finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

subgrid modeling in a galerkin method for the helmholtz equation abstract we improve the accuracy of finite element acoustic wave solutions without introducing additional degrees of freedom with physically inspired subgrid models the models explored here will not adversely impact system matrix storage or bandwidth can be used for structured or unstructured meshes do not require the solution of any auxiliary value problems and have minimal marginal overall computational cost one interpretation of the new models is as momentum transferred to and from modes at length scales below the local mesh dimension we show a method to approximate the effects of all subgrid modes on the grid scale solution and the significant improvements in local accuracy that the new models provide the new subgrid models also improve global or pollution accuracy through a reduction in the element dispersion error in addition we draw comparisons between this proposed method and much of the alternative and related work in this area

on the pollution effect in fe solutions of the 3d helmholtz equation we study convergence of the fem for a 3d problem of rigid scattering using separation of variables analytical results are presented for stability and error estimation for an h version galerkin fem with mesh refinement in radial direction we prove an upper bound that contains a pollution term similar to previous results in 1d and 2d this estimate is evaluated computationally using a 3d fem iem infinite elements implementation the tabulated and plotted errors show the theoretically predicted pollution effect in the fe error

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

numerical solution of the helmholtz equation with high wavenumbers this paper investigates the pollution effect and explores the feasibility of a local spectral method the discrete singular convolution dsc algorithm for solving the helmholtz equation with high wavenumbers fourier analysis is employed to study the dispersive error of the dsc algorithm our analysis of dispersive errors indicates that the dsc algorithm yields a dispersion vanishing scheme the dispersion analysis is further confirmed by the numerical results for one and higher dimensional helmholtz equations the dsc algorithm is shown to be an essentially pollution free scheme furthermore for large scale computation the grid density of the dsc algorithm can be close to the optimal two grid points per wavelength the present study reveals that the dsc algorithm is accurate and efficient for solving the helmholtz equation with high wavenumbers copyright 2003 john wiley sons ltd

neural network for nonsmooth nonconvex constrained minimization via smooth approximation a neural network based on smoothing approximation is presented for a class of nonsmooth nonconvex constrained optimization problems where the objective function is nonsmooth and nonconvex the equality constraint functions are linear and the inequality constraint functions are nonsmooth convex this approach can find a clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation the global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded specially the proposed network does not require 1 the initial point to be feasible 2 a prior penalty parameter to be chosen exactly 3 a differential inclusion to be solved numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network

fast direct solution of the helmholtz equation with a perfectly matched layer or an absorbing boundary condition we consider the efficient numerical solution of the helmholtz equation in a rectangular domain with a perfectly matched layer pml or an absorbing boundary condition abc standard bilinear trilinear finite element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction type fast direct solver we present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a pml and we optimize certain parameters of the layer to minimize the reflection copyright 2003 john wiley sons ltd

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

a survey of finite element methods for time harmonic acoustics many of the current issues and methodologies related to finite element methods for time harmonic acoustics are reviewed the effective treatment of unbounded domains is a major challenge most prominent among the approaches that have been developed for this purpose are absorbing boundary conditions infinite elements and absorbing layers standard computational methods are unable to cope with wave phenomena at short wave lengths due to resolutions required to control dispersion and pollution errors leading to prohibitive computational demands since computation naturally separates the scales of a problem according to the mesh size multiscale considerations provide a useful framework for viewing these difficulties and developing methods to counter them other issues addressed are related to the efficient solution of systems of specialized algebraic equations and inverse problems of acoustics the tremendous progress that has been made in all of the above areas in recent years will surely continue leading to many more exciting developments

killing tensors warped products and the orthogonal separation of the hamilton jacobi equation we study killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the hamilton jacobi equation this work is motivated primarily by the case of spaces of constant curvature where warped products are abundant we first characterize killing tensors which have a natural algebraic decomposition in warped products we then apply this result to show how one can obtain the killing stackel space ks space for separable coordinate systems decomposable in warped products this result in combination with benenti s theory for constructing the ks space of certain special separable coordinates can be used to obtain the ks space for all orthogonal separable coordinates found by kalnins and miller in riemannian spaces of constant curvature next we characterize when a natural hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by benenti can be reduced finally we use this characterization and concircular tensors a special type of torsionless conformal killing tensor to develop a general algorithm to determine when a natural hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature

a discontinuous finite element formulation for helmholtz equation abstract a discontinuous finite element formulation for helmholtz problems is presented with c 0 continuity across inter element boundaries enforced in a weak sense depending on two parameters and whose choice is crucial for the performance of the proposed method these parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error optimal values of these parameters are then adopted in more general situations the accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions

solution of 3d laplace and helmholtz equations in exterior domains using hp infinite elements abstract this work is devoted to a convergence study for infinite element discretizations for laplace and helmholtz equations in exterior domains the proposed approximation applies to separable geometries only combining an hp fe discretization on the boundary of the domain with a spectral like representation resulting from the separation of variables in the radial direction the presentation includes a convergence proof for the laplace equation and a stability analysis for the variational formulation of the helmholtz equation in weighted sobolev spaces the theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain

smoothing projected gradient method and its application to stochastic linear complementarity problems a smoothing projected gradient spg method is proposed for the minimization problem on a closed convex set where the objective function is locally lipschitz continuous but nonconvex nondifferentiable we show that any accumulation point generated by the spg method is a stationary point associated with the smoothing function used in the method which is a clarke stationary point in many applications we apply the spg method to the stochastic linear complementarity problem slcp and image restoration problems we study the stationary point defined by the directional derivative and provide necessary and sufficient conditions for a local minimizer of the expected residual minimization erm formulation of slcp preliminary numerical experiments using the spg method for solving randomly generated slcp and image restoration problems of large sizes show that the spg method is promising

classification of the orthogonal separable webs for the hamilton jacobi and laplace beltrami equations on 3 dimensional hyperbolic and de sitter spaces we review the theory of orthogonal separation of variables on pseudo riemannian manifolds of constant non zero curvature via concircular tensors and warped products we then apply this theory simultaneously to both the three dimensional hyperbolic and de sitter spaces obtaining an invariant classification of the thirty four orthogonal separable webs on each space modulo action of the respective isometry groups the inequivalent coordinate charts adapted to each web are also determined and listed the results obtained for hyperbolic 3 space agree with those in the literature while the results for de sitter 3 space appear to be new we review the theory of orthogonal separation of variables on pseudo riemannian manifolds of constant non zero curvature via concircular tensors and warped products we then apply this theory simultaneously to both the three dimensional hyperbolic and de sitter spaces obtaining an invariant classification of the thirty four orthogonal separable webs on each space modulo action of the respective isometry groups the inequivalent coordinate charts adapted to each web are also determined and listed the results obtained for hyperbolic 3 space agree with those in the literature while the results for de sitter 3 space appear to be new

a parallel fictitious domain method for the three dimensional helmholtz equation the application of a fictitious domain domain embedding method to the three dimensional helmholtz equation with absorbing boundary conditions is considered the finite element discretization is performed by using locally fitted meshes and an algebraic fictitious domain method with a separable preconditioner is used in the iterative solution of the resultant linear systems such a method is based on embedding the original domain into a larger one with a simple geometry with this approach it is possible to realize the gmres iterations in a low dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner an efficient parallel implementation of the iterative algorithm is introduced results of numerical experiments demonstrate good scalability properties on distributed memory parallel computers and the ability to solve high frequency acoustic scattering problems

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

lipschitz behavior of the robust regularization to minimize or upper bound the value of a function robustly we might instead minimize or upper bound the epsilon robust regularization defined as the map from a point to the maximum value of the function within an epsilon radius this regularization may be easy to compute convex quadratics lead to semidefinite representable regularizations for example and the spectral radius of a matrix leads to pseudospectral computations for favorable classes of functions we show that the robust regularization is lipschitz around any given point for all small epsilon 0 even if the original function is non lipschitz like the spectral radius one such favorable class consists of the semi algebraic functions such functions have graphs that are finite unions of sets defined by finitely many polynomial inequalities and are commonly encountered in applications

a superlinearly convergent method for a class of complementarity problems with non lipschitzian functions we consider a class of complementarity problems involving functions which are not lipschitz continuous in this paper we reformulate this class of non lipschitzian complementarity problems into a lipschitzian complementarity problem then we propose an inexact smoothing newton method to solve this lipschitzian complementarity problem we prove that our proposed method converges quadratically and globally under a mild condition numerical results show that this method is promising this method can solve these kinds of complementarity problems with one million variables in reasonable time on a pc with 1 gb of ram

