spectral asymptotics for the third order operator with periodic coefficients abstract we consider the self adjoint third order operator with 1 periodic coefficients on the real line the spectrum of the operator is absolutely continuous and covers the real line we determine the high energy asymptotics of the periodic antiperiodic eigenvalues and of the branch points of the lyapunov function furthermore in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three in the last case the asymptotics of this small interval is determined

a numerical solution for non classical parabolic problem based on chebyshev spectral collocation method a numerical method based on chebyshev polynomials and local interpolating functions is proposed for solving the one dimensional parabolic partial differential equation subject to non classical conditions to assess its validity and accuracy the method is applied to solve several test problem

spectral differentiation matrices for the numerical solution of schrodinger s equation schrodinger s equation is solved numerically using the spectral collocation pseudospectral method based on hermite weighted polynomial interpolants several sets of numerical results from the literature are reproduced using this approach matlab codes that can serve as templates for further exploration are provided both in the paper and online a new proposal is made for solving schrodinger s equation in stokes wedges

a block pseudospectral method for maxwell s equations a block pseudospectral bps method is proposed as a new way to couple pseudospectral discretizations across interfaces in computations for a linear hyperbolic system the coupling is achieved via discretized derivative matching conditions obtained from the system compared to the standard technique of imposing compatibility conditions based on characteristics of the system the bps method offers better stability and accuracy especially in the case where equation coefficients are discontinuous computational examples for maxwell s equations in nonhomogeneous media demonstrate that bps retains high accuracy over times that are orders of magnitude larger than those for not only low order methods such as yee s but also high order methods such as characteristic based spectral elements

fast computation of the spectral differentiation by the fast multipole method abstract this paper is concerned with the application of the fast multipole method fmm to the pseudospectral methods the spectral differentiation in pseudospectral methods involves dense matrix vector products which if evaluated directly entail o n 2 operations the core of this work lies in the fact that the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products the fmm is exploited to estimate the barycentric weights and the pseudospectral differentiation which reduces the complexity to o n furthermore we show the convergence analysis and present the applications to the second order boundary value problems numerical examples confirm the advantages of the method

a time dependent neutron transport method of characteristics formulation with time derivative propagation a new time dependent method of characteristics moc formulation for nuclear reactor kinetics was developed utilizing angular flux time derivative propagation this method avoids the requirement of storing the angular flux at previous points in time to represent a discretized time derivative instead an equation for the angular flux time derivative along 1d spatial characteristics is derived and solved concurrently with the 1d transport characteristic equation this approach allows the angular flux time derivative to be recast principally in terms of the neutron source time derivatives which are approximated to high order accuracy using the backward differentiation formula bdf this approach called source derivative propagation sdp drastically reduces the memory requirements of time dependent moc relative to methods that require storing the angular flux an sdp method was developed for 2d and 3d applications and implemented in the computer code decart in 2d decart was used to model two reactor transient benchmarks a modified twigl problem and a c5g7 transient the sdp method accurately and efficiently replicated the solution of the conventional time dependent moc method using two orders of magnitude less memory

a modified chebyshev pseudospectral dd algorithm for the gbh equation in this paper a chebyshev spectral collocation domain decomposition dd semi discretization by using a grid mapping derived by kosloff and tal ezer in space is applied to the numerical solution of the generalized burger s huxley gbh equation to reduce roundoff error in computing derivatives we use the above mentioned grid mapping in this work we compose the chebyshev spectral collocation domain decomposition and kosloff and tal ezer grid mapping elaborately firstly the theory of application of the chebyshev spectral collocation method with grid mapping and dd on the gbh equation is presented this method yields a system of ordinary differential algebraic equations daes secondly we use a fourth order runge kutta formula for the numerical integration of the system of daes application of this modified method to the gbh equation show that this method m dd is faster and more accurate than the standard chebyshev spectral collocation dd s dd method

trace formulae for the matrix schrodinger equation with energy dependent potential abstract in this paper we consider the eigenvalue problems for the matrix schrodinger equation with energy dependent potential and with separated boundary conditions on the finite interval and find new trace formulae for the matrix schrodinger operator

optimal decomposition of the domain in spectral methods for wave like phenomena a strategy for determining the optimal number of grid points and subdomains in a spectral method with domain decomposition on a serial computer is presented the rapidly growing computational cost for large numbers of grid points in each subdomain is balanced against the exponential convergence for spectral approximation of smooth functions and the optimum is found as the number of grid points and subdomains that gives the minimal computational cost for a given accuracy the typical length scale of the problem is found to influence the number of subdomains but not the number of grid points within each subdomain

a mixed binary non linear programming approach for the numerical solution of a family of singular optimal control problems abstractthis paper presents a new approach for the efficient solution of singular optimal control problems socps a novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution at first the socp is converted into a binary optimal control problem then by utilising the pseudospectral method the resulting problem is transcribed to a mixed binary non linear programming problem this mixed binary non linear programming problem which can be solved by well known solvers allows us to detect the structure of the optimal control and to compute the approximating solution the main advantages of the present method are that 1 without a priori information the structure of optimal control is detected 2 it produces good results even using a small number of collocation points 3 the switching times can be captured accurately these advantages are illustrated through a numerical implementation of the method on four examples

some results on linear rational trigonometric interpolation abstract we present here formulae for calculating the p th derivative of a linear rational trigonometric interpolant written in barycentric form we give sets of interpolating points for which the interpolant converges exponentially towards the interpolated function

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

on the determinant of elliptic differential and finite difference operators in vector bundles over s 1 for an elliptic differential operatora overs 1 a sum limits k 0 n a k x d k witha k x in end r and as a principal angle the regularized determinant det a is computed in terms of the monodromy mapp a associated toa and some invariant expressed in terms ofa n anda n 1 a similar formula holds for finite difference operators a number of applications and implications are given in particular we present a formula for the signature ofa whena is self adjoint and show that the determinant ofa is the limit of a sequence of computable expressions involving determinants of difference approximation ofa

eigenvalue and eigenfunction asymptotics for regular sturm liouville problems abstract in this paper we obtain asymptotic formulas for eigenvalues eigenfunctions and the reciprocals of the eigenfunction norms for eigenvalue problems associated with the general sturm liouville equation pu r q u 0 having regular endpoints the method is based on a conversion to liouville normal form and an iterative procedure of solving the associated volterra integral equation producing an asymptotic expansion of the solution in higher powers of 1 1 2 as

estimation of copula based semiparametric time series models this paper studies the estimation of a class of copula based semiparametric stationary markov models these models are characterized by nonparametric marginal distributions and parametric copula functions while the copulas capture all the scale free temporal dependence of the processes simple estimators of the marginal distribution and the copula parameter are provided and their asymptotic properties are established under easily verifiable conditions these results are used to obtain root n consistent and asymptotically normal estimators of important features of the transition distribution such as the nonlinear conditional moments and conditional quantiles the semiparametric conditional quantile estimators are automatically monotonic across quantiles which is attractive for portfolio conditional value at risk calculations

diagnostic testing and evaluation of maximum likelihood models abstract the paper develops a unified theory of likelihood specification testing based on m estimators of auxiliary parameters the theory is sufficiently general to encompass a wide class of specification tests including moment based tests pearson type goodness of fit tests the information matrix test and the cox test the paper also presents a framework based on frechet differentiation for determining the effects of misspecification on the almost sure limits of parameter estimates and specification test statistics

numerical solution of a non classical parabolic problem an integro differential approach a numerical method based on an integro differential formulation and approximation by local interpolating functions is proposed for solving a one dimensional parabolic partial differential equation subject to non classical conditions some specific test problems are solved using the proposed method numerical results obtained indicate that it can give accurate solutions and that it is an interesting and viable alternative to existing numerical methods for solving the class of problems under consideration

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

an inverse problem for the matrix schrodinger equation abstract by modifying and generalizing some old techniques of n levinson a uniqueness theorem is established for an inverse problem related to periodic and sturm liouville boundary value problems for the matrix schrodinger equation

