spectral convergence for orthogonal polynomials on triangles we study the behavior of orthogonal polynomials on triangles and their coefficients in the context of spectral approximations of partial differential equations in these spectral approximations one studies series expansions u sum k 0 infty hat u k phi k where the phi k are orthogonal polynomials our results show that for any function u in c infty the series expansion converges faster than with polynomial order

on the number of nodes in n dimensional cubature formulae of degree 5 for integrals over the ball in this note cubature formulae of degree 5 are studied for n dimensional integrals over the ball with constant weight function we apply the method of reproducing kernel to show that the existence of such formulae attaining the best known lower bound is equivalent to the existence of tight spherical 5 designs the known results concerning spherical 5 designs show that the lower bound for the integral under consideration will not be attained in general the bound will be attained for n 2 3 7 23 and possibly for n 2p 1 2 2 p 5 in all other cases the bound must be increased at least by 1 in particular stroud s formulae for n 4 5 6 7 are minimal

numerical study of pseudospectral methods in shock wave applications abstract in this paper we use shock capturing spectral methods to simulate compressible flows in the presence of shock waves three applications are considered the first is an interaction of a one dimensional shock with an entropy wave the second case deals with interactions between shock wave and an entropy wave in two space dimensions and the third case is two dimensional shock vortex interactions the first two applications are found in the study of turbulence in high speed flow the last application is a key element in understanding the acoustic dynamics in aeroacoustic and in design of supersonic jet in aerodynamics the purpose of this study is to show the feasibility of simulating shocked flow with spectral methods the numerical methods involved are the chebyshev and fourier collocation methods the euler equations of gas dynamics is discretized by pseudospectral collocation methods in space and a nonlinearly stable third order runge kutta methods in time the fluxes are evaluated pointwise directly and not by the cell averaging technique a significant reduction in cpu time and storage usage are achieved by incorporating several well established numerical techniques such as grid transformation and filtering into the spectral algorithm the results of this study indicate that spectral method is well suited not only for smooth problems but also for those with discontinuity

convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment a new method of imposing boundary conditions in the pseudospectral approximation of hyperbolic systems of equations is proposed it is suggested to collocate the equations not only at the inner grid points but also at the boundary points and use the boundary conditions as penalty terms in the pseudo spectral legrendre method with the new boundary treatment a stability analysis for the case of a constant coefficient hyperbolic system is presented and error estimates are derived

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

spectral simulation of supersonic reactive flows we present in this paper numerical simulations of reactive flows interacting with shock waves we argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long term integrations of complicated flows even in the presence of shock waves a spectral code is described in detail and the theory that leads to each of its components is explained results of interactions of hydrogen jets with shock waves are presented and analyzed and comparisons with eno finite difference schemes are carried out

a spectral viscosity method based on hermite functions for nonlinear conservation laws we consider the approximation by a spectral method of the solution of the cauchy problem for a scalar conservation law in one dimension posed in the whole real line we analyze a spectral viscosity method in which the orthogonal basis considered is the one of hermite functions we prove the convergence of the approximate solution to the unique entropy solution of the problem by using compensated compactness arguments

a spectral viscosity method for correcting the long term behavior of pod models low dimensional flow dynamical systems may converge to erroneous states after long time integration even if they are initialized with the correct state in this paper we investigate the accuracy of such two dimensional models constructed from karhunen loeve expansions for flows past a circular cylinder we first demonstrate that although the short term dynamics may be predicted accurately with only a handful of modes retained drifting of the solution may arise after a few hundred vortex shedding cycles we then propose a dissipative model based on a spectral viscosity sv diffusion convolution operator the parameters of the sv model are selected rigorously based on bifurcation analysis our results show that this is an effective way of improving the accuracy of long term predictions of low dimensional galerkin systems

