efficient computation of the 2 d green s function for 1 d periodic structures using the ewald method the ewald method is applied to accelerate the evaluation of the green s function of an infinite periodic phased array of line sources the ewald representation for a cylindrical wave is obtained from the known representation for the spherical wave and a systematic general procedure is applied to extend previous results only a few terms are needed to evaluate ewald sums which are cast in terms of error functions and exponential integrals to high accuracy singularities and convergence rates are analyzed and a recipe for selecting the ewald splitting parameter spl epsiv is given to handle both low and high frequency ranges indeed it is shown analytically that the choice of the standard optimal splitting parameter spl epsiv sub 0 will cause overflow errors at high frequencies numerical examples illustrate the results and the sensitivity of the ewald representation to the splitting parameter spl epsiv

obstructions to determinantal representability there has recently been ample interest in the question of which sets can be represented by linear matrix inequalities lmis a necessary condition is that the set is rigidly convex and it has been conjectured that rigid convexity is also sufficient to this end helton and vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices we disprove this conjecture by relating the question of finding lmi representations to the problem of determining whether a polymatroid is representable over the complex numbers we find a real zero polynomial such that no power of it admits a determinantal representation the proof uses recent results of wagner and wei on matroids with the half plane property and the polymatroids associated to hyperbolic polynomials introduced by gurvits

accelerating the convergence of series representing the free space periodic green s function a method for improving the convergence of the series representing the doubly periodic free space green s function is presented the method consists of successively applying three different transformations to the green s function spectral representation kummer s transformation is first applied to convert the slowly converging spectral representation into the sum of a rapidly converging series and a slowly converging series the latter series is recognized as the spectral representation of the original periodic source distribution radiating in a medium with an imaginary wavenumber application of the poisson transformation to this series renders it exponentially convergent since it effectively represents propagation of point source contributions through a medium with imaginary wavenumber finally shanks transform is plotted versus the number of terms taken in the series numerical results confirm that an improvement in the convergence rate of the series is achieved for a particular convergence criterion

a fast high order algorithm for the solution of surface scattering problems basic implementation tests and applications we present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three dimensional space this algorithm evaluates scattered fields through fast high order solution of the corresponding boundary integral equation the high order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities the acceleration in turn results from use of a novel approach which based on high order two face equivalent source approximations reduces the evaluation of far interactions to evaluation of 3 d fast fourier transforms ffts this approach is faster and substantially more accurate and it runs on dramatically smaller memories than other fft and k space methods the present algorithm computes one matrix vector multiplication in o n6 5logn to o n4 3logn operations where n is the number of surface discretization points the latter estimate applies to smooth surfaces for which our high order algorithm provides accurate solutions with small values of n the former more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling further our approach exhibits super algebraic convergence it can be applied to smooth and nonsmooth scatterers and it does not suffer from accuracy breakdowns of any kind in this paper we introduce the main algorithmic components in our approach and we demonstrate its performance with a variety of numerical results in particular we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds

data reusing recurrent neural adaptive filters a class of data reusing learning algorithms for real time recurrent neural networks rnns is analyzed the analysis is undertaken for a general sigmoid nonlinear activation function of a neuron for the real time recurrent learning training algorithm error bounds and convergence conditions for such data reusing algorithms are provided for both contractive and expansive activation functions the analysis is undertaken for various configurations that are generalizations of a linear structure infinite impulse response adaptive filter

rapidly convergent two dimensional quasi periodic green function throughout the spectrum including wood anomalies we introduce a new methodology based on new quasi periodic green functions which converge rapidly even at and around wood anomaly configurations for the numerical solution of problems of scattering by periodic rough surfaces in two dimensional space as is well known the classical quasi periodic green function ceases to exist at wood anomalies the approach introduced in this text produces fast green function convergence throughout the spectrum on the basis of a certain finite differencing approach and smooth windowing of the classical green function lattice sum the resulting green function convergence is super algebraically fast away from wood anomalies and it reduces to an arbitrarily high user prescribed algebraic order of convergence at wood anomalies

competitive learning with generalized winner take all activation competitive learning paradigms are usually defined with winner take all wta activation rules the paper develops a mathematical model for competitive learning paradigms using a generalization of the wta activation rule g wta the model is a partial differential equation pde relating the time rate of change in the density of weight vectors to the divergence of a vector field called the neural flux characteristic trajectories are used to study solutions of the pde model over scalar weight spaces these solutions show how the model can be used to design competitive learning algorithms which estimate the modes of unknown probability density functions

an improved dual neural network for solving a class of quadratic programming problems and its k winners take all application this paper presents a novel recurrent neural network for solving a class of convex quadratic programming qp problems in which the quadratic term in the objective function is the square of the euclidean norm of the variable this special structure leads to a set of simple optimality conditions for the problem based on which the neural network model is formulated compared with existing neural networks for general convex qp the new model is simpler in structure and easier to implement the new model can be regarded as an improved version of the dual neural network in the literature based on the new model a simple neural network capable of solving the k winners take all k wta problem is formulated the stability and global convergence of the proposed neural network is proved rigorously and substantiated by simulation results

