the distribution of the finite time lyapunov exponents is shown to be responsible for the pdf evolution

an efficient method for evaluating the integral of a class of highly oscillatory functions highly oscillatory integrals require special techniques for their effective evaluation various studies have been conducted to find computational methods for evaluating such integrals in this paper we present an efficient numerical method to evaluate a class of generalised fourier integrals on a line or a square with integrands of the form f x e i k g x under the assumption that in the domain of integration both f and g are sufficiently smooth and that g does not have any stationary critical points numerical analysis and results are given to illustrate the effectiveness of our method for computing generalised fourier integrals

optimal by accuracy and optimal by order cubature formulae in interpolational classes in this paper the problem of optimal integration for fast oscillatory functions of two variables is solved constructively in the case where a priori information is limited the connection of this problem with the problem of optimal recovery of a function from interpolational classes is analysed using properties of majorants and minorants for these functional classes

coherent sets for nonautonomous dynamical systems we describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems in the autonomous setting such objects are variously known as almost invariant sets metastable sets persistent patterns or strange eigenmodes and have proved to be important in a variety of applications in this current work we explain how to extend existing autonomous approaches to the nonautonomous setting we call the new time dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent the new methods are illustrated via detailed examples in both discrete and continuous time

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

fast computation of a class of highly oscillatory integrals in this paper the problem of numerical evaluation of a class of highly oscillatory integrals a b f x x a b x k 1 n x k k e i x dx where 1 k 1 a k b and 1 has been discussed based on analytic continuation the integrals are transformed into the problems of integrating the forms on 0 with the integrand does not oscillate and decays exponentially fast which can be efficiently computed by using general gauss laguerre quadrature rule theoretical results and numerical experiments perform that the method is very efficient in obtaining very high precision approximations if is sufficiently large

on the convergence of the method for indefinite integration of oscillatory and singular functions we consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in keller 8 to this end we have analysed the properties of the difference operator related to the difference equation for the chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients properties of this operator were never investigated before the obtained results lead us to the conclusion that the studied method is always convergent we also give a rigorous proof of the error estimates

fast integration of rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is presented the method is efficient for integrals involving bessel functions jv rx with large oscillation parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations

on the evaluation of bessel transformations with the oscillators via asymptotic series of whittaker functions in this paper we are concerned with numerical evaluation of bessel transformations with the oscillators first we rewrite the integrals by whittaker functions then based on the asymptotic series of whittaker functions the integrals are transformed into the problems involving the complex exponential functions which can be efficiently computed by using the complex integration method at the same time we obtain the entirely different asymptotic expansions from the references in particular the asymptotic expansions can be denoted by the incomplete gamma function the error analysis shows that the decay of the error drastically improves as the frequency grows the effectiveness and accuracy of the methods are tested for large arguments of bessel functions

efficient method for the computation of oscillatory bessel transform and bessel hilbert transform abstract in this paper we study the numerical methods for the evaluation of two kinds of highly oscillatory bessel transforms firstly we rewrite both integrals as the sum of two integrals by rewriting the bessel function as a linear combination of whittaker w function we then transform one of integrals into the fourier type which can be transformed into the integrals on 0 and can be computed by some proper gaussian quadrature which take into account the asymptotic property of whittaker w function as x 0 the other can be efficiently computed based on the evaluation of special functions in addition error analysis for the presented methods is given moreover we also give an explicit formula for the integral 0 j x x d x in terms of meijer g function and then apply the method for the oscillatory bessel transforms to the computation of highly oscillatory bessel hilbert transforms theoretical results and numerical examples demonstrate the efficiency and accuracy of the proposed methods

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

approximation of highly oscillatory integrals containing special functions abstract a modified form of levin s method based on multi quadric radial basis function is presented for numerical evaluation of integrals having squared oscillatory bessel function of the first kind of order and oscillatory airy function the method converts the oscillatory integrals into a system of coupled ordinary differential equations odes and subsequently numerical solution of the coupled odes is obtained by meshless collocation procedure a multi resolution quadrature is used to tackle singularity of the proposed method theoretical error analysis of the proposed methods is performed in asymptotic sense numerical test problems are included to verify accuracy and efficiency of the new methods in the context of oscillatory airy and bessel integrals