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

fractional burgers equation with nonlinear non locality we consider the viscous burgers equation with a fractional nonlinear term as a model involving non local nonlinearities in conservation laws which surprisingly has an analytical solution obtained by a fractional extension of the hopfcole transformation we use this model and its inviscid limit to develop stable spectral and discontinuous galerkin spectral element methods by employing the concept of spectral vanishing viscosity svv for the global spectral method svv is very effective and the computational cost is o n2 which is essentially the same as for the standard burgers equation we also develop a local discontinuous galerkin ldg spectral element method to improve the accuracy around discontinuities and we again stabilize the ldg method with the svv operator finally we solve numerically the inviscid fractional burgers equation both with the spectral and the spectral element ldg methods we study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters

adaptive control with recursive identification for stochastic linear systems this paper presents a stochastic adaptive control algorithm which is shown to possess the following properties when applied to a possibly unstable inverse stable linear stochastic system with unknown parameters whenever that system satisfies a certain positive real condition on its moving average noise dynamics 1 the adaptive control part of the algorithm stabilizes and asymptotically optimizes the behavior of the system in the sense that the limit of the sample mean square variation of the output around a given demand level equals that of a minimum variance control strategy implemented with known parameters this optimal behavior is subject to an offset 2where 2is the variance of a dither signal added to the control action in order to produce a continually disturbed control for mu 2 0 it is shown that the input output process satisfies a persistent excitation property and hence subject to a simple identifiability condition the next property holds 2 the observed input and output of the controlled system may be taken as inputs to an approximate maximum likelihood algorithm aml which generates strongly consistent estimates of the system s parameters results are presented for the scalar and multivariable cases

a legendre petrov galerkin method for fourth order differential equations in this paper we present a legendre petrov galekin method for one dimensional linear fourth order differential equations a legendre petrov galerkin and chebyshev collocation method is developed for the nonlinear kuramoto sivashinsky equation numerical results are presented to demonstrate the efficiency of the proposed schemes and optimal rates of convergence in the l 2 norm are rigorously derived

cubature formulae and orthogonal polynomials abstract the connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 yr the article j radon published about 50 yr ago j radon zur mechanischen kubatur monatsh math 52 1948 286 300 has been very influential in this text we describe some of the results that were obtained during the search for answers to questions raised by his article

robust estimation without positive real condition the strictly positive real spr condition on the noise model is necessary for a discrete time linear stochastic control system with unmodeled dynamics even so for a time invariant armax system in the past robust analysis of parameter estimation however this condition is hardly satisfied for a high order and or multidimensional system with correlated noise the main work in this paper is to show that for robust parameter estimation and adaptive tracking as well as closed loop system stabilization the spr condition is replaced by a stable matrix polynomial the main method is to design a two step recursive least squares algorithm with or without a weighted factor and with a fixed lag regressive vector and to define an adaptive control with bounded external excitation and with randomly varying truncation

error estimates for the anova method with polynomial chaos interpolation tensor product functions we focus on the analysis of variance anova method for high dimensional function approximation using jacobi polynomial chaos to represent the terms of the expansion first we develop a weight theory inspired by quasi monte carlo theory to identify which functions have low effective dimension using the anova expansion in different norms we then present estimates for the truncation error in the anova expansion and for the interpolation error using multielement polynomial chaos in the weighted korobov spaces over the unit hypercube we consider both the standard anova expansion using the lebesgue measure and the anchored anova expansion using the dirac measure the optimality of different sets of anchor points is also examined through numerical examples

legendre pseudospectral viscosity method for nonlinear conservation laws in this paper the legendre spectral viscosity sv method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied the authors prove that by adding a small amount of sv bounded solutions of the legendre sv method converge to the exact scalar entropy solution the convergence proof is based on compensated compactness arguments and therefore applies to certain 2 times 2 systems finally numerical experiments for scalar as well as the one dimensional system of gas dynamics equations are presented which confirm the convergence of the legendre sv method moreover these numerical experiments indicate that by post processing the sv approximation one can recover the entropy solution within spectral accuracy

