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

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

pullback and pushout constructions in c algebra theory abstract a systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of c algebras particular emphasis is given to the relations with tensor products both with the minimal and the maximal c tensor norm thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group general tensor products between diagrams are also investigated the relations between the theory of extensions and pullback and pushout diagrams are explored in some detail the crowning result is that if three short exact sequences of c algebras are given with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals algebras and quotients then the three new c algebras will again form a short exact sequence under some mild extra conditions as a generalization of a theorem of t a loring it is shown that each morphism between a pair of c algebras combined with its extension to the stabilized algebras gives rise to a pushout diagram this result has applications to corona extendibility and conditional projectivity finally the pullback and pushout constructions are applied to the class of noncommutative cw complexes defined by s eilers t a loring and g k pedersen j reine angew math 1998 499 101 143 to show that this category is stable under tensor products and under restricted direct sums

on the kk theory and the e theory of amalgamated free products of c algebras abstract we establish six terms exact sequences relating the kk theory groups and the e theory groups of an amalgamated free product c algebra a 1 b a 2 to the respective groups of the three constituents a1 a2 and b in the kk theory case we assume that b is finite dimensional but in the e theory case only that b is nuclear or is the range of a conditional expectation in both a1 and a2 in particular we obtain the conjectured six terms exact sequence for the k theory of an amalgamated free product in this generality

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

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

the numerical evaluation of very oscillatory infinite integrals by extrapolation recently the author has given two modifications of a nonlinear extrapolation method due to levin and sidi which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity in this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity the new method is applied to a number of complicated integrals among them the solution to a problem in viscoelasticity some convergence results for this method are presented

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

a priori bounds and stability in the retarded nonlinear wave equation the object of this work is to give sufficient conditions to ensure that the zero solution of certain generalizations of u t 4x g u x t h t x flu o 0 xg x oifx o f b 0 and g x 0 2 a g f and h are at least continuous 3 we impose the boundary conditions u t o u t l o so that u t o u t l o 4 although mixed boundary conditions are generally satisfactory among other things it is crucial to have

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

more on the weeks method for the numerical inversion of the laplace transform most of the numerical methods for the inversion of the laplace transform require the values of several incidental parameters generally these parameters are related to the properties of the algorithm and to the analytical properties of the laplace transform functionf s

uniform asymptotic stability in perturbed differential equations on etudie la stabilite asymptotique uniforme de la solution zero des equations differentielles perturbees x f t x g t x et si t f t x t g t x t

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

a method for the numerical inversion of laplace transforms abstract in this paper a numerical inversion method for laplace transforms based on a fourier series expansion developed by durbin 5 is presented the disadvantage of the inversion methods of that type the encountered dependence of discretization and truncation error on the free parameters is removed by the simultaneous application of a procedure for the reduction of the discretization error a method for accelerating the convergence of the fourier series and a procedure that computes approximately the best choice of the free parameters suitable for a given problem the inversion method allows the adequate application of these procedures therefore in a big range of applications a high accuracy can be achieved with only a few function evaluations of the laplace transform the inversion method is implemented as a fortran subroutine

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

computational error bounds for laplace transform inversion based on smoothing splines abstract in the numerical methods for the laplace transform inversion the errors quantification in computational processes is a crucial issue in this paper we propose two inversion methods based on smoothing splines combined with a procedure for the derivation of error bounds in particular we numerically study the impact of the fitting error amplification through the analysis of several sources of error and their propagation

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

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

algorithms for parameter selection in the weeks method for inverting the laplace transform the weeks method is one of the most efficient numerical techniques for inverting the laplace transform provided two free parameters in the laguerre expansion on which the method is based are chosen judiciously however there appear to be no theoretical estimates or numerical algorithms for computing these parameters the two algorithms presented in this paper fill that gap both algorithms aim to minimize a theoretical error estimate and both are implemented with the fft the first algorithm is restricted to a certain class of transforms and the user is expected to provide information on the singularities of the transform the second algorithm though more expensive is completely general no singularity information is required in challenging numerical tests both algorithms successfully predicted values of the parameters that are near optimal

a modification of the weeks method for numerical inversion of the laplace transform the weeks method for calculating numerical values of f t from its laplace transform f s involves the numerical evaluation of a set of contour integrals in this paper we describe a powerful modification by means of which results of comparable accuracy can be obtained using approximately half the number of complex function values this and other possible minor modifications are discussed at any values of s we like we wish to construct an algorithm which obtains or uses this information to evaluate f t at a given selection of values of t real to a specified accuracy one may assume that f t is real in which case f s f in almost all methods one needs to know or one has to find out first the numerical value of the laplace convergence abscissa ao this may be defined as the limit of the set of values of re s for which this integral converges or alternatively as the maximum of the real parts of the singular points sj of f s a good brief description of the formulas on which the weeks method 2 is based is given in piessens and branders 1 in order to provide a proper background we repeat some of these here briefly we expand f t in terms of laguerre functions thus we assume that

multi precision laplace transform inversion for the numerical inversion of laplace transforms we suggest to use multi precision computing with the level of precision determined by the algorithm we present two such procedures the gaver wynn rho gwr algorithm is based on a special sequence acceleration of the gaver functionals and requires the evaluation of the transform only on the real line the fixed talbot ft method is based on the deformation of the contour of the bromwich inversion integral and requires complex arithmetic both gwr and ft have only one free parameter m which is the number of terms in the summation both algorithms provide increasing accuracy as m increases and can be realized in a few lines using current computer algebra systems copyright 2004 john wiley sons ltd

liapunov razumikhin pairs and the location of positive limit sets for precompact functional differential equations with infinite delay in a recent paper l haddock and terjeki addressed the problem of locating positive limit sets for autonomous functional differential equations with infinite delay the authors obtained invariance principles and other results concerning asymptotic behavior of solutions by employing liapunov razumikhin techniques the autonomous nature of the equations they considered was central to their work the purpose of this paper is to extend their results to a class of nonautonomous equations such a task was undertaken for finite delay equations in 2 here as in 2 the assumption of autonomy is replaced by a continuity condition and our invariance principles involve limiting equations of our original equation we feel that the setting for our theory is quite natural and that our methods illustrate the role of limiting equations in extending results from the autonomous theory to a significantly larger class of equations several examples are given to illustrate our techniques

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

parabolic and hyperbolic contours for computing the bromwich integral some of the most eectiv e methods for the numerical inversion of the laplace transform are based on the approximation of the bromwich contour integral the accuracy of these methods often hinges on a good choice of contour and several such contours have been proposed in the literature here we analyze two recently proposed contours namely a parabola and a hyperbola using a representative model problem we determine estimates for the optimal parameters that dene these contours an application to a fractional diusion equation is presented

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

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

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 hyperrigidity of tensor algebras of c correspondences abstract given a c correspondence x we give necessary and sufficient conditions for the tensor algebra t x to be hyperrigid in the case where x is coming from a topological graph we obtain a complete characterization

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

applications of the wold decomposition to the study of row contractions associated with directed graphs based on a wold decomposition for families of partial isometries and projections of cuntz krieger toeplitz type we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks we prove a szego type factorization theorem for ckt families which leads to information on the structure of the unit ball in free semigroupoid algebras and show that joint similarity implies joint unitary equivalence for such families for each graph we prove a generalization of von neumann s inequality which applies to row contractions of operators on hilbert space which are related to the graph in a natural way this yields a functional calculus determined by quiver algebras and free semigroupoid algebras we establish a generalization of coburn s theorem for the c algebra of a ckt family and prove a universality theorem for c algebras generated by these families in both cases the c algebras generated by quiver algebras play the universal role

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

certain free products of graph operator algebras abstract we develop a notion of a generalized cuntz krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection we associate to these generalized cuntz krieger families a directed graph with a coloring function on the edge set we call such a directed graph an edge colored directed graph we then study the c algebras and the non selfadjoint operator algebras associated to edge colored directed graphs these algebras arise as free products of directed graph algebras with amalgamation we then determine the c envelopes for a large class of the non selfadjoint algebras finally we relate properties of the edge colored directed graphs to properties of the associated c algebra including simplicity and nuclearity using the free product description of these algebras we investigate the k theory of these algebras

c algebras of separated graphs the construction of the c algebra associated to a directed graph e is extended to incorporate a family c consisting of partitions of the sets of edges emanating from the vertices of e these c algebras c e c are analyzed in terms of their ideal theory and k theory mainly in the case of partitions by finite sets the groups k0 c e c and k1 c e c are completely described via a map built from an adjacency matrix associated to e c one application determines the k theory of the c algebras um nnc confirming a conjecture of mcclanahan a reduced c algebra cred e c is also introduced and studied a key tool in its construction is the existence of canonical faithful conditional expectations from the c algebra of any row finite graph to the c subalgebra generated by its vertices differences between cred e c and c e c such as simplicity versus non simplicity are exhibited in various examples related to some algebras studied by mcclanahan

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

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