inertial manifolds and completeness of eigenmodes for unsteady magnetic dynamos we prove the emergence of asymptotic spatial patterns in magnetic dynamos generated by unsteady fluid flows the patterns emerge because solutions of the dynamo equation converge exponentially to a time dependent inertial manifold this inertial manifold exists for general time aperiodic velocity fields under a spectral gap condition on the associated stokes operator for time periodic velocity fields we show that the inertial manifold is spanned by floquet eigenmodes that are analogous to the strange eigenmodes observed in the mixing of diffusive tracers this result gives an affirmative answer to the long standing question of completeness of floquet solutions in time periodic dynamo problems

mixing in fully chaotic flows passive scalar mixing in fully chaotic flows is usually explained in terms of lyapunov exponents i e rates of particle pair separations we present a unified review of this approach which encapsulates also other nonchaotic flows and investigate its limitations during the final stage of mixing when the scalar variance decays exponentially lyapunov exponents can fail to describe the mixing process the failure occurs when another mixing mechanism first introduced by fereday et al phys rev e 65 035301 2002 leads to a slower decay than the mechanism based on lyapunov exponents here we show that this mechanism is governed by the large scale nonuniformities of the flow which are different from the small scale stretching properties of the flow that are captured by the lyapunov exponents however during the initial stage of mixing i e the stage when most of the scalar variance decays lyapunov exponents describe well the mixing process we develop our theory for the incompressible and diffusive baker map a simple example of a chaotic flow nevertheless our results should be applicable to all chaotic flows

communicability in temporal networks a first principles approach to quantify the communicability between pairs of nodes in temporal networks is proposed it corresponds to the imaginary time propagator of a quantum random walk in the temporal network which accounts for unique structural and temporal characteristics of both streaming and nonstreaming temporal networks the influence of the system s temperature on the perdurability of information and how the communicability identifies patterns of communication hidden in the temporal and topological structure of the networks are also studied for synthetic and real world systems

asymptotic expansions and fast computation of oscillatory hilbert transforms in this paper we study the asymptotics and fast computation of the one sided oscillatory hilbert transforms of the form begin aligned h f t e i omega t x int nolimits 0 infty e i omega t frac f t t x dt quad omega 0 quad x ge 0 end aligned where the bar indicates the cauchy principal value and f is a real valued function with analytic continuation in the first quadrant except possibly a branch point of algebraic type at the origin when x 0 the integral is interpreted as a hadamard finite part integral provided it is divergent asymptotic expansions in inverse powers of omega are derived for each fixed x ge 0 which clarify the large omega behavior of this transform we then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms depending on the position of x we classify our discussion into three regimes namely x mathcal o 1 or x gg 1 0 x ll 1 and x 0 numerical experiments show that the convergence of the proposed methods greatly improve when the frequency omega increases some extensions to oscillatory hilbert transforms with bessel oscillators are briefly discussed as well

arity hierarchy for temporal logics a major result concerning temporal logics is kamp s theorem which states that the pair of modalities until and since is expressively complete for the first order fragment of the monadic logic over the linear time canonical model of naturals the paper concerns the expressive power of temporal logics over trees the main result states that in contrast to kamp s theorem for every n there is a modality of arity n definable by a monadic logic formula which is not equivalent over trees to any temporal logic formula which uses modalities of arity less than n its proof takes advantage of an instance of shelah s composition theorem this result has interesting corollaries for instance reproving that ctl and ectl have no finite basis

method for numerical integration of rapidly oscillating functions in diffraction theory the quadrature of general highly oscillatory integrals is a relatively complicated computational problem that occurs in a wide range of practical applications e g in physics chemistry and image analysis it is often necessary to use a high number of nodal points with classical quadrature formulas in order to achieve a required accuracy of numerical integration of rapidly oscillating functions however an increase in integration points leads to a larger computational time our work describes and analyses a method for numerical integration of rapidly oscillating functions which enables to obtain the required accuracy with a smaller number of nodal points than classical integration rules and with a relatively simple integration formula the proposed method is generally suitable for integration of rapidly oscillating functions in various applications the method is demonstrated in examples of calculation of the diffraction integral in optics where the integrand is often a rapidly oscillatory function the results can be readily adapted to similar problems from other fields copyright 2009 john wiley sons ltd

