bootstrap inference in partially identified models defined by moment inequalities coverage of the identified set this paper introduces a novel bootstrap procedure to perform inference in a wide class of partially identi ed econometric models we consider econometric models de ned by nitely many weak moment inequalities y which encompass many applications of economic interest the objective of our inferential procedure is to cover the identi ed set with a prespeci ed probability z we compare our bootstrap procedure a competing asymptotic approximation and subsampling proce dures in terms of the rate at which they achieve the desired coverage level also known as the error in the coverage probability under certain conditions we show that our bootstrap procedure and the asymp totic approximation have the same order of error in the coverage probability which is smaller than the one obtained by using subsampling this implies that inference based on our bootstrap and asymptotic approximation should eventually be more precise than inference based on subsampling a monte carlo study con rms this nding in a small sample simulation

a hamiltonian preserving discontinuous galerkin method for the generalized korteweg de vries equation the invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations in order to minimize phase and amplitude errors after long time simulation in this paper we design analyze and numerically validate a hamiltonian preserving discontinuous galerkin method for solving the korteweg de vries kdv equation for the generalized kdv equation the semi discrete formulation is shown to preserve both the first and the third conserved integrals and approximately preserve the second conserved integral for the linearized kdv equation all the first three conserved integrals are preserved and optimal error estimates are obtained for polynomials of even degree the preservation properties are also maintained by the fully discrete dg scheme our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized kdv equation we also show that the shape of the solution after long time simulation is well preserved due to the hamiltonian preserving property

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

the pseudospectral method for third order differential equations generalized quadrature rules are derived which assist in the selection of collocation points for the pseudospectral solution of differential equations in particular it is shown that for an nth order differential equation in one space dimension with two point derivative boundary conditions an ideal choice of interior collocation points is the set of zeros of a jacobi polynomial the pseudospectral solution of a third order initial boundary value problem is considered and accuracy is assessed by examining how well the discrete eigenproblem approximates the continuous one convergence is established for a special choice of collocation points and numerical results are included to demonstrate the viability of the approach

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

inference for parameters defined by moment inequalities a recommended moment selection procedure this paper is concerned with tests and confidence intervals for parameters that are not necessarily point identified and are defined by moment inequalities in the literature different test statistics critical value methods and implementation methods i e the asymptotic distribution versus the bootstrap have been proposed in this paper we compare these methods we provide a recommended test statistic moment selection critical value and implementation method we provide data dependent procedures for choosing the key moment selection tuning parameter and a size correction factor

a new stream function formulation for the steady euler equations a new stream function formulation is presented for the solution of euler equations in the transonic flow region in this method the stream function and the density are the dependent variables the governing equations for adiabatic flow are then the momentum equations which are solved in the strong conservation law form the method differs from the recent formulation of hafez and lovell which is based on crocco s equation in that it does not require knowledge of the vorticity to study arbitrary geometries the algorithm is combined with the automatic grid solver grape of steger and sorenson results are presented for an naca 0012 airfoil at various mach numbers angles of attack and cylinders comparisons with other solutions of the euler equations show good agreement

the galerkin method for the kdv equation using a new basis of smooth piecewise cubic polynomials abstract in this paper we introduce the galerkin method using a new basis of smooth piecewise cubic polynomials for the one dimensional nonlinear kdv equation the stability analysis shows that the present method is unconditionally stable in l 2 norm numerical experiments show that the proposed method preserves the conservation laws and is effective for simulating the motions and the interactions of solitary waves

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

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

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

convergence analysis of spectral collocation methods for a singular differential equation solutions of partial differential equations with coordinate singularities often have special behavior near the singularities which forces them to be smooth special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency it has been observed numerically in the past that for a scheme to attain high accuracy it is unnecessary to impose all the pole conditions the constraints representing the special solution behavior near singularities in this paper we provide a theoretical justification for this observation specifically we consider an existing approach which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used gauss lobatto collocation scheme spectral convergence of the legendre and chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries

optimal error estimates in jacobi weighted sobolev spaces for polynomial approximations on the triangle spectral approximations on the triangle by orthogonal polynomials are studied in this paper optimal error estimates in weighted semi norms for both the l2 and h 1 0 orthogonal polynomial projections are established by using the generalized koornwinder polynomials and the properties of the sturm liouville operator on the triangle these results are then applied to derive error estimates for the spectral galerkin method for second and fourth order equations on the triangle the generalized koornwinder polynomials and approximation results developed in this paper will be useful for many other applications involving spectral and spectral element approximations in triangular domains

a hybrid numerical method for the kdv equation by finite difference and sinc collocation method abstract in this paper we propose a hybrid numerical method for the kdv equation more precisely we discretize the temporal derivative of kdv equation by a weighted scheme and treat the implicitly nonlinear term with the combination of finite difference and sinc collocation method the stability analysis is presented and numerical experiments illustrate the efficiency and stability of the proposed hybrid method the motions and the interactions of solitary waves relying on particular initial boundary conditions are also simulated

a local discontinuous galerkin method for the korteweg de vries equation with boundary effect a local discontinuous galerkin method for solving korteweg de vries kdv type equations with non homogeneous boundary effect is developed we provide a criterion for imposing appropriate boundary conditions for general kdv type equations the discussion is then focused on the kdv equation posed on the negative half plane which arises in the modeling of transition dynamics in the plasma sheath formation h liu m slemrod kdv dynamics in the plasma sheath transition appl math lett 17 4 2004 401 410 the guiding principle for selecting inter cell fluxes and boundary fluxes is to ensure the l 2 stability and to incorporate given boundary conditions the local discontinuous galerkin method thus constructed is shown to be stable and efficient numerical examples are given to confirm the theoretical result and the capability of this method for capturing soliton wave phenomena and various boundary wave patterns

numerical methods for the solution of the third and fifth order dispersive korteweg de vries equations abstract two numerical methods are proposed for the solution of the third and fifth order korteweg de vries equations the first method is derived using central differences to replace the space derivatives with a predictor corrector time stepping and the second method by linearizing the implicit corrector scheme in which the solution is then found by solving a linear algebraic system at each time step rather than a nonlinear algebraic system which is more usual an important advantage to be gained from the use of the linearized implicit method over the predictor corrector method which is optimally stable is the ability to vary the mesh length the methods are analysed with respect to stability criteria and numerical dispersion numerical results portraying a single soliton solution and the interaction of more than one soliton are reported for the third order korteweg de vries equation numerical results for the fifth order korteweg de vries equation using the linearized implicit method are also reported

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

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

numerical solution of transonic stream function equation the stream function equation in conservation form looks similar to the full potential equation and existing methods e g artificial compressibility can be readily applied rotational flows can be calculated once the vorticity due to shocks or nonuniformity is evaluated there are however two main difficulties first the density is not uniquely determined in terms of the flux there are two solutions the subsonic and the supersonic branch with a square root singularity at the sonic point methods to overcome this difficulty are studied and results are presented with some remarks on inviscid separation and closed stream lines second the need of two stream functions for three dimensional calculations is briefly discussed

consistent moment selection procedures for generalized method of moments estimation this paper considers a generalized method of moments gmm estimation problem in which one has a vector of moment conditions some of which are correct and some incorrect the paper introduces several procedures for consistently selecting the correct moment conditions the procedures also can consistently determine whether there is a sufficient number of correct moment conditions to identify the unknown parameters of interest the paper specifies moment selection criteria that are gmm analogues of the widely used bic and aic model selection criteria the latter is not consistent the paper also considers downward and upward testing procedures all of the moment selection procedures discussed in this paper are based on the minimized values of the gmm criterion function for different vectors of moment conditions the procedures are applicable in time series and cross sectional contexts application of the results of the paper to instrumental variables estimation problems yields consistent procedures for selecting instrumental variables

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

transonic flow calculations using the euler equations an implicit finite difference method with implicit boundary conditions is employed to solve the steady euler equations for flows past arbitrary geometries the resulting code is used to investigate in a systematic way various aspects of flow past airfoils at transonic speeds such as method of solution boundary conditions grid stretching and generation shock and sonic point operators the kiitta condition and smoothing results ob tained are in good agreement with results of other codes moreover it appears that the method of solution employed is such that kutta s condition need not be invoked this statement appears to be valid for other existing schemes employed in the solution of euler equations he primitive variable form of the system of partial differential equations governing the steady motion of an inviscid fluid euler equations are first order and of mixed elliptic and hyperbolic type because of this two approaches are used in obtaining steady state solutions for subcritical and supercritical flows past bodies the first seeks temporally asymptotic solutions of the time dependent equations1 7 while the second exploits relaxation methods developed for the solution of second order equations by embedding euler equations in a second order system8 or by using a stream function formulation 9 the purpose of this investigation is to develop an efficient algorithm for the steady euler equations in primitive variable form and use it to carry out numerical experiments designed to investigate in a systematic way the various aspects of flows past airfoils at transonic speeds these include method of solution i e peaceman rachford10 pr vs douglas gunn1 1 dg boundary conditions grid generation and stretching shock and sonic point operators smoothing and the kutta condition the resulting algorithm solves the strong con servation form of the euler equations with the energy equation replaced by the statement that the total enthalpy is constant using an adi finite difference scheme and implicit boundary conditions to study arbitrary geometries the algorithm is combined with the automatic grid solver grape of steger and sorenson 12 with this capability the code provides a general and efficient method for the solution of flowfields past arbitrary geometries for all speed ranges the achieved efficiency of this code can be traced to two main reasons the resulting block tridiagonal matrix for two dimensional flows is 3 x 3 and not 4 x 4 moreover the code incorporates a procedure that selects an optimum relaxation factor for each iteration formulation of the problem

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

confidence sets for partially identified parameters that satisfy a finite number of moment inequalities this paper proposes a computationally simple way to construct confidence sets for a parameter of interest in models comprised of moment inequalities building on results from the literature on multivariate one sided tests i show how to test the hypothesis that any particular parameter value is logically consistent with the maintained moment inequalities the associated test statistic has an asymptotic chi bar square distribution and can be inverted to construct an asymptotic confidence set for the parameter of interest even if that parameter is only partially identified critical values for the test are easily computed and a monte carlo study demonstrates implementation and finite sample performance

a new bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis in this paper a new matrix approach for solving second order linear partial differential equations pdes under given initial conditions has been proposed the basic idea includes integrating from the considered pdes and transforming them to the associated integro differential equations with partial derivatives therefore bernoulli operational matrices of differentiation and integration together with the completeness of bernoulli polynomials can be used for transforming integro differential equations to the corresponding algebraic equations a rigorous error analysis in the infinity norm is given provided that the known functions and the exact solution are sufficiently smooth and bounded a numerical example is included to demonstrate the validity and the applicability of the technique the results confirm the theoretical prediction

efficient spectral petrov galerkin methods for the integrated forms of third and fifth order elliptic differential equations using general parameters generalized jacobi polynomials abstract this article analyzes some algorithms for solving numerically the integrated forms of third and fifth order differential equations in one variable subject to homogenous and nonhomogeneous boundary conditions using a dual petrov galerkin method two new families of general parameters generalized jacobi polynomials are introduced and used for this purpose numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

a direct discontinuous galerkin method for the generalized korteweg de vries equation energy conservation and boundary effect a direct discontinuous galerkin method for solving the generalized korteweg de vries kdv equation with both periodic data and non homogeneous boundary data is developed a class of numerical fluxes is identified so that the method preserves both momentum and energy for the initial value problem the method with such a property is numerically shown robust with less error in both phase and amplitude after long time simulation numerical examples are given to confirm the theoretical result and the capacity of this method for capturing soliton wave phenomena and various boundary wave patterns

integration preconditioning of pseudospectral operators i basic linear operators this paper develops a family of preconditioners for pseudospectral approximations of pth order linear differential operators subject to various types of boundary conditions the approximations are based on ultraspherical polynomials with special attention being paid to legendre and chebyshev polynomial methods based on gauss lobatto quadrature points the eigenvalue spectrum of the preconditioned operators are obtained in closed analytic form and the weakly enforced boundary conditions are shown to result in a rank 2p perturbation of the identity operator i e the majority of the preconditioned eigenvalues are unity the spectrum of the preconditioned advective operator is shown to be bounded independent of the order of the approximation n however the preconditioned diffusive operator is in general indefinite with four real eigenvalues for dirichlet boundary conditions the spectral radius grows as sqrt n while it scales as n for the case of neumann boundary conditions these results are shown to be asymptotically optimal within the present framework generalizations to higher order differential operators general boundary conditions and arbitrary polynomial basis and quadrature nodes are discussed

fourier methods with extended stability intervals for the korteweg de vries equation a full leap frog fourier method for integrating the korteweg de vries kdv equation u t uu x varepsilon u xxx 0 results in an o n 3 stability constraint on the time step where n is the number of fourier modes used this stability limit is much more restrictive than the accuracy limit for many applications in this paper we propose a method for which the stability limit is extended by treating the linear dispersive u xxx term implicitly thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit the implicit method is implemented without solving linear systems by integrating in time in the fourier space and discretizing the nonlinear uu x term by leap frog a second method we propose uses basis functions which solve the linear part of the kdv equation and leap frog for time integration a linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

numerical solution of stochastic fractional integro differential equation by the spectral collocation method fractional calculus is used to model various different phenomena in nature today the aim of this paper is to propose the shifted legendre spectral collocation method to solve stochastic fractional integro differential equations sfides in this approach we consider the p panels m point newtoncotes rules with m fixed for estimating it integrals the main characteristic of the presented method is that it reduces sfides into a system of algebraic equations thus we can solve the problem by newtons method furthermore the convergence analysis of the approach is considered the method is computationally attractive and numerical examples confirm the validity and efficiency of the proposed method