infinite phased array analysis using fdtd periodic boundary conditions pulse scanning in oblique directions unit cell analysis of infinite phased arrays in the finite difference time domain fdtd is performed by implementation of periodic boundary conditions the technique allows for pulse excitation and oblique scan directions in both the cardinal and intercardinal planes to our knowledge this is the first paper presenting fdtd computations for intercardinal pulse scanning in oblique directions the ordinary yee lattice is used which makes the algorithm easy to incorporate in an already existing fdtd code nonperiodic boundaries are truncated by berenger s see j comput phys vol 127 p 363 79 1996 perfectly matched layer pml active impedance of an infinite dipole array is calculated with the new method and validation is performed via the element by element approach i e by a conventional fdtd simulation of a corresponding large finite array excellent agreement is found and the technique has been numerically stable in all cases analyzed

efficient numerical solution of acoustic scattering from doubly periodic arrays of axisymmetric objects we present a high order accurate boundary based solver for three dimensional 3d frequency domain scattering from a doubly periodic grating of smooth axisymmetric sound hard or transmission obstacles we build the one obstacle solution operator using separation into p azimuthal modes via the fft the method of fundamental solutions with n proxy points lying on a curve and dense direct least squares solves the effort is o n 3 p with a small constant periodizing then combines fast multipole summation of nearest neighbors with an auxiliary global helmholtz basis expansion to represent the distant contributions and enforcing quasiperiodicity and radiation conditions on the unit cell walls eliminating the auxiliary coefficients and preconditioning with the one obstacle solution operator leaves a well conditioned square linear system that is solved iteratively the solution time per incident wave is then o n p at fixed frequency our scheme avoids singular quadratures periodic green s functions and lattice sums and its convergence rate is unaffected by resonances within obstacles we include numerical examples such as scattering from a grating of period 13 13 comprising highly resonant sound hard cups each needing n p 64800 surface unknowns to 10 digit accuracy in half an hour on a desktop

maxwell s equations in a periodic structure consider a diffraction of a beam of particles in r3 when the dielectric coefficient is a constant cl above a surface s and a constant e2 below a surface s and the magnetic permeability is constant throughout rii s is assumed to be periodic in the xl direction and of the form x1 f1 s x3 f3 s x2 arbitrary we prove that there exists a unique solution to the time harmonic maxwell equations in r 3 having the form of refracted waves for x3 1 and of transmitted waves for x3 1 if and only if there exists a unique solution to a certain system of two coupled fredholm equations thus in particular for all the e s except for a discrete number there exists a unique solution to the maxwell equations introduction in this paper we consider the maxwell equations for time harmonic solutions in the entire space r3 with piecewise constant dielectric coefficient having jump across a periodic surface the magnetic permeability u is assumed to be constant whereas the dielectric coefficient e is given by e 8 above a surface s x3 f xl and e e2 below the surface s e and 82 are different constants if s is a half space x3 0 then the solution eo ho can be computed explicitly we assume in this paper that s is periodic i e f xl l f xl vx1 e r l 0 we wish to find a solution e h such that e eo and h ho are superpositions of transmitted waves 0 1 in x3 a where a maxlfl in 1 7 we assume that f e c2 and we reduce the solution of the maxwell equations to a fredholm system of four integral equations in 8 we reduce it further to a fredholm system of two integral equations thus for all but a discrete sequence of values of the physical parameters there exists a unique solution to the integral equations yielding a solution of the maxwell equations received by the editors january 25 1989 1980 mathematics subject classification 1985 revision primary 78a10 78a45 secondary 35p25 47a40

perfectly matched layer boundary integral equation method for wave scattering in a layered medium for scattering problems of time harmonic waves the boundary integral equation bie methods are highly competitive since they are formulated on lower dimension boundaries or interfaces and can automatically satisfy outgoing radiation conditions for scattering problems in a layered medium standard bie methods based on green s function of the background medium need to evaluate the expensive sommerfeld integrals alternative bie methods based on the free space green s function give rise to integral equations on unbounded interfaces which are not easy to truncate since the wave fields on these interfaces decay very slowly we develop a bie method based on the perfectly matched layer pml technique the pmls are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space our pml based bie method uses the pml transformed free space green s function to define the boundary integral operators the method is efficient since the pml transformed free space green s func

a least squares finite element analysis for diffraction problems the diffraction of a time harmonic wave incident upon a grating or periodic structure is treated by a least squares finite element method that incorporates the jump conditions at interfaces into the objective functional two fundamental polarizations are considered coercivity of the quadratic functional is established and optimal discretization error estimates are obtained in both cases the theoretical results indicate that for sufficiently smooth interfaces the error estimates are superior to those of standard finite element methods

one and two dimensional lattice sums for the three dimensional helmholtz equation the accurate and efficient computation of lattice sums for the three dimensional helmholtz equation is considered for the cases where the underlying lattice is one or two dimensional we demonstrate using careful numerical computations that the reduction method in which the sums for a two dimensional lattice are expressed as a sum of one dimensional lattice sums leads to an order of magnitude improvement in performance over the well known ewald method in the process we clarify and improve on a number of results originally formulated by twersky in the 1970s