laplace transforms for approximation of highly oscillatory volterra integral equations of the first kind abstract this paper focuses on laplace and inverse laplace transforms for approximation of volterra integral equations of the first kind with highly oscillatory bessel kernels where the explicit formulae for the solution of the first kind integral equations are derived from which the integral equations can also be efficiently calculated by the clenshaw curtis filon type methods furthermore by applying the asymptotics of the solution some simpler formulas for approximating the solution for large values of the parameters are deduced preliminary numerical results are presented based on the approximate formulae and the explicit formulae which are compared with the convolution quadrature and numerical inverse laplace transform methods all these methods share that the costs the same independent of large values of frequencies

the construction of cubature rules for multivariate highly oscillatory integrals we present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points a general method is presented to identify these points and to compute the weights of the corresponding rule the accuracy of the constructed rules increases with increasing frequency of the integrand for a fixed frequency the accuracy can be improved by incorporating more derivatives of the integrand the results are illustrated numerically for fourier integrals on a circle and on the unit sphere and for more general oscillators on a rectangular domain

on precise integration method the numerical integration for the ordinary differential equations is extremely important for applications so far almost all the numerical integration methods apply the finite difference approximation even for a time invariant system the precise integration method pim solves the time step integration for the time invariant system first for such a problem precise integration gives a highly precise numerical result which approaches the full computer precision after the integration of time invariant system is solved various approximate methods can be applied to other problems such as time variant or nonlinear time integration the numerical integration of two point boundary value problems tpbvp is also very important in applications such as wave propagation optimal control structural mechanics electro magnetic wave guide problems etc the pim can also be applied to solve the tpbvp the tpbvp induced initial value problems such as the lyapunov differential equation the riccati differential equation and the kalman bucy filter equation etc can also be solved along the same way

algorithm for the computation of bessel function integrals abstract a fortran subroutine is given for the computation of integrals of the form c 0 f x j v x d x where v 0 1 10

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

asymptotic expansion and quadrature of composite highly oscillatory integrals we consider in this paper asymptotic and numerical aspects of highly oscillatory integrals of the form b a f x g sin x dx where w 1 such integrals occur in the simulation of electronic circuits but they are also of independent mathematical interest the integral is expanded in asymptotic series in inverse powers of this expansion clarifies the behaviour for large w and also provides a powerful means to design effective computational algorithms in particular we introduce and analyse filon type methods for this integral

computation of oscillatory integrals with an exponential kernel and jacobi type singularities abstract in this communication two rules are considered for computing oscillatory integrals with an exponential kernel and jacobi type singularities an asymptotic rule is presented with the aid of integration by parts an asymptotic filon type quadrature rule is constructed on the basis of a two point taylor interpolation polynomial error analysis on the proposed rules in inverse powers of large frequency is provided numerical examples are provided to illustrate that the two rules are very efficient in obtaining very high precision approximations if is sufficiently large

letter to the editor on the filon and levin methods for highly oscillatory integral this paper shows that for any suitably smooth function f x and arbitrarily selected interpolation nodes c1 c2 cv in a b the filon method and the levin method for with the polynomial interpolation approach are identical when g x is a linear function based on this result a new efficient levin quadrature for is presented

on the implementation of the asymptotic filon type method for infinite integrals with oscillatory bessel kernels in this paper we consider the implementation of the asymptotic filon type method for the semi infinite highly oscillatory bessel integrals of the form 1 f x c v wx dx where c v wx denotes bessel function j v wx of the first kind y v wx of the second kind h v 1 wx and h v 2 wx of the third kind and modified bessel function k v wx of the second kind respectively f is a smooth function on 1 lim x f k x 0 k 0 1 2 and w is large by approximating f by a linear combination of negative integer powers so that the moments can be expressed by some special functions we complete the implementation of the method furthermore we give the error analysis of the method for computing the integrals the method is very efficient in obtaining very high precision approximations if w is sufficiently large numerical examples are provided to confirm our analysis

novel perceptually uniform chromatic space chromatically perceptive observers are endowed with a sense of similarity between colors for example two shades of green that are only slightly discriminable are perceived as similar whereas oth

on the use of the pulsed convection approach for modelling advection diffusion in chaotic flows a prototypical example and direct numerical simulations abstract this article addresses the application of pulsed system models in which the advection operator is decoupled from the diffusion operator for investigating the physics of dispersion homogenization in deterministic chaotic flows the analysis is organized along to main directions i the development of a simplified time continuous model which can be viewed as a generalization in a time continuous frame of the baker s transformation and which is amenable to analytical investigation and ii the comparison of the results deriving from several typical pulsed system models with the direct numerical simulation of the advection diffusion equation both these approaches reveal the intrinsic ambiguity of the pulsed system approach in describing advection diffusion problems

on the levin iterative method for oscillatory integrals this paper considers and gives error analysis for levin iteration method to approximate bessel trigonometric transformation i f email protected a bf x cos r 1x j m rx dx for generalized fourier transformation i f email protected a bf x e i w g x dx under the condition that g x 0 for all email protected a b levin iteration method with the initial u 0 x 0 is identical to the asymptotic method

on generalized quadrature rules for fast oscillatory integrals abstract in this paper we present numerical analysis for the generalized quadrature rule for a b f x w r x d x where w r x is an oscillatory function and derive higher order generalized quadrature rules the results show that the generalized quadrature rules are efficient for bessel trigonometric transformations and the accuracy increases when oscillation becomes faster

fast integration of highly oscillatory integrals with exotic oscillators in this paper we present an efficient filon type method for the integration of systems containing bessel functions with exotic oscillators based on a diffeomorphism transformation and give applications to airy transforms preliminary numerical results show the effectiveness and accuracy of the quadrature for large arguments of integral systems

numerical solutions to volterra integral equations of the second kind with oscillatory trigonometric kernels this paper explores quadrature methods for weakly singular volterra integral equation of the second kind with highly oscillatory trigonometric kernels and presents piecewise constant and linear collocation methods for approximation of the solution the evaluation of the highly oscillatory kernels of integral equation will give rise to the computation of oscillatory integrals which can be done by using efficient filon type methods moreover by using asymptotic analysis of the solution the convergence rates for piecewise constant and linear collocation methods which show the higher the frequency the more accurate the numerical results the effectiveness and accuracy of the proposed method are tested by numerical examples

mixing enhancement by optimal flow advection we consider the problem of optimal mixing control our objective is to best enhance mixing by flow advection while the flow is optimized in the sense that it is almost steady and is of the least magnitude and the least rotation for this we define a mixing efficiency functional by penalizing the average of variance of a diffusive scalar the average of the flow and the average of its acceleration and strain tensor by variational principles we prove the existence of an optimal flow and derive optimality conditions that consist of a system of nonlinear advection diffusion equations wave equations and laplace s equation

the linear kdv equation with an interface the interface problem for the linear korteweg de vries kdv equation in one dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain the location of the interface is known and a number of compatibility conditions at the boundary are imposed we provide an explicit characterization of sufficient interface conditions for the construction of a solution using fokas s unified transform method the problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives

derivation of human chromatic discrimination ability from an information theoretical notion of distance in color space the accuracy with which humans detect chromatic differences varies throughout color space for example we are far more precise when discriminating two similar orange stimuli than two similar green stimuli in order for two colors to be perceived as different the neurons representing chromatic information must respond differently and the difference must be larger than the trial to trial variability of the response to each separate color photoreceptors constitute the first stage in the processing of color information many more stages are required before humans can consciously report whether two stimuli are perceived as chromatically distinguishable therefore although photoreceptor absorption curves are expected to influence the accuracy of conscious discriminability there is no reason to believe that they should suffice to explain it here we develop information theoretical tools based on the fisher metric that demonstrate that photoreceptor absorption properties explain about 87 of the variance of human color discrimination ability as tested by previous behavioral experiments in the context of this theory the bottleneck in chromatic information processing is determined by photoreceptor absorption characteristics subsequent encoding stages modify only marginally the chromatic discriminability at the photoreceptor level

efficient evaluation of oscillatory bessel hilbert transforms in this paper we study the asymptotics and evaluation of the oscillatory bessel hilbert transform 0 f x email protected n wx dx with 0 t the singularity of the hilbert transform is transferred to an individual oscillatory integral independent of f x for this singular integral we present two methods one is the combination of a filon type method and a complex integration method the other is the combination of a filon type method and an adaptive clenshaw curtis quadrature the remaining integral which is nonsingular can be well calculated with a combination of a filon type method and a gauss laguerre quadrature the efficiency and accuracy of the proposed methods are illustrated by numerical examples