synchronization of multi agent systems without connectivity assumptions multi agent systems arise from diverse fields in natural and artificial systems such as schooling of fish flocking of birds coordination of autonomous agents in multi agent systems a typical and basic situation is the case where each agent has the tendency to behave as other agents do in its neighborhood through computer simulations vicsek et al 1995 showed that such a simple local interaction rule can lead to a certain kind of cooperative phenomenon synchronization of the overall system if the initial states are randomly distributed and the size of the system population is large since this model is of fundamental importance in understanding the multi agent systems it has attracted much research attention in recent years in this paper we will present a comprehensive theoretical analysis for this class of multi agent systems under a random framework with large population but without imposing any connectivity assumptions as did in almost all of the previous investigations to be precise we will show that for any given and fixed model parameters concerning with the interaction radius r and the agents moving speed v the overall system will synchronize as long as the population size n is large enough furthermore to keep the synchronization property as the population size n increases both r and v can actually be allowed to decrease according to certain scaling rates

an efficient chebyshev algorithm for the solution of optimal control problems in this paper we derive an efficient chebyshev algorithm for solving optimal control problems the chebyshev expansions are employed to approximate both the control and the state functions the discretizing process and the related techniques are unique compared to existing methods the optimal control problems are transformed into the resulting mathematical programming problems theoretical analysis is given to support the method further numerical examples and comparisons are presented to illustrate the efficiency of the method

subspace based identification of infinite dimensional multivariable systems from frequency response data a new identification algorithm which identifies low complexity models of infinite dimensional systems from equidistant frequency response data is presented the new algorithm is a combination of the fourier transform technique with the recent subspace techniques given noisefree data finite dimensional systems are exactly retrieved by the algorithm when noise is present it is shown that identified models strongly converge to the balanced truncation of the identified system if the measurement errors are covariance bounded several conditions are derived on consistency illustrating the trade offs in the selection of certain parameters of the algorithm two examples are presented which clearly illustrate the good performance of the algorithm

state observability and observers of linear time invariant systems under irregular sampling and sensor limitations state observability and observer designs are investigated for linear time invariant systems in continuous time when the outputs are measured only at a set of irregular sampling time sequences the problem is primarily motivated by systems with limited sensor information in which sensor switching generates irregular sampling sequences state observability may be lost and the traditional observers may fail in general even if the system has a full rank observability matrix it demonstrates that if the original system is observable the irregularly sampled system will be observable if the sampling density is higher than some critical frequency independent of the actual time sequences this result extends shannon s sampling theorem for signal reconstruction under periodic sampling to system observability under arbitrary sampling sequences state observers and recursive algorithms are developed whose convergence properties are derived under potentially dependent measurement noises persistent excitation conditions are validated by designing sampling time sequences by generating suitable switching time sequences the designed state observers are shown to be convergent in mean square with probability one and with exponential convergence rates schemes for generating desired sampling sequences are summarized

on minimal cubature formulae of small degree for spherically symmetric integrals the aim of this paper is to develop the existence and nonexistence problem of a cubature formula of degree 4k 1 for a spherically symmetric integral which attains the moller lower bound for this purpose we bring together the theory of euclidean design in combinatorics and that of reproducing kernels in numerical analysis we show that if there exists a minimal formula of degree 4k 1 then the nodes are distributed into k 1 concentric spheres including the origin and the weights in the formula are a constant on each concentric sphere we particularly focus on the cases of degrees 5 and 9 we show the equivalence between a minimal formula of degrees 5 and 4 and a spherical tight 5 and 4 design we also prove that there exists no minimal formula of degree 9 for some classical weights such as the gaussian weight on mathbb r d the radial exponential weight on mathbb r d and the jacobi weight on b d

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

new method of order estimation for arma armax processes let the observation y k be generated by the multivariate arma process a z y k c z w k with unknown coefficients theta a theta c and orders p r where w k is a sequence of independent and identically distributed i i d random vectors with zero mean and unknown covariance matrix r w 0 a new method for estimating the orders p r is introduced in contrast to most of the existing results the new method is not based on optimizing a certain criterion and the order estimates given in the paper are rather easy to update computationally in comparison with the criterion optimization based methods when new data arrive the method is then extended to determining the orders of armax processes under reasonable conditions the estimates are proved to converge to the true orders with probability one as time tends to infinity

constructing cubature formulae by the method of reproducing kernel we examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently several new cubature formulae are derived

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

on the gibbs phenomenon iv recovering exponential accuracy in a subinterval from a gegenbauer partial sum of a piecewise analytic function we continue the investigation of overcoming gibbs phenomenon i e obtaining exponential accuracy at all points including at the discontinuities themselves from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function we show that if we are given the first n expansion coefficients of an l 2 function f x in terms of either the trigonometrical polynomials or the chebyshev or legendre polynomials we can construct an exponentially convergent approximation to the point values of f x in any sub interval in which it is analytic

on the gibbs phenomenon v recovering exponential accuracy from collocation point values of a piecewise analytic function this paper presents a method to recover exponential accuracy at all points including at the discontinuities themselves from the knowledge of an approximation to the interpolation polynomial or trigonometrical polynomial we show that if we are given the collocation point values or a highly accurate approximation at the gauss or gauss lobatto points we can reconstruct an uniform exponentially convergent approximation to the function f x in any sub interval of analyticity the proof covers the cases of fourier chebyshev legendre and more general gegenbauer collocation methods a numerical example is also provided

a multidomain spectral method for supersonic reactive flows this paper has a dual purpose it presents a multidomain chebyshev method for the solution of the two dimensional reactive compressible navier stokes equations and it reports the results of the application of this code to the numerical simulations of high mach number reactive flows in recessed cavity the computational method utilizes a newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations the results of the simulations are relevant to recessed cavity flame holders

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

spectral viscosity approximations to multidimensional scalar conservation laws the authors study the spectral viscosity sv method in the context of multidimensional scalar conservation laws with periodic boundary conditions they show that the spectral viscosity which is sufficiently small to retain the formal spectral accuracy of the underlying fourier approximation is large enough to enforce the correct amount of entropy dissipation which is otherwise missing in the standard fourier method moreover they prove that because of the presence of the spectral viscosity the truncation error in this case becomes spectrally small independent of whether the underlying solution is smooth or not consequently the sv approximation remains uniformly bounded and converges to a measure valued solution satisfying the entropy condition that is the unique entropy solution they also show that the sv solution has a bounded total variation provided that the total variation of the initial data is bounded thus confirming its strong convergence to the entropy solution they obtain an l sup 1 convergence rate of the usual optimal order one half 22 refs

simultaneous identification and stabilization of nonlinearly parameterized discrete time systems by nonlinear least squares algorithm this paper addresses the challenging problem of designing an adaptive feedback strategy for simultaneous identification and stabilization for a class of nonlinearly parameterized uncertain systems in discrete time the nonlinear least squares nls algorithm is applied to estimate the unknown parameters and it turns out to be the standard least squares ls algorithm whenever the model is linearly parameterized based on this algorithm both the strong consistency of the estimator and the global stability of the system are achieved with the output feedback design for the scalar parameter case

constructing cubature formulae of degree 5 with few points this paper is devoted to construct a family of fifth degree cubature formulae for n cube with symmetric measure and n dimensional spherically symmetrical region the formula forn cube contains at most n 2 5n 3 points and for n dimensional spherically symmetrical region contains only n 2 3n 3 points moreover the numbers can be reduced to n 2 3n 1 and n 2 n 1 if n 7 respectively the latter of which is minimal

convergence rate of least squares identification and adaptive control for stochastic systems the strong consistency and the convergence rate of least squares identification for the multidimensional armax model are established under some decaying excitation conditions which are satisfied if both input and output do not grow too fast and the attenuating excitation technique is applied the parameter identification results are applied to adaptive control systems with a quadratic loss function the rate of convergence of the loss function to its minimum is also obtained