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

a note on stability of pseudospectral methods for wave propagation in this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation we suppose that spatial derivatives are approximated by chebyshev pseudospectral discretizations through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time however the resulting stability and hence efficiency properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated specifically mathematically equivalent well posed approaches are not equivalent numerically an analysis of the spectrum of the resultant system operator predicts these properties

numerical solution of the heat equation with nonlocal boundary conditions abstract the method for the heat equation with nonlocal boundary conditions is discussed in this paper the unconditional stability is proved for 1 2 subject to a condition that is much weaker than the one assumed in a paper by ekolin due to the nonlocal boundary conditions the systems of linear equations generated by the method have a coefficient matrix that is tridiagonal except its first and last rows three efficient algorithms for solving this kind of linear systems are presented a simple numerical example is given to compare their efficiency

specification testing and quasi maximum likelihood estimation abstract this paper develops robust regression based conditional moment tests for models estimated by quasi maximum likelihood using a density in the linear exponential family the tests are relatively simple to compute while being robust to distributional assumptions other than those being explicitly tested the conditional mean tests assume only that the conditional mean is correctly specified under the null hypothesis but are asymptotically equivalent to standard forms of the tests if the conditional variance is correctly specified tests for conditional variance misspecification assume only that the first two moments are correctly specified under the null hypothesis an empirical application to individual arrest data is included

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

numerical solution of a parabolic equation with non local boundary specifications the parabolic partial differential equations with non local boundary specifications model various physical problems numerical schemes are developed for obtaining approximate solutions to the initial boundary value problem for one dimensional second order linear parabolic partial differential equation with non local boundary specifications replacing boundary conditions the method of lines semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the spatial derivative in the pde is approximated by a finite difference approximation the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the new algorithms are tested on two problems from the literature the central processor unit times needed are also considered

likelihood based estimation in a panel setting robustness redundancy and validity of copulas abstract this paper considers the estimation of likelihood based models in a panel setting that is we have panel data and for each time period separately we have a correctly specified model that could be estimated by mle we want to allow non independence over time this paper shows how to improve on the qmle it then considers mle based on joint distributions constructed using copulas it discusses the efficiency gain from using the true copula and shows that knowledge of the true copula is redundant only if the variance matrix of the relevant set of moment conditions is singular it also discusses the question of robustness against misspecification of the copula and proposes a test of the validity of the copula gmm methods are argued to be useful analytically and also for reasons of efficiency if the copula is robust but not correct

a pseudospectral fictitious point method for high order initial boundary value problems when pseudospectral approximations are used for space derivatives one often encounters spurious eigenvalues these can lead to severe time stepping difficulties for pdes this is especially the case for equations with high order derivatives in space requiring multiple conditions at one or both boundaries we note here that a very simple to implement fictitious point approach circumvents most of these difficulties the new approach is tested on the kuramoto sivashinsky equation and on a dispersive linear pde featuring a time space corner singularity

numerical algorithm for parabolic problems with non classical conditions the parabolic problems with non classical conditions are discussed in a reproducing kernel space in this paper a reproducing kernel space is constructed in which the non classical conditions of the parabolic problems are satisfied based on the reproducing kernel space a new technique for solving the non classical parabolic problems is presented some examples are displayed to demonstrate the validity and applicability of the proposed method

redundancy of moment conditions it is well known that adding moment conditions cannot decrease asymptotic efficiency however sometimes additional moment conditions are redundant in the sense that they do not increase the asymptotic efficiency of estimation for some or all of the parameters of interest this paper gives a general treatment of redundancy of moment conditions it provides necessary and sufficient conditions for redundancy in several equivalent forms the paper also provides interesting results for the case that there are three or more sets of moment conditions

a trace formula for multidimensional schrodinger operators we prove multidimensional analogs of the trace formula obtained previously for one dimensional schrodinger operators for example letvbe a continuous function on 0 1 r fora 1 let abe the laplace operator on 0 1 with mixed dirichlet neumann boundary conditions formula let a number of points ina then we ll prove that formula with v the average ofvat the 2 corners of 0 1

on the derivation of highest order compact finite difference schemes for the one and two dimensional poisson equation with dirichlet boundary conditions the primary aim of this paper is to answer the question what are the highest order five or nine point compact finite difference schemes to answer this question we present several simple derivations of finite difference schemes for the one and two dimensional poisson equation on uniform quasi uniform and nonuniform face to face hyperrectangular grids and directly prove the existence or nonexistence of their highest order local accuracies our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights for the one dimensional problem the derivation using the three point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high order local accuracy however for the two dimensional problem the derivation using the corresponding five point stencil on uniform and quasi uniform grids yields a scheme with at most second order local accuracy and on nonuniform grids yields at most first order local accuracy when expanding the five poin

numerical solution of the orr sommerfeld equation using the viscous green function and split gaussian quadrature we continue our study of the construction of numerical methods for solving two point boundary value problems using green functions building on the successful use of split gauss type quadrature schemes here we adapt the method for eigenvalue problems in particular the orr sommerfeld equation of hydrodynamic stability theory use of the green function for the viscous part of the problem reduces the fourth order ordinary differential equation to an integro differential equation which we then discretize using the split gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators as the latter are only second order the resulting discrete equations are much more stable than those obtained from the original differential equation this permits us to obtain results for the standard test problem plane poiseuille flow at unit streamwise wavenumber and reynolds number 10000 that we believe are the most accurate to date

mesh independence of the generalized davidson algorithm abstract we give conditions under which the generalized davidson algorithm for eigenvalue computations is mesh independent in this case mesh independence means that the iteration statistics residual norms convergence rates for example of a sequence of discretizations of a problem in a banach space converge the statistics for the infinite dimensional problem we illustrate the result with several numerical examples

accuracy and speed in computing the chebyshev collocation derivative we study several algorithms for computing the chebyshev spectral derivative and compare their roundoff error for a large number of collocation points the elements of the chebyshev differentiation matrix if constructed in the usual way are not computed accurately a subtle cause is found to account for the poor accuracy when computing the derivative by the matrix vector multiplication method methods for accurately computing the elements of the matrix are presented and we find that if the entries of the matrix are computed accurately the roundoff error of the matrix vector multiplication is as small as that of the transform recursion algorithm furthermore results of the cpu time usage are shown for several different algorithms for computing the derivative by the chebyshev collocation method for a wide variety of two dimensional grid sizes on both an ibm mainframe and a cray 2 computer we find that which algorithm is fastest on a particular machine depends not only on the grid size but also on small

a new robust line search technique based on chebyshev polynomials abstract newton s method is an important and basic method for solving nonlinear univariate and unconstrained optimization problems in this study a new line search technique based on chebyshev polynomials is presented the proposed method is adaptive where it determines a descent direction at each iteration and avoids convergence to a maximum point approximations to the first and the second derivatives of a function using high order pseudospectral differentiation matrices are derived the efficiency of the new method is analyzed in terms of the most popular and widely used criterion in comparison with newton s method using seven test functions

goodness of fit test for specification of semiparametric copula dependence models this paper concerns goodness of fit tests for semiparametric copula models our contribution is two fold we first propose a new test constructed via the comparison between in sample and out of sample pseudo likelihoods under the null hypothesis that the copula model is correctly specified we show that the proposed test statistic converges in probability to a constant equal to the dimension of the parameter space we establish the asymptotic normality and investigate the local power of the test we also extend the proposed test to the specification test of a class of multivariate time series models and propose a new bootstrap procedure to establish the finite sample null distribution which is shown to have better control of type i error than the commonly used bootstrap secondly we introduce a bonferroni based hybrid mechanism to combine several test statistics which yields a useful test this hybrid method is particularly appealing when there exists no single dominant optimal test we conduct comprehensive simulation experiments to compare the proposed new test and hybrid approach with two of the best blanket tests in the literature for illustration we apply the proposed tests to analyze two real datasets

adaptive solution of bvps in singularly perturbed second order odes by the extended numerov method combined with an iterative local grid h refinement the extended numerov scheme of chawla is one of high order compact finite difference discretisations applicable to bvps in second order odes and to pdes the use of such discretisations in connection with adaptive grids remains largely unexplored in order to make up this deficiency we study the combination of the extended numerov discretisation with the iterative local adaptive grid h refinement previously tested by the present author at the assumption of the conventional finite difference discretisation a detailed formalism enabling a posteriori error analysis of the extended numerov scheme is developed together with several alternative choices for error estimators grid refinement indicators and regridding strategies the adaptive algorithms obtained are examined in calculations on a collection of 15 examples of singularly perturbed odes including equations relevant for electrochemistry the most satisfactory algorithm involves deferred approach to error estimation based on virtual grid coarsening for truncation error evaluation local discretisation errors as refinement indicators and regridding strategy that uses the mean indicator values as the indicator thresholds for refinement the extended numerov scheme proves more efficient than the conventional discretisation when used together with the iterative h refinement

the periodic euler bernoulli equation we continue the study of the floquet spectral theory of the beam equation namely the fourth order eigenvalue problem a x u x x u x x where the functions a and p are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described by a and p here we develop a theory analogous to the theory of the hill operator d dx 2 q x we first review some facts and notions from our previous works including the concept of the pseudospectrum or spectrum our new analysis begins with a detailed study of the zeros of the function f a k for any given quasimomentum k c where f k 0 is the floquet bloch variety of the beam equation the hill quantity corresponding to f k is 2 cos kb where a a is the discriminant and b the period of q we show that the multiplicity m of any zero of f a k can be one or two and m 2 for some k if and only if is also a zero of another entire function d a independent of k furthermore we show that d a has exactly one zero in each gap of the spectrum and two zeros counting multiplicities in each gap if is a double zero of f a k it may happen that there is only one floquet solution with quasimomentum k thus there are exceptional cases where the algebraic and geometric multiplicities do not agree next we show that if is an open gap of the pseudospectrum i e then the floquet matrix t a has a specific jordan anomaly at a a and a we then introduce a multipoint dirichlet type eigenvalue problem which is the analogue of the dirichlet problem for the hill equation we denote by n n z the eigenvalues of this multipoint problem and show that n n z is also characterized as the set of values of a for which there is a proper floquet solution f x a such that f 0 a 0 we also show theorem 7 that each gap of the l 2 r spectrum contains exactly one n and each gap of the pseudospectrum contains exactly two n s counting multiplicities here when we say gap or gap we also include the endpoints so that when two consecutive bands or bands touch the in between collapsed gap or gap is a point we believe that n n z can be used to formulate the associated inverse spectral problem as an application of theorem 7 we show that if is a collapsed closed gap then the floquet matrix t is diagonalizable some of the above results were conjectured in our previous works however our conjecture that if all the gaps are closed then the beam operator is the square of a second order hill type operator is still open

transport schemes in spherical geometries using spline based rbf fd with polynomials abstract this work presents a numerical algorithm for using radial basis function generated finite differences rbf fd to solve partial differential equations pdes on s 2 using polyharmonic splines with added polynomials defined in a 2d plane phs poly we introduce a novel method for calculating rbf fd phs poly differentiation weights on s 2 using first a householder reflection and then a projection onto the tangent plane the new phs poly rbf fd method is implemented on two standard test cases 1 solid body rotation on s 2 and 2 3d tracer transport within earth s atmosphere compared to existing methods including those in the rbf literature at similar resolutions this approach requires fewer degrees of freedom and is algorithmically much simpler a matlab code to implement the method is included in the appendix

on the norms of inverses of pseudospectral differentiation matrices in this paper we give integral expressions for the elements of the inverses of second order pseudospectral differentiation matrices simple upper bounds are given for the maximum norms of these inverse matrices when chebyshev collocation points are used comment is made on the failure to obtain upper bounds that are uniform in the number of collocation points when the points are evenly spaced we also give integral expressions for inverses of first order chebyshev pseudospectral differentiation matrices

equations with infinite delay numerical bifurcation analysis via pseudospectral discretization abstract we address the problem of the numerical bifurcation analysis of general nonlinear delay equations including integral and integro differential equations for which no software is currently available pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations whose properties can be numerically studied with well developed software we explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations

the lyapunov function for schrodinger operators with a periodic 2 2 matrix potential abstract we consider the schrodinger operator on the real line with a 2 2 matrix valued 1 periodic potential the spectrum of this operator is absolutely continuous and consists of intervals separated by gaps we define a lyapunov function which is analytic on a two sheeted riemann surface on each sheet the lyapunov function has the same properties as in the scalar case but it has branch points which we call resonances we prove the existence of real as well as non real resonances for specific potentials we determine the asymptotics of the periodic and the anti periodic spectrum and of the resonances at high energy we show that there exist two type of gaps 1 stable gaps where the endpoints are the periodic and the anti periodic eigenvalues 2 unstable resonance gaps where the endpoints are resonances i e real branch points of the lyapunov function we also show that periodic and anti periodic spectrum together determine the spectrum of the matrix hill operator

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

a fast spectrally accurate homotopy based numerical method for solving nonlinear differential equations abstract we present an algorithm for constructing numerical solutions to one dimensional nonlinear variable coefficient boundary value problems this scheme is based upon applying the homotopy analysis method ham to decompose a nonlinear differential equation into a series of linear differential equations that can be solved using a sparse spectrally accurate gegenbauer discretisation uniquely for nonlinear methods our scheme involves constructing a single sparse matrix operator that is repeatedly solved in order to solve the full nonlinear problem as such the resulting scheme scales quasi linearly with respect to the grid resolution we demonstrate the accuracy and computational scaling of this method by examining a fourth order nonlinear variable coefficient boundary value problem by comparing the scheme to newton iteration and the spectral homotopy analysis method which is the most commonly used implementation of the ham

generation of pseudospectral differentiation matrices i we present a simple method for computing n times n pseudospectral differentiation matrices of order p in cal o pn 2 operations for the case of quasi polynomial approximation the algorithm is based on fornberg s finite difference algorithm and is numerically stable a fortran implementation is included a necessary and sufficient condition for d p d p 1 is also given

a fast numerical method to price american options under the bates model we consider the problem of pricing american options in the framework of a well known stochastic volatility model with jumps the bates model according to this model the price of an american option can be obtained as the solution of a linear complementarity problem governed by a partial integro differential equation in this paper a numerical method for solving such a problem is proposed in particular first of all using a bermudan approximation and a richardson extrapolation technique the linear complementarity problem is reduced to a set of standard linear partial differential problems see for example ballestra and sgarra 2010 chang et al 2007 2012 then these problems are solved using an ad hoc pseudospectral method which efficiently combines the chebyshev polynomial approximation an implicit explicit time stepping and an operator splitting technique numerical experiments are presented showing that the novel algorithm is very accurate and fast and significantly outperforms other methods that have recently been proposed for pricing american options under the bates model

3d weak dispersion reverse time migration using a stereo modeling operator reliable 3d imaging is a required tool for developing models of complex geologic structures reverse time migration rtm as the most powerful depth imaging method has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts finite difference methods are among the most popular numerical approaches used for rtm however these methods often encounter a serious issue of numerical dispersion which is typically suppressed by reducing the grid interval of the propagation model resulting in large computation and memory requirements in addition even with small grid spacing numerical anisotropy may degrade images or worse provide images that appear to be focused but position events incorrectly recently stereo operators have been developed to approximate the partial differential operator in space these operators have been used to develop several weak dispersion and efficient stereo modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy we generalized one stereo modeling method fourth order nearly analytic central difference nacd from 2d to 3d and applied it to 3d rtm the rtm results for the 3d seg eage phase a classic data set 1 and the seg advanced modeling project model demonstrated that even when using a large grid size the nacd method can handle very complex velocity models and produced better images than can be obtained using the fourth order and eighth order lax wendroff correction lwc schemes we also applied 3d nacd and fourth order lwc to a field data set and illustrated significant improvements in terms of structure imaging horizon layer continuity and positioning we also investigated numerical dispersion and found that not only does the nacd method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for lwc