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

three dimensional dispersion analysis and stabilized finite element methods for acoustics abstract galerkin least squares and galerkin gradient least squares stand out among several approaches designed to improve the numerical solution accuracy and counteract the pollution effect by adding terms to the standard galerkin formulation these added terms are multiplied by a stability parameter which must be properly defined in this paper an original three dimensional dispersion analysis is performed for the helmholtz equation together with the determination of the three dimensional stability parameters for structured and unstructured meshes numerical experiments show the relative efficiency of the proposed methods for solving acoustic problems arising from industry

classification of hamilton jacobi separation in orthogonal coordinates with diagonal curvature we find all orthogonal metrics where the geodesic hamilton jacobi equation separates and the riemann curvature tensor satisfies a certain equation called the diagonal curvature condition all orthogonal metrics of constant curvature satisfy the diagonal curvature condition the metrics we find either correspond to a benenti system or are warped product metrics where the induced metric on the base manifold corresponds to a benenti system furthermore we show that most metrics we find are characterized by concircular tensors these metrics called kalnins eisenhart miller metrics have an intrinsic characterization which can be used to obtain them on a given space in conjunction with other results we show that the metrics we found constitute all separable metrics for riemannian spaces of constant curvature and de sitter space

first order conditions for nonsmooth discretized constrained optimal control problems this paper studies first order conditions karush kuhn tucker conditions for discretized optimal control problems with nonsmooth constraints we present a simple condition which can be used to verify that a local optimal point satisfies the first order conditions and that a point satisfying the first order conditions is a global or local optimal solution of the optimal control problem

the discontinuous enrichment method for multiscale analysis computation naturally separates scales of a problem according to the mesh size a variety of improved numerical methods are described by multiscale considerations differing in the treatment of the unresolved fine scales the discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework the method thus combines relative ease of implementation with improved numerical performance these properties are demonstrated for both multiscale wave and transport problems pointing to the potential of considerable savings in computational resources

remarks on the connection between the additive separation of the hamilton jacobi equation and the multiplicative separation of the schrodinger equation i the completeness and robertson conditions the fundamental elements of the variable separation theory are revisited including the eisenhart and robertson theorems kalnins miller theory and the intrinsic characterization of the separation of the hamilton jacobi equation in a unitary and geometrical perspective the general notion of complete integrability of first order normal systems of pdes leads in a natural way to completeness conditions for separated solutions of the schrodinger equation and to the robertson condition two general types of multiplicative separation for the schrodinger equation are defined and analyzed they are called free and reduced separation respectively in the free separation the coordinates are necessarily orthogonal while the reduced separation may occur in nonorthogonal coordinates but only in the presence of symmetries killing vectors

spectral approximation of the helmholtz equation with high wave numbers a complete error analysis is performed for the spectral galerkin approximation of a model helmholtz equation with high wave numbers the analysis presented in this paper does not rely on the explicit knowledge of continuous discrete green s functions and does not require any mesh condition to be satisfied furthermore new error estimates are also established for multi dimensional radial and spherical symmetric domains illustrative numerical results in agreement with the theoretical analysis are presented

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

a fourth order magnus scheme for helmholtz equation for wave propagation in a slowly varying waveguide it is necessary to solve the helmholtz equation in a domain that is much larger than the typical wavelength standard finite difference and finite element methods must resolve the small oscillatory behavior of the wave field and are prohibitively expensive for practical applications a popular method is to approximate the waveguide by segments that are uniform in the propagation direction and use separation of variables in each segment for a slowly varying waveguide it is possible that the length of such a segment is much larger than the typical wavelength to reduce memory requirements it is advantageous to reformulate the boundary value problem of the helmholtz equation as an initial value problem using a pair of operators such an operator marching scheme can also be solved with the piecewise uniform approximation of the waveguide this is related to the second order midpoint exponential method for a system of linear odes in this paper we develop a fourth order operator marching scheme for the helmholtz equation using a fourth order magnus method

separation of variables for complex riemannian spaces of constant curvature i orthogonal separable coordinates for s _n c and e _n c a complete classification of all orthogonal coordinate systems that admit a separation of variables for the null hamilton jacobi equation in conformally flat complex riemannian spaces is presented this is a first step towards the complete solution of the problem for complex riemannian spaces when in general the coordinates need not be orthogonal a detailed prescription for constructing all such orthogonal coordinate systems is presented

how to deal with the unbounded in optimization theory and algorithms the aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions but also for constructing new methods

equivalence problem for the orthogonal webs on the 3 sphere we solve the equivalence problem for the orthogonally separable webs on the 3 sphere under the action of the isometry group this continues a classical project initiated by olevsky in which he solved the corresponding canonical forms problem the solution to the equivalence problem together with the results by olevsky forms a complete solution to the problem of orthogonal separation of variables for the hamilton jacobi equation defined on the 3 sphere via orthogonal separation of variables it is based on invariant properties of characteristic killing tensors in addition to properties of the corresponding algebraic curvature tensor and the associated ricci tensor the result is illustrated by a nontrivial application to a natural hamiltonian defined on the 3 sphere

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem