statistical fitting and validation of non linear simulation metamodels a case study abstract linear regression metamodels have been widely used to explain the behavior of computer simulation models although they do not always provide a good global fit to smooth response functions of arbitrary shape in the case study discussed in this paper the use of several linear regression polynomial results in a poor fit the use of a non linear regression metamodeling methodology provides simple functions that adequately approximate the behavior of the target simulation model the importance of metamodel validation is emphasized by using the generalization of rao s test to non linear metamodels and double cross validation

hierarchical divergence conforming nedelec elements for volumetric cells a new family of hierarchical divergence conforming vector bases is proposed for tetrahedral prism and brick cells these functions span the divergence conforming reduced curl spaces of nedelec the bases are constructed from orthogonal scalar polynomials to enhance their linear independence which is a simpler process than an orthogonalization applied to the final vector functions specific functions are tabulated to order 6 5 these basis are intended for use with numerical solutions of volume integral equations or differential equations containing divergence operators

p adaptive boundary elements this paper presents the implementation of an adaptive philosophy to plane potential problems using the direct boundary element method after some considerations about the state of the art and a discussion of the standard approach features the possibility of separately treating the modelling of variables and their interpolation through hierarchical shape functions is analysed then the proposed indicators and estimators are given followed by a description of a small computer program written for an ibm pc finally some examples show the kind of results to be expected

the generalized method of moments for electromagnetic boundary integral equations the generalized method of moments gmm is a partition of unity based technique for solving electromagnetic and acoustic boundary integral equations past work on gmm for electromagnetics was confined to geometries modeled by piecewise flat tessellations and suffered from spurious internal line charges in the present article we redesign the gmm scheme and demonstrate its ability to model scattering from pec scatterers composed of mixtures of smooth and non smooth geometrical features furthermore we demonstrate that because the partition of unity provides both functional and effective geometrical continuity between patches gmm permits mixtures of local geometry descriptions and approximation function spaces with significantly more freedom than traditional moment methods

restricted subset selection procedures for simulation this paper develops two procedures for screening a set of normal populations with unknown moments in order that the final subset of selected populations satisfies the following requirements involving the user specified parameters p and m with probability at least p the selected subset will contain a population whose mean lies less than the distance from the largest mean although the size of the selected subset is random at most m populations will finally be chosen where m is usually taken small enough to reserve adequate resources for a more intensive follow up study the exact procedure ve is a two stage random sampling scheme that is designed to compare transient or steady state simulation models based on independent replications of each model the heuristic procedure vs is a modification of ve that is designed to compare steady state simulation models based on a single prolonged run of each model a complete justification is given for ve together with tables of the constants required by bo

symmetries and constants of motion for new hierarchy of the kp equations abstract two sets of infinite numbers of symmetries and constants of motion for a class of 2 1 dimensional nonlinear evolution equations nees are derived these nees we call them the new hierarchy of the kp equations are generated by a set of additional symmetries of the kp and higher order kp equations and explicitly depend on the space variables x and y the expressions for their constants of motion in terms of the scattering data are given and the time evolution of the scattering data corresponding to the new hierarchy of the kp equations are computed

bi hamiltonian formulation of the kadomtsev petviashvili and benjamin ono equations it was shown recently that the kadomtsev petviashvili kp equation an integrable equation in 2 1 i e in two spatial and one temporal dimensions admits a bi hamiltonian formulation this was achieved by considering kp as a reduction of a 3 1 dimensional system in the variables x y1 y2 t it is shown here using the kp as a concrete example that equations in 2 1 possess two bi hamiltonian formulations and two recursion operators both hamiltonian operators associated with the x direction are local in contrast only one of the hamiltonian operators associated with the y direction is local furthermore using the benjamin ono equation as a concrete example it is shown that intergrodifferential equations in 1 1 admit an algebraic formulation analogous to that of equations in 2 1

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

minimizing the bias and variance of the gradient estimate in rsm simulation studies abstract to minimize the variance of the response estimator in response surface methodology rsm based simulations most users extend the design points to the limits of the subregion under investigation our analysis however shows that when common pseudorandom numbers are used this practice likely increases the variance of the estimated gradient under the criterion of the variance of the gradient and the mahalanobis distance function we show that smaller subregions reduce both the bias and the variance of the estimated gradient furthermore we discuss how users may take advantage of the magnitude of the induced correlations to draw inferences about the search process and to manipulate the subregion size our analysis offers a new direction for future research

explicit local error estimators for electromagnetic integral equations several local cell by cell error estimation techniques for use with the electric field integral equation efie for te to z polarization are proposed these error estimators are evaluated for use in adaptive refinement in the locally corrected nystrom lcn solution of the efie for various problems with localized high error regions the error estimation schemes are shown to correctly identify the high error regions

nonlinear superposition formula of the novikov veselov equation a nonlinear superposition formula of the novikov veselov equation is proved under certain conditions some particular solutions of the novikov veselov equation are given as an illustrative application of the obtained result

adaptive h p and hp versions for boundary integral element methods recently very promising results in a so called hp version of the finite element method have been obtained the basic idea is a balanced combination of mesh refinement and increase of the polynomial degree of the shape functions this idea is applied to a boundary collocation method in this paper the new method is compared with adaptive h and p versions and it is shown in numerical examples that even in the presence of singularities in the exact solution exponential convergence is obtained

applications of differential forms to boundary integral equations in this paper we discuss the application of differential forms to integral equations arising in the study of electromagnetic wave propagation the usual stratton chu integral equations are derived in terms of differential forms and corresponding galerkin formulations are constructed all numerical schemes require the specification of basis functions and the use of differential forms provides a very general method for the construction of arbitrary order basis functions on curvilinear geometries it is noted that the lowest order approximations on flat geometries reduce to forms essential equivalent to the standard rao wilton glisson functions the effect on accuracy is investigated for electric field integral equation and magnetic field integral equation formulations for a range of bases hierarchical classes of functions are also developed as are transition elements useful in p adaptive schemes where variable orders of approximation are sought

volume surface integral equation method based on higher order hierarchical vector basis functions for em scattering and radiation from composite metallic and dielectric structures this paper presents a novel galerkin type method of moments solution of the volume surface integral equation vsie which is developed for the analysis of electromagnetic scattering and radiation from composite metallic and dielectric structures our new scheme utilizes curved tetrahedral elements and curved triangular patches for geometric modeling and the associated higher order hierarchical vector hohv basis functions for volume surface current modeling in contrast to the previous studies on higher order vsie our scheme which is reported for the first time in vsie is much more flexible in comparison with the conventional low order scheme our scheme requires much less memory and computational time for the same accuracy level numerical results are provided to demonstrate the accuracy efficiency and flexibility of our scheme

integrable equations in 2 1 dimensions associated with symmetric and homogeneous spaces generalizations of the n wave davey stewartson and kadomtsev petviashvili equations associated with homogeneous and symmetric spaces are presented these equations are 2 1 dimensional generalizations of those presented by fordy and kulish commun math phys 89 427 1983 and athorne and fordy j phys a 20 1377 1987 examples are explicitly presented that are associated with the simplest spaces in particular a new single component 2 1 dimensional generalization of the kdv equation is presented

validation of regression metamodels in simulation bootstrap approach abstract simulation experiments are often analyzed through a linear regression model of their input output data such an analysis yields a metamodel or response surface for the underlying simulation model this metamodel can be validated through various statistics this article studies 1 the coefficient of determination r square for generalized least squares and 2 a lack of fit f statistic originally formulated by rao biometrika 46 1959 49 who assumed multivariate normality to derive the distributions of these two validation statistics this paper shows how to apply bootstrapping without assuming normality to illustrate the performance of these bootstrapped validation statistics the paper uses monte carlo experiments with simple models for these models i r square is a conservative statistic rejecting a valid metamodel relatively rarely so its power is low ii rao s original statistic may reject a valid metamodel too often iii bootstrapping rao s statistic gives only slightly conservative results so its power is relatively high