multivariate stable polynomials theory and applications univariate polynomials with only real roots while special do occur often enough that their properties can lead to interesting conclusions in diverse areas due mainly to the recent work of two young mathematicians julius borcea and petter branden a very successful multivariate generalization of this method has been developed the first part of this paper surveys some of the main results of this theory of multivariate stable polynomials the most central of these results is the characterization of linear transformations preserving stability of polynomials the second part presents various applications of this theory in complex analysis matrix theory probability and statistical mechanics and combinatorics

a fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions we present a new integral equation method for the calculation of two dimensional scattering from periodic structures involving triple points multiple materials meeting at a single point the combination of a robust and high order accurate integral representation and a fast direct solver permits the efficient simulation of scattering from fixed structures at multiple angles of incidence we demonstrate the performance of the scheme with several numerical examples

electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate we consider a two dimensional problem of scattering of a time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate the magnetic permeability is assumed to be a fixed positive constant in the media the material properties of the media are characterized completely by an index of refraction which is a bounded measurable function in the layer and a positive constant above the layer corresponding to a homogeneous dielectric medium in this paper we only examine the tm transverse magnetic polarization case a radiation condition is introduced and equivalence with a second kind lippmann schwinger type integral equation is shown with additional assumptions on the index of refraction in the layer uniqueness of solution is proved existence of solution is then established by employing a form of fredholm alternative using a general result on the solvability of integral equations on unbounded domains published earlier by chandler wilde and zhang an approximate analytic solution for the case of a thin inhomogeneous layer is obtained from the integral equation formulation and is used to show that if the index of refraction is appropriately chosen the scattered field can grow with distance from the plate

lattice sums for off axis electromagnetic scattering by gratings we consider both the spatial domain and spectral domain forms of the green s function appropriate in the electromagnetic diffraction of a plane wave incident at a general angle in the xy plane on a singly periodic structure or grating oriented along the x axis we equate the spatial and spectral forms of the green s function and so establish expressions from which grating lattice sums can be evaluated for oblique incidence we also obtain a set of identities among the lattice sums we use these lattice sums in an expression for the green s function which we show to be computationally fast if knowledge of this function at several points is required for small values of y

certified reduced basis method for electromagnetic scattering and radar cross section estimation we study nontrivial applications of the reduced basis method rbm for electromagnetic applications with an emphasis on scattering and the estimation of radar cross section rcs the method and several extensions are explained with two examples with different characteristics parameters that are allowed to vary within the model include frequency incident angle and measurement angle as well as the geometry of the scatterers with appropriate applications of the empirical interpolation method eim transformation of the domain configuration of perfectly matched layer exponential convergence of the reduced basis solution over the entire parameter domain is achieved moreover we demonstrate that this approach allows for the effective capture of the critical behavior in this case through shapes that minimize scattering this further highlights the robustness and quality of the greedy approximation and the reduced basis method approach

polynomials with the half plane property and matroid theory a polynomial f is said to have the half plane property if there is an open half plane h subset of c whose boundary contains the origin such that f is non zero whenever all the variables are in h

three dimensional quasi periodic shifted green function throughout the spectrum including wood anomalies this work part ii in a series presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called wood anomaly frequencies at these frequencies there is a grazing rayleigh wave and the quasi periodic green function ceases to exist we present a modification of the green function by adding two types of terms to its lattice sum the first type are transversely shifted green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence the second type are quasi periodic plane wave solutions of the helmholtz equation which reinstate certain necessary grazing modes without leading to blow up at wood anomalies using the new quasi periodic green function we establish for the first time that the dirichlet problem of scattering by a smooth doubly periodic scattering surface at a wood frequency is uniquely solvable we also present an efficient high order numerical method based on this new green function for scattering by doubly periodic surfaces at and around wood frequencies we believe this is the first solver able to handle wood frequencies for doubly periodic scattering problems in three dimensions we demonstrate the method by applying it to acoustic scattering

global asymptotic stability of recurrent neural networks with multiple time varying delays in this paper several sufficient conditions are established for the global asymptotic stability of recurrent neural networks with multiple time varying delays the lyapunov krasovskii stability theory for functional differential equations and the linear matrix inequality lmi approach are employed in our investigation the results are shown to be generalizations of some previously published results and are less conservative than existing results the present results are also applied to recurrent neural networks with constant time delays

a comparison of acceleration procedures for the two dimensional periodic green s function alternate acceleration procedures are evaluated for the two dimensional free space periodic green s function two of these techniques yield exponential convergence rates and are preferable if high accuracies are required numerical aspects of these procedures are described

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

windowed green function method for layered media scattering this paper introduces a new windowed green function wgf method for the numerical integral equation solution of problems of electromagnetic scattering by obstacles in the presence of dielectric or conducting half planes the wgf method which is based on the use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free space green function does not require evaluation of expensive sommerfeld integrals the proposed approach is fast accurate flexible and easy to implement in particular straightforward modifications of existing accelerated or unaccelerated integral equation solvers suffice to incorporate the wgf capability the method relies on a certain integral equation posed on the union of the boundary of the obstacle and a small flat section of the interface between the penetrable media our analysis and numerical experiments demonstrate that both the near and far field errors resulting from the proposed approach decrease faster than any negative power of the window size in the examples considered in this paper the proposed method is up to thousands of times faster for a given accuracy than a corresponding method based on use of sommerfeld integrals

non representable hyperbolic matroids the generalized lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices hyperbolic polynomials give rise to a class of hyperbolic matroids which properly contains the class of matroids representable over the complex numbers this connection was used by the first author to construct counterexamples to algebraic stronger versions of the generalized lax conjecture by considering a non representable hyperbolic matroid the va mos matroid and a generalization of it are to this day the only known instances of non representable hyperbolic matroids we prove that the non pappus and non desargues matroids are non representable hyperbolic matroids by exploiting a connection due to jordan between euclidean jordan algebras and projective geometries we further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non representable finally we explore consequences to algebraic versions of the generalized lax conjecture

on analyticity of linear waves scattered by a layered medium abstract the scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering in this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high order perturbation of surfaces methods for simulating such waves more specifically we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a multiply layered periodic structure in three dimensions this result provides hypotheses under which a rigorous numerical analysis could be conducted for recent generalizations to the methods of operator expansions field expansions and transformed field expansions

a novel recurrent neural network with one neuron and finite time convergence for k winners take all operation in this paper based on a one neuron recurrent neural network a novel k winners take all k wta network is proposed finite time convergence of the proposed neural network is proved using the lyapunov method the k wta operation is first converted equivalently into a linear programming problem then a one neuron recurrent neural network is proposed to get the kth or k 1 th largest inputs of the k wta problem furthermore a k wta network is designed based on the proposed neural network to perform the k wta operation compared with the existing k wta networks the proposed network has simple structure and finite time convergence in addition simulation results on numerical examples show the effectiveness and performance of the proposed k wta network

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

corner singularities for elliptic problems integral equations graded meshes quadrature and compressed inverse preconditioning we take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points use of non standard integral equations graded meshes interpolatory quadrature and compressed inverse preconditioning are techniques that are explored developed mixed and tested on some familiar problems in materials science the recursive compressed inverse preconditioning the major novelty of the paper turns out to be particularly powerful and when it applies eliminates the need for mesh grading completely in an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 1 2 in another example five times as large but with a conductivity ratio of only a hundred we achieve an estimated relative error of 10 1 4

multivariate polya schur classification problems in the weyl algebra a multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts we classify all linear partial differential operators in the weyl algebra a n that preserve stability an important tool that we develop in the process is the higher dimensional generalization of polya schur s notion of multiplier sequence we characterize all multivariate multiplier sequences as well as those of finite order next we establish a multivariate extension of the cauchy poincare interlacing theorem and prove a natural analog of the lax conjecture for real stable polynomials in two variables using the latter we describe all operators in a 1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in a n preserves stability if and only if its fischer fock adjoint does these are powerful multivariate extensions of the classical hermite poulain jensen theorem polya s curve theorem and schur malo szegocomposition theorems examples and applications to strict stability preservers are also discussed

complete monotonicity for inverse powers of some combinatorially defined polynomials we prove the complete monotonicity on 0 infty n for suitable inverse powers of the spanning tree polynomials of graphs and more generally of the basis generating polynomials of certain classes of matroids this generalizes a result of szeg and answers among other things a long standing question of lewy and askey concerning the positivity of taylor coefficients for certain rational functions our proofs are based on two ab initio methods for proving that p beta is completely monotone on a convex cone c the determinantal method and the quadratic form method these methods are closely connected with harmonic analysis on euclidean jordan algebras or equivalently on symmetric cones we furthermore have a variety of constructions that given such polynomials can create other ones with the same property among these are algebraic analogues of the matroid operations of deletion contraction direct sum parallel connection series connection and 2 sum the complete monotonicity of p beta for some beta 0 can be viewed as a strong quantitative version of the half plane property hurwitz stability for p and is also related to the rayleigh property for matroids

stable and efficient evaluation of periodized green s functions for the helmholtz equation at high frequencies a difficulty that arises in the context of infinite d periodic rough surface scattering relates to the effective numerical evaluation of the corresponding quasi periodic green function g q p due to its relevance in a variety of applications this problem has generated significant interest over the last 40 years and a variety of numerical methods have been devised for this purpose none of these methods to evaluate g q p however were designed for high frequency calculations as a result in this regime these methods become prohibitively expensive and or unstable here we present a novel scheme that can be shown to outperform every alternative numerical evaluation procedure and is especially effective for high frequency calculations our new algorithm is based on the use of some exact integrals that arise on judicious manipulation of the integral representation of g q p and which reduce the overall problem to that of evaluation of a sequence of simpler integrals that can be effectively handled by standard quadrature formulas we include a variety of numerical results that confirm that indeed our algorithm compares favorably with alternative methods

lattice sums and the two dimensional periodic green s function for the helmholtz equation many algorithms that are currently used for the solution of the helmholtz equation in periodic domains require the evaluation of the green s function g x x0 the fact that the natural representat

polya schur master theorems for circular domains and their boundaries we characterize all linear operators on finite or infinite dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region omega subset of c for arbitrary

numerical analysis of diffraction by periodic structures tm polarization the diffraction of a time harmonic wave in tm polarization by some periodic inhomogeneous material is studied in this paper the diffraction problem is modeled by a generalized helmholtz equation with transparent boundary conditions existence and uniqueness are proved for the model problem uniform error estimates for the finite element approximation of the solution are established error estimates are also obtained when the truncation of the nonlocal boundary operators takes place

superalgebraically convergent smoothly windowed lattice sums for doubly periodic green functions in three dimensional space this work part i in a two part series presents i a simple and highly efficient algorithm for evaluation of quasi periodic green functions as well as ii an associated boundary integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three dimensional space except for certain wood frequencies at which the quasi periodic green function ceases to exist the proposed approach which is based on smooth windowing functions gives rise to tapered lattice sums which converge superalgebraically fast to the green function that is faster than any power of the number of terms used this is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing the wood frequency problem is treated in part ii this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums a variety of numerical results demonstrate the practical efficiency of the proposed approach

a fast and robust solver for the scattering from a layered periodic structure containing multi particle inclusions we present a solver for plane wave scattering from a periodic dielectric grating with a large number m of inclusions lying in each period of its middle layer such composite material geometries have a growing role in modern photonic devices and solar cells the high order scheme is based on boundary integral equations and achieves many digits of accuracy with ease the usual way to periodize the integral equation via the quasi periodic green s function fails at wood s anomalies we instead use the free space green s kernel for the near field add auxiliary basis functions for the far field and enforce periodicity in an expanded linear system this is robust for all parameters inverting the periodic and layer unknowns we are left with a square linear system involving only the inclusion scattering coefficients preconditioning by the single inclusion scattering matrix this is solved iteratively in o m time using a fast matrix vector product numerical experiments show that a diffraction grating containing m 1000 inclusions per period can be solved to 9 digit accuracy in under 5 minutes on a laptop

a bk inequality for randomly drawn subsets of fixed size the bk inequality van den berg and kesten in j appl probab 22 556 569 1985 says that for product measures on 0 1 n the probability that two increasing events a and b occur disjointly is at most the product of the two individual probabilities the conjecture in van den berg and kesten 1985 that this holds for all events was proved by reimer combin probab comput 9 27 32 2000 several other problems in this area remained open for instance although it is easy to see that non product measures cannot satisfy the above inequality for all events there are several such measures which intuitively should satisfy the inequality for all increasing events one of the most natural candidates is the measure assigning equal probabilities to all configurations with exactly k 1 s and probability 0 to all other configurations the main contribution of this paper is a proof for these measures we also point out how our result extends to weighted versions of these measures and to products of such measures

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

solving the assignment problem using continuous time and discrete time improved dual networks the assignment problem is an archetypal combinatorial optimization problem in this brief we present a continuous time version and a discrete time version of the improved dual neural network idnn for solving the assignment problem compared with most assignment networks in the literature the two versions of idnns are advantageous in circuit implementation due to their simple structures both of them are theoretically guaranteed to be globally convergent to a solution of the assignment problem if only the solution is unique

a stable high order perturbation of surfaces method for numerical simulation of diffraction problems in triply layered media the accurate numerical simulation of linear waves interacting with periodic layered media is a crucial capability in engineering applications in this contribution we study the stable and high order accurate numerical simulation of the interaction of linear time harmonic waves with a periodic triply layered medium with irregular interfaces in contrast with volumetric approaches high order perturbation of surfaces hops algorithms are inexpensive interfacial methods which rapidly and recursively estimate scattering returns by perturbation of the interface shape in comparison with boundary integral element methods the stable hops algorithm we describe here does not require specialized quadrature rules periodization strategies or the solution of dense non symmetric positive definite linear systems in addition the algorithm is provably stable as opposed to other classical hops approaches with numerical experiments we show the remarkable efficiency fidelity and accuracy one can achieve with an implementation of this algorithm

greedy algorithm and symmetric matroids symmetric matroids are set systems which are obtained in some sense by a weakening of the structure of a matroid these set systems are characterized by a greedy algorithm and they are suitable for dealing with autodual properties of matroids applications are given to the eulerian tours of 4 regular graphs and the theory ofg matroids

lattice sums for the helmholtz equation a survey of different representations for lattice sums for the helmholtz equation is made these sums arise naturally when dealing with wave scattering by periodic structures one of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d lambda lattice sums are related to and can be calculated from the quasi periodic green s function and this object serves as the starting point of the analysis

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

finite time convergent recurrent neural network with a hard limiting activation function for constrained optimization with piecewise linear objective functions this paper presents a one layer recurrent neural network for solving a class of constrained nonsmooth optimization problems with piecewise linear objective functions the proposed neural network is guaranteed to be globally convergent in finite time to the optimal solutions under a mild condition on a derived lower bound of a single gain parameter in the model the number of neurons in the neural network is the same as the number of decision variables of the optimization problem compared with existing neural networks for optimization the proposed neural network has a couple of salient features such as finite time convergence and a low model complexity specific models for two important special cases namely linear programming and nonsmooth optimization are also presented in addition applications to the shortest path problem and constrained least absolute deviation problem are discussed with simulation results to demonstrate the effectiveness and characteristics of the proposed neural network

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

taylor expansion based fast multipole method for 3 d helmholtz equations in layered media abstract in this paper we develop a taylor expansion te based fast multipole method fmm for low frequency 3d helmholtz green s function in layered media two forms of taylor expansions with either non symmetric or symmetric derivatives of layered media green s functions are used for the implementations of the proposed te fmm in the implementation with non symmetric derivatives an algorithm based on discrete complex image approximations and recurrence formulas is shown to be very efficient and accurate in computing the high order derivatives meanwhile the implementation based on symmetric derivatives is more robust and pre computed tables for the high order derivatives in translation operators are used numerical tests in layered media have validated the accuracy and o n complexity of the proposed algorithms

a new integral representation for quasi periodic fields and its application to two dimensional band structure calculations in this paper we consider band structure calculations governed by the helmholtz or maxwell equations in piecewise homogeneous periodic materials methods based on boundary integral equations are natural in this context since they discretize the interface alone and can achieve high order accuracy in complicated geometries in order to handle the quasi periodic conditions which are imposed on the unit cell the free space green s function is typically replaced by its quasi periodic cousin unfortunately the quasi periodic green s function diverges for families of parameter values that correspond to resonances of the empty unit cell here we bypass this problem by means of a new integral representation that relies on the free space green s function alone adding auxiliary layer potentials on the boundary of the unit cell itself an important aspect of our method is that by carefully including a few neighboring images the densities may be kept smooth and convergence rapid this framework results in an integral equation of the second kind avoids spurious resonances and achieves spectral accuracy because of our image structure inclusions which intersect the unit cell walls may be handled easily and automatically our approach is compatible with fast multipole acceleration generalizes easily to three dimensions and avoids the complication of divergent lattice sums

a one layer recurrent neural network with a discontinuous hard limiting activation function for quadratic programming in this paper a one layer recurrent neural network with a discontinuous hard limiting activation function is proposed for quadratic programming this neural network is capable of solving a large class of quadratic programming problems the state variables of the neural network are proven to be globally stable and the output variables are proven to be convergent to optimal solutions as long as the objective function is strictly convex on a set defined by the equality constraints in addition a sequential quadratic programming approach based on the proposed recurrent neural network is developed for general nonlinear programming simulation results on numerical examples and support vector machine svm learning show the effectiveness and performance of the neural network

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

universal quadratures for boundary integral equations on two dimensional domains with corners we describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two dimensional domains with corner points the resulting quadrature rules allow for the rapid high accuracy solution of dirichlet boundary value problems for laplace s equation and the helmholtz equation on such domains under a mild assumption on the boundary data our approach can be adapted to other boundary value problems and certain aspects of our scheme generalize to the case of surfaces with singularities in three dimensions the performance of the quadrature rules is illustrated with several numerical examples

negative dependence and the geometry of polynomials we introduce the class of strongly rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter this class covers important models such as determinantal measures e g product measures and uniform random spanning tree measures and distributions for symmetric exclusion processes we show that strongly rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to liggett pemantle and wagner respectively moreover we extend lyons recent results on determinantal measures and we construct counterexamples to several conjectures of pemantle and wagner on negative dependence and ultra log concave rank sequences

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

effect of input noise and output node stochastic on wang s k rm wta recently an analog neural network model namely wang s kwta was proposed in this model the output nodes are defined as the heaviside function subsequently its finite time convergence property and the exact convergence time are analyzed however the discovered characteristics of this model are based on the assumption that there are no physical defects during the operation in this brief we analyze the convergence behavior of the wang s kwta model when defects exist during the operation two defect conditions are considered the first one is that there is input noise the second one is that there is stochastic behavior in the output nodes the convergence of the wang s kwta under these two defects is analyzed and the corresponding energy function is revealed

a high order perturbation of surfaces method for scattering of linear waves by periodic multiply layered gratings in two and three dimensions abstract the capability to rapidly and robustly simulate the scattering of linear waves by periodic multiply layered media in two and three dimensions is crucial in many engineering applications in this regard we present a high order perturbation of surfaces method for linear wave scattering in a multiply layered periodic medium to find an accurate numerical solution of the governing helmholtz equations for this we truncate the bi infinite computational domain to a finite one with artificial boundaries above and below the structure and enforce transparent boundary conditions there via dirichlet neumann operators this is followed by a transformed field expansion resulting in a fourier collocation legendre galerkin taylor series method for solving the problem in a transformed set of coordinates assorted numerical simulations display the spectral convergence of the proposed algorithm

a tensor product finite element method for the diffraction grating problem with transparent boundary conditions abstract we consider the diffraction grating problem in optics which has been modeled by a boundary value problem governed by a helmholtz equation with transparent boundary conditions a tensor product finite element method is proposed to numerically solve the problem an fft based matrix decomposition algorithm is developed to solve the linear system arising in the vertically layered medium case which can be used as a preconditioning technique for the general case numerical examples are presented to illustrate the accuracy and efficiency of the method

efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane this paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance the solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary and the discussion of the paper is motivated by this application the paper starts by deriving representations as laplace type integrals of the solution and its first partial derivatives the evaluation of these integral representations by gauss laguerre quadrature is discussed and theoretical bounds on the truncation error are obtained specific approximations are proposed which are shown to be accurate except in the very near field for all angles of incidence and a wide range of values of surface impedance the paper finishes with derivations of partial results and analogous laplace type integral representations for the case of a point source

the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

fdtd local grid with material traverse often a finite difference time domain fdtd calculation requires a relatively higher mesh resolution in only small subvolumes of the total mesh space by locally applying finer grids local grids to these volumes the necessary resolution can be obtained computation time and memory requirements may be far less than for an fdtd space with the smaller mesh resolution throughout in many situations it is important that these local grids function when materials traverse the main grid local grid mg lg boundary surfaces since the volumes that require local grids may not be isolated in a homogeneous medium a local grid method which allows dielectric and or conducting materials to traverse the boundaries is developed three different fdtd problems that utilize the local grid method are used as validation tests results are compared with uniform mesh fdtd solutions

efficient computation of the 3d green s function for the helmholtz operator for a linear array of point sources using the ewald method the ewald method is applied to accelerate the evaluation of the green s function gf of an infinite equispaced linear array of point sources with linear phasing only a few terms are needed to evaluate ewald sums which are cast in terms of error functions and exponential integrals to high accuracy it is shown analytically that the choice of the standard optimal ewald splitting parameter e0 causes overflow errors at high frequencies period large compared to the wavelength and convergence rates are analyzed a recipe for selecting the ewald splitting parameter is provided

a new eulerian method for the computation of propagating short acoustic and electromagnetic pulses a new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics the pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections the method has many of the advantages of lagrangian ray tracing but is completely eulerian typically using a uniform cartesian grid accordingly it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse also information describing the pulses which are treated as continuous surfaces can be available throughout the computational grid rather than only at isolated individual markers the method uses a new type of representation which we call dynamic surface extension the basic idea is to propagate or broadcast defining fields from each pulse surface through a computational grid these fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes such as amplitude thus the emphasis is on the dynamics of these propagating defining fields which obey only local eulerian equations at each node the dynamic surface extension representation can be thought of as dual to level set representation the defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces in this way the new method overcomes the inability of level set or eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other while still using only single valued variables propagating thin pulse surfaces in 1 d 2 d and 3 d that can reflect from boundaries and pass through each other are computed using the new method the method was first presented as a new general representation of surfaces filaments and particles by j steinhoff and m fan 1998 eulerian computation of evolving surfaces curves and discontinuous fields utsi preprint

an efficient numerical evaluation of the green s function for the helmholtz operator on periodic structures abstract numerical methods which are based on boundary integral formulations require the numerical evaluation of the green s function associated with the problem in the case of periodic structures the green s function is often an infinite series this infinite series may converge slowly making numerical evaluation expensive here we present a practical computer implementation of a technique which dramatically speeds up the convergence of the infinite series green s function associated with the helmholtz operator to show the consequences of using this technique we include some numerical examples

k winners take all circuit with o n complexity presents a k winners take all circuit that is an extension of the winner take all circuit by lazzaro et al 1989 the problem of selecting the largest k numbers is formulated as a mathematical programming problem whose solution scheme based on the lagrange multiplier method is directly implemented on an analog circuit the wire length in this circuit grows only linearly with the number of elements and the circuit is more suitable for real time processing than the hopfield networks because the present circuit produces the solution almost instantaneously in contrast to the hopfield network which requires transient convergence to the solution from a precise initial state the selection resolution in the present circuit is however only finite in contrast to the almost infinite resolution in the hopfield networks

hyperbolic polynomials interlacers and sums of squares hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids the innermost of which is convex these polynomials appear in many areas of mathematics including optimization combinatorics and differential equations here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it this set of interlacers is a convex cone which we write as a linear slice of the cone of nonnegative polynomials in particular this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials using a sums of squares relaxation we then approximate a hyperbolicity cone by the projection of a spectrahedron a multiaffine example coming from the vamos matroid shows that this relaxation is not always exact using this theory we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists

partial facial reduction simplified equivalent sdps via approximations of the psd cone we develop a practical semidefinite programming sdp facial reduction procedure that utilizes computationally efficient approximations of the positive semidefinite cone the proposed method simplifies sdps with no strictly feasible solution a frequent output of parsers by solving a sequence of easier optimization problems and could be a useful pre processing technique for sdp solvers we demonstrate effectiveness of the method on sdps arising in practice and describe our publicly available software implementation we also show how to find maximum rank matrices in our psd cone approximations which helps us find maximal simplifications and we give a post processing procedure for dual solution recovery that generally applies to facial reduction based pre processing techniques finally we show how approximations can be chosen to preserve problem sparsity

some experiences from fdtd analysis of infinite and finite multi octave phased arrays some experiences from finite differences time domain fdtd analysis of infinite and finite multi octave phased arrays are presented first a more unified derivation of equations suitable for unit cell analysis of phased arrays or other types of periodic structures using fdtd is presented second results from fdtd calculations of small to very large multi octave finite arrays are summarized in order to answer the question of how large an array model must be in order to capture the characteristics of both the interior and the edge elements of a large multi octave phased array it is found that a considerably larger number of elements is required in the broad band case than in the normal narrow band case and it is also found that fdtd is well suited for such calculations third simple methods to save computer memory using locally fine grids in an otherwise coarse fdtd grid to model finite phased arrays are explored the two local grid methods tested were found in our application to suffer from numerical instabilities

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

a new recurrent neural network for solving convex quadratic programming problems with an application to the k winners take all problem in this paper a new recurrent neural network is proposed for solving convex quadratic programming qp problems compared with existing neural networks the proposed one features global convergence property under weak conditions low structural complexity and no calculation of matrix inverse it serves as a competitive alternative in the neural network family for solving linear or quadratic programming problems in addition it is found that by some variable substitution the proposed network turns out to be an existing model for solving minimax problems in this sense it can be also viewed as a special case of the minimax neural network based on this scheme a k winners take all k wta network with o n complexity is designed which is characterized by simple structure global convergence and capability to deal with some ill cases numerical simulations are provided to validate the theoretical results obtained more importantly the network design method proposed in this paper has great potential to inspire other competitive inventions along the same line

efficient calculation of the free space periodic green s function electromagnetic scattering from periodic structures can be formulated in terms of an integral equation that has as its kernel a periodic green s function the periodic green s function can be derived as a response to an array of line point sources spatial domain or as a response to series of current sheets spectral domain these responses are a fourier transform pair and are slowly convergent summations the convergence problems in each domain arise from unavoidable singularities in the reciprocal domain a method is discussed to overcome the slow convergence by using the poisson summation formula and summing in a combination of spectral and spatial domains a parameter study is performed to determine an optimum way to weigh the combination of domains simple examples of scattering from a one dimensional array of strips and two dimensional array of plates are used to illustrate the concepts

least squares finite element approximations to solutions of interface problems a least squares finite element method for second order elliptic boundary value problems having interfaces due to discontinuous media properties is proposed and analyzed both dirichlet and neumann boundary data are treated the boundary value problems are recast into a first order formulation to which a suitable least squares principle is applied among the advantages of the method are that nonconforming with respect to the interface approximating subspaces may be used moreover the grids used on each side of an interface need not coincide along the interface error estimates are derived that improve on other treatments of interface problems and a numerical example is provided to illustrate the method and the analyses

stepsize range and optimal value for taylor zhang discretization formula applied to zeroing neurodynamics illustrated via future equality constrained quadratic programming in this brief future equality constrained quadratic programming fecqp is studied via a zeroing neurodynamics method a continuous time zeroing neurodynamics ctzn model is presented by using taylor zhang discretization formula to discretize the ctzn model a taylor zhang discrete time zeroing neurodynamics tz dtzn model is presented to perform fecqp furthermore we focus on the critical parameter of the tz dtzn model i e stepsize by theoretical analyses we obtain an effective range of the stepsize which guarantees the stability of the tz dtzn model in addition we further discuss the optimal value of the stepsize which makes the tz dtzn model possess the optimal stability i e the best stability with the fastest convergence finally numerical experiments and application experiments for motion generation of a robot manipulator are conducted to verify the high precision of the tz dtzn model and the effective range and optimal value of the stepsize for fecqp

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

solution of a boundary value problem for the helmholtz equation via variation of the boundary into the complex domain in this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials this corresponds to a two dimensional quasi periodic boundary value problem for the helmholtz equation we prove that solutions behave analytically with respect to variations of the interface the interest of this result is both theoretical the legitimacy of power series expansions in the parameters of the problem has indeed been questioned and perhaps more importantly practical we have found that the solution can be computed on the basis of this observation the simple algorithm that results from such boundary variations is described to establish the property of analyticity of the solution for the grating with respect to the height we present a holomorphic formulation of the problem using surface potentials we show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary or in other words that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables x y this permits us to conclude in particular that the partial derivatives of u with respect to at 0 satisfy certain boundary value problems for the helmholtz equation in regions with plane boundaries which can be solved in a closed form

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

a domain decomposition method with nonconformal meshes for finite periodic and semi periodic structures we present a domain decomposition method as a preconditioner for krylov type solvers to model complex electromagnetic problems containing periodicities the method reduces memory requirements by decomposing the original problem into several nonoverlapping sub domains the 1st order transmission condition is employed on interfaces between adjacent sub domains to enforce continuity of electromagnetic fields and to ensure the sub domain problems are well posed by following the spirit of duality paring a symmetric system is obtained to reduce the computational burdens of the present method the finite element tearing and interconnecting like algorithm is adopted this algorithm results in the computation of the so called numerical green s function which can be compressed efficiently via a rank revealing matrix factorization algorithm the final system matrix is solved by krylov solvers instead of classical stationary solvers to improve the convergence of iterative solvers several robust implementation details are discussed and the choice of some popular krylov subspace solvers is studied through numerical examples

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

finite element approximation of time harmonic waves in periodic structures consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material it is shown that the scattering diffraction problem may be modeled by a helmholtz equation with transparent boundary conditions the diffraction problem may be solved by a finite element method in this paper optimal error estimates for the finite element method are established the error estimates are also established when the truncation of the nonlocal transparent boundary operators takes place