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

numerical approximations for highly oscillatory bessel transforms and applications abstract this paper presents an efficient numerical method for approximating highly oscillatory bessel transforms based on analytic continuation we transform the integrals into the problems of integrating the forms on 0 with the integrand that does not oscillate and decays exponentially fast which can be efficiently computed by using gauss laguerre quadrature rule we then derive the error of the method depending on the frequency and the node number moreover we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations preliminary numerical results show the efficiency and accuracy of numerical approximations

choice of specific interpolation points in quadrature formula for fast oscillating integrals integrals containing trigonometric functions with non linear argument having a large multiplier are treated with the help of a specific choice of interpolation points this allows excluding the contribution of internal points of interpolation and determining upper and low bounds of the considered integral

analysis of a collocation method for integrating rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is analyzed the method is efficient for integrals involving bessel functions j rx with a large oscillation frequency parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations the analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases ko words oscillatory integrals collocation analysis

footprints of lagrangian flow structures in eulerian concentration distributions in periodic mixing flows abstract transport of passive tracers may be described through the spatio temporal evolution of eulerian concentration distributions or via the geometrical composition of the lagrangian flow structure the present study seeks to deepen insight into the connection between the eulerian and lagrangian perspectives by investigating the role of lagrangian coherent structures lcss in the eulerian concentration distributions in time periodic and spatially periodic mixing flows representation of the eulerian transport by the mapping method describing realistic transport problems by distribution matrices admits a generic analysis based on matrix and graph theory this reveals that lcss and the transport barriers that separate them leave a distinct footprint in the eigenmode spectrum of the distribution matrix and by proxy of the underlying eulerian transport operator transport barriers impart a block diagonal structure upon the mapping matrix where each block matrix a corresponds with a given lcs its kind is reflected in the spectrum of a higher order periodicity yields a distinct permutation within a the composition of the distribution matrix versus the lagrangian flow structure thus predicted is demonstrated by way of examples these findings increase fundamental understanding of transport phenomena and have great practical potential for e g flow and mixing control

boundary feedback stabilization of homogeneous equilibria in unstable fluid mixtures we consider the problem of boundary feedback stabilization of homogeneous equilibria in unstable fluid mixtures that are governed by unstable linear reaction convection diffusion equations we extend boundary feedback control laws designed for the one dimensional reaction diffusion equation using the backstepping method to this higher dimensional case we show that under certain mathematical conditions on the velocity field boundary feedback controls similar to the ones for one dimensional equations also works for the higher dimensional case and exponentially stabilize the homogeneous equilibrium zero at any given decay rate

strange eigenmodes and decay of variance in the mixing of diffusive tracers we prove the existence of asymptotic spatial patterns for diffusive tracers advected by unsteady velocity fields the asymptotic patterns arise from convergence to a time dependent inertial manifold in the underlying advection diffusion equation for time periodic velocity fields we find that the inertial manifold is spanned by a finite number of floquet solutions the strange eigenmodes observed first numerically by pierrehumbert these strange eigenmodes only admit a regular asymptotic expansion in the diffusivity if the velocity field is completely integrable

global parametric solutions of scalar transport passive scalar transport involves complex interactions between advection and diffusion where the global transport rate depends upon scalar diffusivity and the values of the possibly large set of parameters controlling the advective flow although computation of a single solution of the advection diffusion equation ade is simple in general it is prohibitively expensive to compute the parametric variation of solutions over the full parameter space q even though this is crucial for e g optimization parameter estimation and elucidating the global structure of transport by decomposing the flows within q so as to exploit symmetries we derive a spectral method that solves the ade over q three orders of magnitude faster than other methods of similar accuracy solutions are expressed in terms of the exponentially decaying natural periodic patterns of the ade sometimes called strange eigenmodes we apply the method to the experimentally realisable rotated arc mixer chaotic flow both to establish numerical properties and to calculate the fine scale structure of the global solution space for transport in this chaotic flow over 10 5 solutions within q are resolved and spatial pattern locking a symmetry breaking transition to disordered spatial patterns and fractally distributed optima in transport rate are observed the method exhibits exponential convergence and efficiency increases with resolution of q

efficient computation of highly oscillatory integrals with hankel kernel in this paper we consider the evaluation of two kinds of oscillatory integrals with a hankel function as kernel we first rewrite these integrals as the integrals of fourier type by analytic continuation these fourier type integrals can be transformed into the integrals on 0 the integrands of which are not oscillatory and decay exponentially fast consequently the transformed integrals can be efficiently computed by using the generalized gauss laguerre quadrature rule moreover the error analysis for the presented methods is given the efficiency and accuracy of the methods have been demonstrated by both numerical experiments and theoretical results

on numerical computation of integrals with integrands of the form f x sin w xr on 0 1 with existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form 0 1f x sin wx r dxor 0 1f x cos wx r dx r 0 where f is a sufficiently smooth function on 0 1 gautschi has developed software as scripts in matlab for computing these integrals for the special case r w 1 in this paper an algorithm as a mathematica program is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range numerical examples are given of testing the performance of the algorithm program

efficient numerical methods for bessel type of oscillatory integrals in this paper new quadrature rules are proposed for numerical evaluation of highly oscillatory integrals containing first kind of bessel functions j x meshless procedure with uniform and scattered nodes is used to cope with frequent irregular oscillations caused by bessel and bessel trigonometric functions in addition multi resolution quadrature rules based on hybrid functions and haar wavelets are used as supporting tools to handle the case of singularity of the meshless collocation method error bounds of the proposed methods are calculated and numerically verified by solving some benchmark test problems

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

on graded meshes for weakly singular volterra integral equations with oscillatory trigonometric kernels we compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular volterra integral equation of the second kind with oscillatory trigonometric kernels due to in general unbounded derivatives at the left endpoint of the interval of integration we should approximate the solution of the integral equation by collocation methods on graded meshes however we show that this non smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory numerical examples are given to confirm the proposed results

an infinite hierarchy of temporal logics over branching time many temporal logics have been suggested as branching time specification formalisms during the past 20 years these logics were compared against each other for their expressive power model checking complexity and succinctness yet unlike the case for linear time logics no canonical temporal logic of branching time was agreed upon we offer an explanation for the multiplicity of temporal logics over branching time and provide an objective quantified yardstick to measure these logics we define an infinite hierarchy btlk of temporal logics and prove its strictness we examine the expressive power of commonly used branching time temporal logics we show that ctl has no finite base and that almost all of its many sublogics suggested in the literature are inside the second level of our hierarchy we introduce new ehrenfeucht fraisse games on trees and use them as our main tool to prove inexpressibility 2001 elsevier science

psychophysically tuned divisive normalization approximately factorizes the pdf of natural images the conventional approach in computational neuroscience in favor of the efficient coding hypothesis goes from image statistics to perception it has been argued that the behavior of the early stages of biological visual processing e g spatial frequency analyzers and their nonlinearities may be obtained from image samples and the efficient coding hypothesis using no psychophysical or physiological information in this work we address the same issue in the opposite direction from perception to image statistics we show that psychophysically fitted image representation in v1 has appealing statistical properties for example approximate pdf factorization and substantial mutual information reduction even though no statistical information is used to fit the v1 model these results are complementary evidence in favor of the efficient coding hypothesis

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

advected fields in maps i magnetic flux growth in the stretch fold shear map abstract the behaviour of magnetic field in the stretch fold shear sfs dynamo map is considered for zero magnetic diffusion it is shown by a mixture of analytical and numerical approaches that the sfs map is a perfect dynamo for sufficiently large shear the adjoint operator has smooth growing eigenfunctions and so smooth flux averages grow exponentially with time for zero diffusion in the paper first a number of numerical discretisations are presented that give differing results for growth rates and indicate the need to develop systematic theory then magnetic fields that are only required to be square integrable are considered and the spectral properties of the sfs dynamo operator and its adjoint are discussed as operators in l 2 adjoint eigenfunctions are typically not smooth however to obtain smooth growing adjoint eigenfunctions attention is restricted to a subset of magnetic fields that are analytic in a disc in the complex plane restricted to this subset and using a supremum norm the sfs adjoint operator is compact and this allows a numerical treatment of eigenvalues and eigenfunctions with systematic error estimates these estimates show that for sufficiently large shear there are smooth growing adjoint eigenfunctions and so perfect dynamo action is established

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

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

a unified framework for asymptotic analysis and computation of finite hankel transform abstract in this paper we present a unified framework for asymptotic analysis of the finite hankel transform this framework enables us to derive asymptotic expansions of the transform including the cases where the oscillator has zeros and stationary points as a consequence two efficient and affordable methods for computing the transform numerically are discussed and a detailed analysis of their asymptotic error estimate is carried out numerical examples are provided to confirm our analysis

universal long time properties of lagrangian statistics in the batchelor regime and their application to the passive scalar problem we consider the transport of dynamically passive quantities in the batchelor regime of a smooth in space velocity field for the case of arbitrary temporal correlations of the velocity we formulate the statistics of relevant characteristics of lagrangian motion this allows us to generalize many results obtained previously for strain ensuremath delta correlated in time thus answering a question about the universality of these results

btl 2 and the expressive power of ectl we show that ectl the classical extension of ctl with fairness properties is expressively equivalent to btl2 a natural fragment of the monadic logic of order btl2 is the branching time logic with arbitrary quantification over paths and where path formulac are restricted to quantifier depth 2 first order formulae in the monadic logic of order this result linking ectl to a natural fragment of the monadic logic of order provides a characterization that other branching time logics e g ctl lack we then go on to show that ectl and btl2 are not finitely based i e they cannot be defined by a finite set of temporal modalities and that their model checking problems are of the same complexity

optimal by order quadrature formulae for fast oscillatory functions with inaccurately given a priori information abstract in this article the authors construct optimal by order quadrature formulae for integration of fast oscillatory functions in interpolational classes c 1 l n 1 and c 1 l n e 1 the construction of efficient formulae for numerical integration of fast oscillatory functions is based on the application of the residual method and the method of quasi solutions both cases weak and strong oscillations are considered results of numerical examples are presented

quadrature methods for multivariate highly oscillatory integrals using derivatives while there exist effective methods for univariate highly oscillatory quadrature this is not the case in a multivariate setting in this paper we embark on a project extending univariate theory to more variables inter alia we demonstrate that in the absence of critical points and subject to a nonresonance condition an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices a direct counterpart of the univariate case this provides a convenient avenue towards the generalization of asymptotic and filon type methods as formerly introduced by the authors in a single dimension to simplices and more generally to polytopes the nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth in effect its violation is equivalent to the presence of stationary points in a single dimension we further explore this issue and propose a technique that often can be used in this situation yet much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals

fast computation of seabed spherical wave reflection coefficients in geoacoustic inversion this paper develops a fast numerical approach to computing spherical wave reflection coefficients swrcs for layered seabeds which provides substantial savings in computation time when used as the forward model for geoacoustic inversion of broadband seabed reflectivity data the approach exploits the sommerfeld integral representation of swrcs as the hankel transform of a function proportional to the plane wave reflection coefficient pwrc and applies levin integration to the rapidly oscillating integrand cast as the product of a pre computed media independent matrix and a vector involving pwrcs at a sparse sampling of integration angles compared to conventional simpson s rule integration for computation of the swrc the levin integration yields speed up factors of an order of magnitude or more further it results in reduced memory requirements for storage of pre computed quantities a desirable property when a graphics processing unit gpu is used for parallel computation of swrcs the paper applies trans dimensional bayesian inversion to investigate the impact of forward modeling in terms of pwrcs and swrcs on the estimation of geoacoustic parameters and uncertainties model comparisons are quantified in simulated and measured data inversions by comparing the estimated geoacoustic parameters to the true parameters or core measurements respectively and by calculating the deviance information criterion for model selection

extraction and prediction of coherent patterns in incompressible flows through space time koopman analysis abstract we develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time dependent fluid flows driven by ergodic dynamical systems our approach is based on representations of the generators of the koopman and perron frobenius groups of operators governing the evolution of observables and probability measures on lagrangian tracers respectively in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm these operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place and as a result their eigenfunctions correspond to global space time coherent patterns under a skew product dynamical system moreover using this data driven representation of the generators in conjunction with leja interpolation for matrix exponentiation we construct model free prediction schemes for the evolution of observables and probability densities defined on the tracers we present applications to periodic gaussian vortex flows and aperiodic flows generated by lorenz 96 systems

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