hierarchal vector boundary elements and p adaption for 3 d electromagnetic scattering this paper describes the development of new vector boundary elements for solving electromagnetic em scattering problems the new elements are suitable for the magnetic field integral equation mfie electrical field integral equation efie or the combination field integral equation cfie the basis functions are assigned to the edges of an element rather than to its nodes the new element guarantees the continuity of the normal component of the surface current across element edges furthermore the basis functions are hierarchical from linear to higher order which enables one to use the new elements in a p adaption scheme

simulation designs for quadratic response surface models in the presence of model misspecification this article considers the selection of experimental designs for the estimation of second order response surface metamodels in a simulation environment rather than construct designs based on the premise that the postulated model exactly represents the simulated response as is the case in optimal design theory we assume that the estimation process may be biased by the presence of unfitted third order terms we therefore seek to specify experimental plans that address the bias due to possible model misspecification as well as traditional variance considerations the performance measure used for this fit protect scenario is box and draper s design criterion of average mean squared error of predicted response four important classes of response surface experimental plans are examined 1 central composite designs 2 box behnken plans 3 three level factorial experiments and 4 small central composite designs each design class is studied in conjunction with three pseudorandom number assignment str

regression metamodels for simulation with common random numbers comparison of validation tests and confidence intervals linear regression analysis is important in many fields in the analysis of simulation results a regression metamodel can be applied even when common pseudorandom numbers are used to test the validity of the specified regression model rao 1959 generalized the f statistic for lack of fit whereas kleijnen 1983 proposed a cross validation procedure using a student s t statistic combined with bonferroni s inequality this paper reports on an extensive monte carlo experiment designed to compare these two methods under the normality assumption cross validation is conservative whereas rao s test realizes its nominal type i error and has high power robustness is investigated through lognormal and uniform distributions when simulation responses are distributed lognormally then cross validation using ordinary least squares is the only technique that has acceptable type i error uniform distributions give results similar to the normal case once the regression model is validated confidence intervals for the individual regression parameters are computed the monte carlo experiment compares several confidence interval procedures under normality rao s procedure is preferred since it has good coverage probability and acceptable half length under lognormality ordinary least squares achieves nominal coverage probability uniform distributions again give results similar to the normal case

a hierarchic p version boundary element method for axisymmetric acoustic scattering and radiation a rapidly convergent boundary element method that has a high accuracy capability is developed for the solution of the exterior neumann problem for the helmholtz equation the approach makes use of the boundary operator combination idea of burton and miller a j burton and g f miller proc r soc london ser a 323 201 210 1971 to avoid the classical irregular frequency difficulties together with p version boundary elements to obtain a high rate of convergence the entire algorithm is designed to be fully hierarchic to minimize the cost of multiple solutions which are necessary for a posteriori assessment of accuracy a new hierarchic singular kernel quadrature rule is developed for this purpose numerical examples demonstrate the accuracy and convergence rate of the method

curl conforming hierarchical vector bases for triangles and tetrahedra a new family of hierarchical vector bases is proposed for triangles and tetrahedra these functions span the curl conforming reduced gradient spaces of nedelec the bases are constructed from orthogonal scalar polynomials to enhance their linear independence which is a simpler process than an orthogonalization applied to the final vector functions specific functions are tabulated to order 6 5 preliminary results confirm that the new bases produce reasonably well conditioned matrices

integrable systems associated with the schrodinger spectral problem in the plane associated with the schrodinger spectral problem in the plane the nonlinear integrable systems are investigated the commutative properties of equations of the flow and their symmetries and mastersymmetries are derived in spite of the noncanonicality of the related extended recursion operator by applying the inverse scattering transform the systems can be solved and then the correspondences between commuting resp noncommuting flows and isospectral resp nonisospectral deformations of the spectral problem are found

a comparison of higher order nodal and edge basis functions in the mfie on rational bezier geometries higher order nodal basis functions for representing equivalent surface currents on antennas and scatterers are introduced the performance of the nodal basis is evaluated by comparing two existing higher order edge bases using the magnetic field integral equation mfie formulation for scattering by a perfect electric conductor pec sphere and icosahedron as test problems both nodal and edge bases are implemented on rational bezier patches giving an exact representation of the surfaces free from geometrical error the accuracy of the numerical solutions obtained with the three different bases for both the surface current and the radar cross section rcs are compared and it is shown that in general the nodal bases give better accuracy than the edge bases for equal computational cost

analyzing simulation experiments with common random numbers to analyze simulation runs which use the same random numbers the blocking concept of experimental design is not needed instead this paper applies a linear regression model with a nondiagonal covariance matrix this covariance matrix does not need to have a specific pattern such as constant covariances a simple example yields surprising results the paper proposes a new framework for the error analysis this framework consists of three factors namely common random numbers replication model validity each with three levels

symmetries and hierarchies of equations for the 2 1 dimensional n wave interaction abstract infinite sets of symmetries for the 2 1 dimensional n wave interaction are derived and two hierarchies of equations are given the lax pairs for these hierarchies are derived by an algebraic approach consequently two hierarchies are proved to be respectively the isospectral and non isospectral flows thus the inverse scattering transform of the multidimensional akns spectral problem can be applied and the corresponding time evolution of scattering data are obtained

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

adaptive multiresolution antenna modeling using hierarchical mixed order tangential vector finite elements hierarchical mixed order tangential vector finite elements tvfes for tetrahedra are attractive for accurate and efficient finite element or hybrid finite element boundary integral fe bi simulation of complicated electromagnetic problems they provide versatility in the geometrical modeling of physical structures guarantee solutions free of spurious modes and allow a local increase of resolution by combination of mixed order tvfes of different orders within a computational domain regions with higher order mixed order tvfes can be selected a priori or be found adaptively via methods for a posteriori error estimation or indication this paper demonstrates the merits of various explicit residual methods for a posteriori error indication via adaptive refinement of fe bi solutions of electromagnetic radiation problems using hierarchical mixed order tvfes for tetrahedra

singularity subtraction technique for high order polynomial vector basis functions on planar triangles in this paper a singularity subtraction technique is developed for computing the impedance matrix elements of various electromagnetic surface integral equation formulations with the galerkin method and high order basis functions analytical closed form formulas for computing surface integrals with r r sup n n spl ges 3 singularities times polynomial nodal shape functions of arbitrary order on a planar triangle are presented

symmetry algebras of the potential nizhnik novikov veselov model two sets of generalized symmetries of the potential nizhnik novikov veselov equation are given in a simple formula by developing a generalized formal series method for searching symmetries of one type of generalized partial differential equations these two sets of symmetries constitute an infinite dimensional lie algebra which contains three types of virasoro algebras and three w type algebras as its special cases

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

evaluation of local error estimators for the rwg based efie several error estimators suitable for use with the adaptive refinement solution of the electric field integral equation efie are investigated the estimators are evaluated using a series of test problems for which correlation coefficients and scatter plots are computed in addition two new discontinuity estimators are introduced based on the combination of the charge discontinuity with the current or current discontinuity the evaluations show that simple discontinuity estimators are as accurate for adaptive refinement as the efie residual estimator and are far more computationally efficient

integrability of the modified generalised vakhnenko equation integrability of the modified generalised vakhnenko equation is investigated systematically based on binary bell polynomials its bilinear representation n soliton solutions bilinear backlund transformation and lax pair are succinctly constructed moreover the conservation laws of the modified generalised vakhnenko equation are discussed by using corresponding lax pair furthermore the quasiperiodic solution of the modified generalised vakhnenko equation is presented by applying hirota direct method and riemann theta function the asymptotic behavior of the one periodic wave is analyzed in details it is shown that the one periodic wave solution tends to the one soliton solution under a small amplitude limit 0 finally the new n soliton solutions of the standard vakhnenko equation are presented it would be specially mentioned that all the results of modified generalised vakhnenko equation can be reduced to the generalised vakhnenko equation and standard vakhnenko equation under the special case of 1 and 1 0 respectively

integrable non isospectral flows associated with the kadomtsev petviashvili equations in 2 1 dimensions the symmetries of the kadomtsev petviashvili kp equations in 2 1 dimensions yield two hierarchies of integrable non linear evolution equations nee one is the familiar family of isospectral flows the kp hierarchy the other is non isospectral and its flows have coefficients which depend linearly on x and y the spectral methods used to solve kp can be used to solve all these nee an underlying infinite dimensional lie algebra is used to determine all the lax pairs for both families and it also determines their symplectic structures constants of the motion are constructed for the non isospectral cases

on the infinite hierarchies of symmetries and constants of motion for the kadomtsev petviashvili equation abstract a set of infinite hierarchies of symmetries and constants of motion for the kadomtsev petviashvili equation is derived through the newly proposed integrability condition which takes the form l t bl la moreover a graded lie algebra can be constructed for these symmetries and constants of motion and enables us to generate the whole set of symmetries and constants of motion from only three seed elements

two dimensional schro dinger operator inverse scattering transform and evolutional equations abstract the inverse problem for the two dimensional schrodinger operator on the data from one energy level is solved in a special class of finite zone or algebraic operators this class seems to be dense among all smooth periodic schrodinger operators evolutional equations associated with this problem are constructed

a p mus preconditioner for the efie this paper considers the solution of the electric field integral equation efie in electromagnetics as with associated finite element methods their solution relies upon the construction of conforming bases while lowest order rwg spaces are near ubiquitous their extension to higher order offers potentially a number of benefits in terms of accuracy and efficiency which has been well documented in both finite elements and integral equation formulations a further evolution of higher order conforming bases is the hierarchical basis these have demonstrated considerable gains in efficiency in finite element applications such bases allow for the development of effective acceleration schemes for instance the multilevel schwarz type preconditioner p mus an obvious question arises as to the applicability of such hierarchical bases and their associated acceleration schemes to integral equations it is seen that the conclusions as to their efficacy depend strongly on the scattering regime in particular high frequency problems those where the wavelength is the principal determinant of mesh size are shown to benefit little from hierarchical functions on the other hand for low frequency problems where geometry is the main determinant of mesh size there are significant improvements in performance over corresponding interpolatory schemes

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

some new exact solutions of the novikov veselov equation the novikov veselov nv equation is considered using backlund transformations and a nonlinear superposition formula for the nv equation some new exact solutions to this equation are found among which are the so called one and two lump solutions

an application of the boundary element method to the magnetic field integral equation a boundary element method bem for the solution of electromagnetic scattering problems using the magnetic field integral equation mfie is discussed the discretized form of the mfie is written in indicial notation with no limitations placed on the order of either the geometric or functional approximation by considering several different types of boundary elements it is determined that geometric errors can be significant and degrade the accuracy of the numerical solution it is shown that a higher order approximation for the current could significantly improve the accuracy of the numerical solution the superparametric boundary element in which the geometry was given quadratic approximation and the current was given linear approximation was more efficient than elements using lower order approximations the bem results are compared to the results obtained using the dielectric bodies of revolution dbr code

on a new hierarchy of symmetries for the kadomtsev petviashvili equation abstract we present a new hierarchy of symmetries for the kadomtsev petviashvili equation these new symmetries depend on the space and time variables explicitly together with the previously known classical symmetries they constitute an infinite dimensional lie algebra