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

quadrature discretization method in tethered satellite control a pseudospectral method for solving the tethered satellite retrieval problem based on nonclassical orthogonal and weighted interpolating polynomials is presented traditional pseudospectral methods expand the state and control trajectories using global lagrange interpolating polynomials based on a specific class of orthogonal polynomials from the jacobi family such as legendre or chebyshev polynomials which are orthogonal with respect to a specific weight function over a fixed interval although these methods have many advantages the location of the collocation points are more or less fixed the method presented in this paper generalizes the existing methods and allows a much more flexible selection of grid points by the arbitrary selection of the orthogonal weight function and interval the trajectory optimization problem is converted to set of algebraic equations by discretization of the necessary conditions using a nonclassical pseudospectral method

spectral differencing with a twist spectral collocation methods have become very useful in providing highly accurate solutions to differential equations a straightforward implementation of these methods involves the use of spectral differentiation matrices to obtain optimal accuracy these matrices must be computed carefully we demonstrate that naive algorithms for computing these matrices suffer from severe loss of accuracy due to roundoff errors several improvements are analyzed and compared a number of numerical examples are provided demonstrating significant differences between the sensitivity of the forward problem and the inverse problem

trace formulae for the schrodinger equation with energy dependent potential we consider the eigenvalue problems for the schrodinger equation with energy dependent potential subject to the eigenparameter boundary conditions and quasiperiodic boundary conditions respectively two new formulae of the regularized trace for this class of schrodinger equation with boundary conditions are obtained in an explicit form

the errors in calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points abstract we discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points we propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices

efficient estimation of semiparametric multivariate copula models we propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions a joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions this class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable and its circumvention of the curse of dimensionality associated with purely nonparametric multivariate distributions we show that the plug in sieve maximum likelihood estimators mles of all smooth functionals including the finite dimensional copula parameters and the unknown marginal distributions are semiparametrically efficient and that their asymptotic variances can be estimated consistently moreover prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve fu

efficient and stable generation of higher order pseudospectral integration matrices the main purpose of this work is to provide new higher order pseudospectral integration matrices hpims for the chebyshev type points and present an exact efficient and stable approach for computing the hpims the essential idea is to reduce the computation of hpims to that of higher order chebyshev integration matrices hcims and take a very simple and recursive way to compute the hcims efficiently and stably extensive numerical results show that the new approach for computing the hpims has better stability than that of the recently derived elgindy s approach for large number of chebyshev type points

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

eigenvalue estimates and trace formulas for the matrix hill s equation abstract estimates for floquet multipliers and periodic eigenvalues are developed for the matrix hill s equation basic estimates and a description of the periodic eigenvalues as roots of an entire function are adequate for establishing trace formulas with a residue computation refined estimates and power sum formulas are available for restricted pairs of hill s operators

conformal spectral theory for the monodromy matrix for any n x n monodromy matrix we define the lyapunov function which is analytic on an associated n sheeted riemann surface on each sheet the lyapunov function has the standard properties of the lyapunov function for the hill operator the lyapunov function has real or complex branch points which we call resonances we determine the asymptotics of the periodic anti periodic spectrum and of the resonances at high energy we show that the endpoints of each gap are periodic anti periodic eigenvalues or resonances real branch points moreover the following results are obtained 1 we define the quasimomentum as an analytic function on the riemann surface of the lyapunov function various properties and estimates of the quasimomentum are obtained 2 we construct the conformal mapping with imaginary part given by the lyapunov exponent and we obtain various properties of this conformal mapping which are similar to the case of the hill operator 3 we determine various new trace formulae for potentials and the lyapunov exponent 4 we obtain a priori estimates of gap lengths in terms of the dirichlet integral we apply these results to the schrodinger operators and to first order periodic systems on the real line with a matrix valued complex self adjoint periodic potential

a fast algorithm for spectral differentiation a simple algorithm is presented for matrix multiplication in just over half the number of operations entailed by the conventional algorithm in cases where the matrix possesses the degree of symmetry widely exhibited by derivative matrices the algorithm is used to multiply the chebyshev derivative matrix by a vector for the larger values of n the ratio of execution times approached the expected value of 2