extended least squares and their applications to adaptive control and prediction in linear systems herein strong consistency of recursive extended least squares is established under considerably weaker assumptions than previously assumed in the literature the argument used to establish consistency also leads to certain basic properties of adaptive predictors based on these recursive estimators making use of these properties of the adaptive predictors simple modifications of the astrom wittenmark self tuning regulator are proposed and shown to be asymptotically optimal

least squares identification for armax models without the positive real condition recursive identification problems of linear stochastic feedback control systems described by armax models are studied without imposing the strictly positive real condition on the noise model an increasing lag least squares is used in the first step to estimate the noise process while the parameter estimate is formed in the second step by an extended least squares in the algorithm there is an increase in computational cost in comparison to traditional algorithms however the results of this note do not need any a priori information or conditions on the noise model except that of stability and they are applicable to the identification of general feedback control systems

approximate solution of the sturm liouville problems with legendre galerkin chebyshev collocation method abstract in this paper we use the legendre galerkin chebyshev collocation method lgcc to approximate the eigenvalues of a regular sturm liouville problem with three kinds of boundary conditions unlike the classical chebyshev collocation method this method preserves the symmetry of the problem numerical results show that the method is of high accuracy

a new approach for the existence problem of minimal cubature formulas based on the larman rogers seidel theorem in this paper we consider the existence problem of minimal cubature formulas of degree 4k 1 for spherically symmetric integrals we prove that for a minimal formula in sufficiently high dimensional space there exists some concentric sphere on which the inner product of any two distinct points is rational by using this result we prove that for any d ge 2 there exist no d dimensional minimal formulas of degrees 13 and 21 for some special integral

some constructions of spherical 5 designs abstract spherical designs were introduced by delsarte goethals and seidel in 1977 a spherical t design in rn is a finite set x sn 1 with the property that for every polynomial p with degree t the average value of p on x equals the average value of p on sn 1 this paper contains some existence and nonexistence results mainly for spherical 5 designs in r3 delsarte goethals and seidel proved that if x is a spherical 5 design in r3 then x 12 and if xz sfnc 12 then x consists of the vertices of a regular icosahedron we show that such designs exist with cardinality 16 18 20 22 24 and every integer 26 if x is a spherical 5 design in rn then x n n 1 if x n n 1 then x has been called tight tight spherical 5 designs in rn are known to exist only for n 2 3 7 23 and possibly n u2 2 for odd u 7 any tight spherical 5 design in rn must consist of n n 1 2 antipodal pairs of points we show that for n 3 there are no spherical 5 designs in rn consisting of n n 1 2 1 antipodal pairs of points

optimal error estimates of the chebyshev legendre spectral method for solving the generalized burgers equation in this paper the chebyshev legendre collocation method is applied to the generalized burgers equation optimal error estimate of the method is proved for the problem with the dirichlet boundary conditions also a legendre galerkin chebyshev collocation method is given for the generalized burgers equation the scheme is basically formulated in the legendre spectral form but with the nonlinear term being treated by the chebyshev collocation method so that the scheme can be implemented at chebyshev gauss lobatto points efficiently optimal order convergence is also obtained through coupling estimates in the l2 norm and the h1 norm

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

on one sided filters for spectral fourier approximations of discontinuous functions in this paper the existence of one sided filters is proved for spectral fourier ap proximations of discontinuous functions which can recover spectral accuracy up to the discontinuity from one side a least square procedure is used to construct such a filter and is tested on several discontinuous functions numerically

spectral vanishing viscosity method for nonlinear conservation laws we propose a new spectral viscosity sv scheme for the accurate solution of nonlinear conservation laws it is proved that the sv solution converges to the unique entropy solution under appropriate reasonable conditions the proposed sv scheme is implemented directly on high modes of the computed solution this should be compared with the original nonperiodic sv scheme introduced by maday ould kaber and tadmor in siam j numer anal 30 1993 321 342 where sv is activated on the derivative of the sv solution the new proposed sv method could be viewed as a correction of the former and it offers an improvement which is confirmed by our numerical experiments a postprocessing method is implemented to greatly enhance the accuracy of the computed sv solution the numerical results show the efficiency of the new method

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows