analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

on wave boundary elements for radiation and scattering problems with piecewise constant impedance discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction etc however these methods including the finite element and boundary element methods can require a prohibitively large number of elements as the wavelength becomes progressively shorter in this paper a new type of interpolation for the wave field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions including such a plane wave basis in a boundary element formulation is found in this paper to be highly successful results are shown for a variety of scattering radiating problems from convex and nonconvex obstacles on which are prescribed piecewise constant robin conditions notable results include a conclusion that using this new formulation only approximately three degrees of freedom per wavelength are required

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

a periodic fmm for maxwell s equations in 3d and its applications to problems related to photonic crystals this paper presents an fmm fast multipole method for periodic boundary value problems for maxwell s equations in 3d the effect of periodicity is taken into account with the help of the periodised moment to local expansion m2l transformation formula which includes lattice sums we verify the proposed method by comparing the obtained numerical results with analytic solutions for models of the multi layered dielectric slab we then apply the proposed method to scattering problems for periodic two dimensional arrays of dielectric spheres and compare the obtained energy transmittances with those from the previous studies we also consider scattering problems for woodpile crystals where we find a passband related to a localised mode through these numerical tests we conclude that the proposed method is efficient and accurate

acceleration of fast multipole method for large scale periodic structures with finite sizes using sub entire domain basis functions an acceleration technique to the fast multipole method fmm has been proposed to handle large scale problems of periodic structures in free space with finite sizes based on the accurate sub entire domain basis functions in the proposed algorithm only nine or 27 elements in the whole impedance matrix are required to be computed and stored for a two dimensional or three dimensional periodic structure and the matrix vector multiply can be performed efficiently using the combination of fast fourier transform and fmm the theoretical analysis and numerical results show that both the memory requirement and computational complexity are only of the order of o n with small constants where n is the total number of unknowns

an efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures an efficient multilevel fast multipole algorithm mlfma formalism to model radiation and scattering by from large planar microwave structures is presented the technique relies on an electric field integral equation efie formulation and a series expansion for the green dyadic based on the use of perfectly matched layers pml in this way a new pml mlfma is developed to efficiently evaluate matrix vector multiplications arising in the iterative solution of the scattering problem the computational complexity of the new algorithm scales down to o n for electrically large structures the theory is validated by means of several illustrative numerical examples

approximation properties of bernstein durrmeyer type operators abstract this paper deals with the approximation of continuous functions by sequences of some modified bernstein durrmeyer type operators that reproduce certain test functions the orders of approximation of the new versions turn to be at least as good as the one of the genuine bernstein durrmeyer operators moreover by extrapolating techniques recently applied to the classical bernstein operators we present a one parameter family of modified sequences of operators that reproduce certain polynomials and possess that popular genuine sequence as a limit case comparisons and some illustrative graphics are also presented

on the computation of power in volume integral equation formulations we present simple and stable formulas for computing power including absorbed radiated scattered and extinction power in current based volume integral equation formulations the proposed formulas are given in terms of vector matrix vector products of quantities found solely in the associated linear system in addition to their efficiency the derived expressions can guarantee the positivity of the computed power we also discuss the application of poynting s theorem for the case of sources immersed in dissipative materials the formulas are validated against results obtained both with analytical and numerical methods for scattering and radiation benchmark cases

simulation of near ground long distance radiowave propagation over terrain using nystrom method with phase extraction technique and fmm acceleration a 2d surface integral equation based nystroumlm solver in which a phase extraction technique is utilized to reduce the number of surface unknowns is described as forward scattering is the dominant mechanism for the near ground wave propagation scenario the associated rapidly varying phase components of the integral equation kernel and solution unknowns are deduced and isolated in advance and subsequently built into the solver it is shown that by applying this method when combining with an adaptive surface segmentation routine as few as one to two average unknowns per wavelength is adequate in obtaining accurate solutions this significantly reduces the memory storage and computational expense for the simulation of long distance propagation effects the efficiency of this method is further improved by incorporating it into the framework of the fast multipole scheme the full details of the algorithm are discussed along with performance comparisons of the new solver to a regular nystroumlm solver for terrain surfaces in terms of solution convergence

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

quantum versus deterministic counter automata this paper focuses on quantum analogues of various models of counter automata and almost completely proves the relation between the classes of languages recognizable by bounded error quantum ones and classical deterministic ones in every model of counter automata it is proved that i there are languages that can be recognized by two way quantum one counter automata with bounded error but cannot be recognized by two way deterministic one counter automata ii under some reasonable restriction every language that can be recognized by two way deterministic one counter automata can also be recognized by two way reversible one counter automata and hence by bounded error two way quantum one counter automata and iii for any fixed k quantum ones and deterministic ones are incomparable in one way k counter automata

improved impedance matrix localization method em problems moment method calculations have the well known limitations of requiring excessive storage and execution times for even modestly large electromagnetics problems the impedance matrix localization iml method was introduced as a modification to standard moment method calculations to ease these limitations it utilizes a matrix transformation which effectively changes the basis testing functions into ones resembling traveling waves an improved method that uses an orthogonal transformation to generate standing wave like basis functions is presented here remarkable improvements are achieved in the numerical stability of the method and in its compatibility with iterative solvers furthermore the correspondence of the large elements in this matrix to geometrical theory of diffraction gtd terms is strengthened as is the possibility of further increasing the speed of iterative solutions by constructing preconditioners based on the pattern of nonzero matrix elements

an o 1 integration scheme for three dimensional surface scattering problems we present an accurate method of o 1 complexity with respect to frequency i e a method that to achieve a prescribed error tolerance requires a bounded computational cost for arbitrarily high frequencies for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three dimensional space like the two dimensional counterpart of this algorithm which we introduced recently and which is applicable to scattering by curves in the plane the present method is based on a combination of two main elements 1 a high frequency ansatz for the unknown density in a boundary integral formulation of the problem and 2 an extension of the ideas of the method of stationary phase to allow for o 1 high order accurate integration of oscillatory functions the techniques we introduce to implement an efficient o 1 integrator in the present three dimensional context differ significantly from those used in the earlier two dimensional algorithm in particular in the present text we introduce an efficient canonical hybrid analytic numerical algorithm which in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two dimensional scattering surface we include numerical results illustrating the behavior of the integration algorithm on sound soft spheres with diameters of up to 5000 wavelengths in such cases for a single integral the algorithm yields accuracies of the order of three digits in computational times of less than two seconds in a preliminary full scattering simulation we present a solution with two digits of accuracy in the surface density was obtained in about three hours running time in a single 1 5ghz amd athlon processor for a sphere of 500 wavelengths in diameter

on two way fa with monotonic counters and quadratic diophantine equations we show an interesting connection between two way deterministic finite automata with monotonic counters and quadratic diophantine equations the automaton m operates on inputs of the form a1i1 anin for some fixed n and distinct symbols a1 an where i1 in are nonnegative integers we consider the following reachability problem given a machine m a state q and a presburger relation e over counter values is there i1 in such that m when started in its initial state on the left end of the input a1i1 anin with all counters initially zero reaches some configuration where the state is q and the counter values satisfy e in particular we look at the case when the relation e is an equality relation i e a conjunction of relations of the form ci cj we show that this case and variations of it are equivalent to the solvability of some special classes of systems of quadratic diophantine equations we also study the nondeterministic version of two way finite automata augmented with monotonic counters with respect to the reachability problem finally we introduce a technique which uses decidability and undecidability results to show separation between language classes

iterative solution of high order boundary element method for acoustic impedance boundary value problems abstract a high order boundary element method for time harmonic acoustic impedance boundary value problems is presented the method is based on the galerkin type formulation of the burton miller integral equation bmie with high order polynomial basis and testing functions this formulation has several important advantages it is free of the interior resonance problem the hypersingular integral operator of the traditional bmie formulation can be avoided it leads to faster convergence in terms of the number of unknowns than the low order methods and it shows a good performance with iterative solvers to avoid the numerical difficulties associated to the implementation of the singular surface integral equations the singularity extraction technique is applied to evaluate the integrals in the singular and near singular cases the resulting matrix equation is solved iteratively with the generalized minimal residual method gmres and a simple preconditioner based on the incomplete lu factorization is applied to expedite the convergence numerical results indicate that the bmie formulation with galerkin method and high order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the gmres method is applied to solve the matrix equation iteratively this in turn indicates that the formulation is well suited for an efficient application of fast solution procedures such as the fast multipole method

the method of the local parabolic approximation for rough surface scattering a method for evaluation of scattering from rough surfaces which is similar to the kirchhoff approximation is considered however it is based on a local parabolic approximation of the surface irregularities rather than a tangent plane approximation and two iterations of the surface field integral equation the method first proposed by belobrov and fuks izv vuz radiofiz 29 1083 1089 1986 sov phys acoust 31 442 445 1985 accounts for local diffraction effects important modifications to the original method are introduced and extensive numerical results for gaussian one dimensional randomly rough surfaces for both the dirichlet and neumann problems are provided the validity of the local parabolic approximation is assessed by comparison with results from monte carlo simulations it is demonstrated that the local parabolic approximation improves the kirchhoff approximation for large and intermediate values of the surface correlation length especially in the backscattering region

solution of arbitrarily dimensional matrix equation in computational electromagnetics by fast lifting wavelet like transform a new wavelet matrix transform wmt operated by lifting wavelet like transform lwlt is applied to the solution of matrix equations in computational electromagnetics the method can speedup the wmt without allocating auxiliary memory for transform matrices and can be implemented with the absence of the fast fourier transform furthermore to handle the matrix equation of arbitrarily dimension a new in space preprocessing technique based on lwlt is constructed to eliminate the limitation in matrix dimension complexity analysis and numerical simulation show the superiority of the proposed algorithm in saving cpu time numerical simulations for scattering analysis of differently shaped objects are considered to validate the effectiveness of the proposed method in particular due to its generality the proposed preprocessing technique can be extended to other engineering areas and therefore can pave a broad way for the application of the wmt

electromagnetic scattering from a rough layer propagation inside layer expansion method combined to an updated bmia cag approach an efficient method is developed to calculate the bistatic scattering coefficient bsc from a stack of two one dimensional rough interfaces separating homogeneous media the propagation inside layer expansion pile method recently published by dechamps was efficient with a complexity o n2 n being the number of samples per interface to reduce this complexity a fast method valid for a single rough surface the banded matrix iterative approach canonical grid bmia cag is combined to the pile method furthermore the calculation of the coupling interactions between both interfaces are also accelerated using a new method similar to the bmia cag the pile method reaches then a complexity of only o nlogn a study of the convergence of the pile is done and compared to another efficient method the forward backward

a wideband fast multipole accelerated boundary integral equation method for time harmonic elastodynamics in two dimensions summary this article presents a wideband fast multipole method fmm to accelerate the boundary integral equation method for two dimensional elastodynamics in frequency domain the present wideband fmm is established by coupling the low frequency fmm and the high frequency fmm that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by nishimura s group for each of the two fmms we estimated the approximation parameters that is the expansion order for the low frequency fmm and the quadrature order for the high frequency fmm according to the requested accuracy considering the coexistence of the derivatives of the helmholtz kernels for the longitudinal and transcendental waves in the burton muller type boundary integral equation of interest in the numerical tests the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised also the computational complexity of the present fast boundary integral equation method agreed with the theory that is nlog n where n is the number of boundary elements in a series of scattering problems the present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures as an example the wave propagation along a waveguide fabricated in a finite size phononic crystal was demonstrated copyright 2012 john wiley sons ltd

a contribution to scattering calculation for small wavelengths the high frequency panel method numerical treatment of electromagnetic scattering in the case of small wavelengths too small for reasonable matrix sizes when using discretization methods and too large for traditional asymptotic methods is considered an approach which takes advantage of the user friendly nature of matrix discretization methods while avoiding the explosive increase in computing effort as the wavelength decreases is suggested it appears that no sensible increase of matrix size is required as the wavelength decreases since this is an untried technique applied in an inaccessible computational domain the preliminary work described is confined to two dimensional conducting circular cylinders where the answers are well known and well understood the overall idea however appears to be quite general

a fast high order solver for problems of scattering by heterogeneous bodies a new high order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented here a scatterer is described by a continuously or discontinuously varying refractive index n x within a two dimensional 2d bounded region solutions of the associated helmholtz equation under given incident fields are then obtained by high order inversion of the lippmann schwinger integral equation the algorithm runs in o nlog n operations where n is the number of discretization points a wide variety of numerical examples provided include applications to highly singular geometries high contrast configurations as well as acoustically electrically large problems for which supercomputing resources have been used recently our method provides highly accurate solutions for such problems on small desktop computers in cpu times of the order of seconds

asymptotically derived boundary elements for the helmholtz equation in high frequencies we present an asymptotically derived boundary element method for the helmholtz equation in exterior domains each basis function is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase factor is determined a priori by using arguments from geometrical optics and the geometrical theory of diffraction while the smooth amplitude is represented by high order splines this yields a high order method in which the number of unknowns is virtually independent of the wavenumber k two types of diffracted basis functions are presented the first accounts for the dominant oscillatory behavior in the shadow region while the second also accounts for the decay of the amplitude there we show that the matrix a associated with the discrete problem has only o n significant entries as k where n is the number of basis functions hence it can be approximated with a matrix having o n terms and the relative error between a and rapidly converges to zero as k although the method is applicable to a variety of scatterers we focus our attention here on scattering from smooth closed convex bodies in two dimensions computations on a circular cylinder illustrate our results

discretized boundary equation method for two dimensional scattering problems a unified approach named discretized boundary equation dbe method is introduced for two dimensional 2 d scattering problems it is based on the discretization of field expressions for one or two components of the scattered field the dbes can be used either on the object surface to obtain the solution directly or on the truncation boundary of a finite difference fd or finite element fe mesh as termination conditions this paper describes the general theory of the dbe method and key points or limitations for its implementation a new on surface formulation for the solution of scattering by perfectly conducting cylinders is presented as an application of the two component version of the dbe method and validated through numerical examples mesh termination conditions for the fd or fe method are derived based on the one component formulation of the dbe method and their equivalence and difference to the measured equation of invariance are discussed in particular the dbe obtained with the minimum norm least squares solution is investigated thoroughly and its validity and features are demonstrated through numerical results generated together with the fd method for scattering by cylinders with various material properties

a fast integral equation approach to acoustic scattering from three dimensional objects using a natural basis set boundary value integral equations and the moment method solution technique provide a succinct and elegant strategy for the solution of acoustic surface scattering problems this paper explores the application of a subdomain basis set based on the phase of the incident radiation which is used to solve for the unknown normal pressure gradient on the surface of the scatterer using the moment method the advantage of this basis set lies in the fact that it can accurately represent the fast phase character of the pressure gradient on the surface of the scatterer this allows for far fewer discretizations basis elements in the moment method formulation and therefore significantly reduces the density of unknowns that must be numerically computed the main contribution of this paper is to show how to efficiently employ this natural basis and to explore the physical justification for using this approach the method is compared with other techniques including the kirchhoff solution technique high frequency single scattering techniques the nussenzveig fock technique and the parabolic equation approximation unlike these methods no approximation to the physical system need be introduced

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

scattering at a rough boundary extensions of the kirchhoff approximation a helmholtz integral formula gives the acoustic field reflected from a rough surface in terms of the values of the acoustic field and of its normal derivative at all points on the surface for surface reflection problems in which one or the other of these surface field values are specified we can make use of this formula to derive a boundary surface integral equation on the unspecified field values further a series solution of this equation can be constructed by iteration the zeroth order term of the series being the result predicted by a kirchhoff approximation the nth order iterate of this series requires an n fold integration over the region of the rough surface an integration that does not in general converge absolutely for unboundedly large surfaces for a pressure release surface we consider the first two iterates in detail and using stationary phase approximations replace the required integration by summations in this way the source of the convergence difficulty is made clear and is demo

the hybridization of fast multipole method with asymptotic waveform evaluation for the fast monostatic rcs computation the fast multipole method hybridized with the asymptotic waveform evaluation is presented in this paper to efficiently calculate the scattering from electrically large objects a novel interpolation method for the hybridization is proposed which provides a more accurate result for the scattering pattern calculation than the original extrapolation method the accuracy and efficiency of this proposed hybrid method are examined through numerical examples

the equivalence problem and correctness formulas for a simple class of programs this paper is concerned with the semantics or computational power of very simple loop programs over different sets of primitive instructions recently a complete and consistent hoare axiomatics for the class of x 0 x y x x 1 x x 1 do x end programs which contain no nested loops was given where the allowable assertions were those formulas in the logic of presburger arithmetic the class of functions computable by such programs is exactly the class of presburger functions thus the resulting class of correctness formulas has a decidable validity problem in this paper we present simple loop programming languages which are computationally strictly more powerful i e which can compute more than the class of presburger functions furthermore using a logical assertion language that is also more powerful than the logic of presburger arithmetic we present a class of correctness formulas over such programs that also has a decidable validity problem in related work we examine the expressive power of loop programs over different sets of primitive instructions in particular we show that an x 0 x y x x 1 do x end if x 0 then y z program which contains no nested loops can be transformed into an equivalent x 0 x y x x 1 do x end program also without nested loops in exponential time and space this translation was earlier claimed in the literature to be doable in polynomial time but then this was subsequently shown to imply that pspace ptime consequently the question of translatability was left unanswered also we show that the class of functions computable by x 0 x y x x 1 x x 1 do x end if x 0 then x c programs is exactly the class of presburger functions when the conditional instruction is changed to if x 0 then x y 1 then the class of computable functions is significantly enlarged enough so in fact as to render many decision problems e g equivalence undecidable

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

on the use of fourier series fft s as global basis functions in the solution of boundary integral equations for em scattering current methods for the solution of the surface integral equations of interest to practical electromagnetic scattering appear to have reached an impasse plagued by disputes concerning the suitability and performance of various competing choices for piecewise basis functions and facing horribly complicated piecewise quadratures for the construction of influence matrix elements it has become quite commonplace for these methods to be matrix build time limited rather than matrix solve time limited an extremely simple alternative to all this is described below and demonstrated in an introductory sense on some 2 d problems in this procedure the fft method the basis functions are all global fourier series in a suitably chosen parameter and one attempts to estimate the coefficients describing the unknown surface current directly from the integral equation this format circumvents all of the shortcomings discussed above there are no singular integrals to evaluate furthermore since the fft algorithm is used throughout the computation is extremely efficient the results shown below suggest that so far this approach is very effective

application of the integral equation asymptotic phase method to two dimensional scattering a hybrid procedure called the integral equation asymptotic phase ie ap method is investigated for scattering from perfectly conducting cylinders of arbitrary cross section shape the ie ap approach employs an asymptotic solution to predict the relatively rapid phase dependence of the unknown current distribution to leave a slowly varying residual function that can be represented by a coarse density of unknowns in the present investigation the current density appearing within the combined field integral equation is replaced by the product of a rapidly varying phase function obtained from the physical optics current and a residual function the resulting equation is discretized by the method of moments using subsectional quadratic polynomial basis functions defined on curved cells to represent the residual function results show that the required density of unknowns can often be as few as one per wavelength on average without a significant loss of accuracy in the computed current density even for scatterers with corners

incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems we consider the preconditioned iterative solution of large dense linear systems where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering for some scattering structures this matrix can be poorly conditioned the main purpose of this study is to evaluate the efficiency of a class of incomplete lu ilu factorization preconditioners for solving this type of matrices we solve the electromagnetic wave equations using the bicg method with an ilu preconditioner in the context of a multilevel fast multipole algorithm mlfma the novelty of this work is that the ilu preconditioner as constructed using the near part block diagonal submatrices generated from the mlfma experimental results show that the ilu preconditioner reduces the number of bicg iterations substantially compared to the block diagonal preconditioner the preconditioned iteration scheme also maintains the computational complexity of the mlfma and consequently reduces the total cpu time

on iterative approaches for electromagnetic rough surface scattering problems iterative techniques developed for solving general systems of linear equations have been applied to systems resulting from electromagnetic rough surface scattering problems previously used iterative procedures that model the multiple scattering of the electromagnetic energy are shown to be mathematically equivalent to the application of stationary iterative procedures to the system of equations resulting from the standard moment method convergence difficulties that are sometimes observed with these approaches are due to the inherent limitations of the stationary techniques the performances of the stationary approaches are compared with that of several conjugate direction based nonstationary iterative procedures through the application to a series of scattering surfaces that yield rapidly changing conditioning of the moment method interaction matrix the stationary algorithms give the quickest convergence when applied to the systems with the best conditioning but the nonstationary techniques prove much more robust in other more ill conditioned situations

on two way nondeterministic finite automata with one reversal bounded counter we show that the emptiness problem for two way nondeterministic finite automata augmented with one reversal bounded counter i e the counter alternates between nondecreasing and nonincreasing modes for a fixed number of times operating on bounded languages i e subsets of w 1 w k for some nonnull words w1 wk is decidable resolving an open problem the proof is a rather involved reduction to the solution of a special class of diophantine systems of degree 2 via a class of programs called two phase programs

reflection of a spherical wave by acoustically hard concave cylindrical walls based on the tangential plane approximation the tangential plane approximation tpa is introduced to investigate the spherical wave reflection from smooth concave cylindrical walls and as a practical calculation scheme asymptotic expression of the reflected sound from that surface with a relatively large dimension is derived the physical condition under which tpa holds is derived for the spherical wave incidence moreover the essential properties of the reflected sound from the curved walls which cannot be treated by the geometrical acoustics are discussed through the numerical calculation for infinitely long cylinders a formula of the reflection factor of the two dimensional curved surface is obtained for the purpose of the room acoustical design this formula coincides with that based on the geometrical acoustics when the frequency is infinitely high

volumetric fast multipole method for modeling schrodinger s equation a volume integral equation method is presented for solving schrodinger s equation for three dimensional quantum structures the method is applicable to problems with arbitrary geometry and potential distribution with unknowns required only in the part of the computational domain for which the potential is different from the background two different green s functions are investigated based on different choices of the background medium it is demonstrated that one of these choices is particularly advantageous in that it significantly reduces the storage and computational complexity solving the volume integral equation directly involves o n2 complexity in this paper the volume integral equation is solved efficiently via a multi level fast multipole method mlfmm implementation requiring o nlogn memory and computational cost we demonstrate the effectiveness of this method for rectangular and spherical quantum wells and the quantum harmonic oscillator and present preliminary results of interest for multi atom quantum phenomena

multistep and multistage convolution quadrature for the wave equation algorithms and experiments we describe how a time discretized wave equation in a homogeneous medium can be solved by boundary integral methods the time discretization can be a multistep runge kutta or a more general multistep multistage method the resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of lubich the aim of this work is twofold it describes an efficient robust and easily parallelizable method for solving the semidiscretized system the resulting algorithm has the main advantages of time stepping methods and of fourier synthesis at each time step a system of linear equations with the same system matrix needs to be solved yet computations can easily be done in parallel the computational cost is almost linear in the number of time steps and only the laplace transform of the time domain fundamental solution is needed the new aspect of the algorithm is that all this is possible without ever explicitly constructing the weights of the convolution quadrature this approach also readily allows the use of modern data sparse techniques to efficiently perform computation in space we investigate theoretically and numerically to which extent hierarchical matrix mathcal h matrix techniques can be used to speed up the space computation the second aim of this article is to perform a series of large scale 3d experiments with a range of multistep and multistage time discretization methods the backward difference formula of order 2 bdf2 the trapezoid rule and the 3 stage radau iia methods are investigated in detail one of the conclusions of the experiments is that the radau iia method often performs overwhelmingly better than the linear multistep methods especially for problems with many reflections yet in connection with hyperbolic problems bdfs have so far been predominant in the literature on convolution quadrature

approximation by complex phillips stancu operators in compact disks under exponential growth conditions the present article deals with the approximation properties of the complex phillips stancu operators here we study a voronovskaja kind asymptotic formula with quantitative estimates for these operators attached to analytic functions and having growth of exponential order on compact disks also we obtain the exact order of approximation

adaptive multiscale moment method ammm for analysis of scattering from three dimensional perfectly conducting structures the adaptive multiscale moment method ammm is presented for the analysis of scattering from three dimensional 3d perfectly conducting bodies this algorithm employs the conventional moment method mm using the subsectional triangular patch basis functions and a special matrix transformation which is derived from solving the fredholm equation of the first kind by the multiscale technique this methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time the widely used triangular patch vector basis functions developed by rao et al 1982 are used for expansion and testing functions in the conventional mm the objective here is to compress the unknowns in existing mm codes which solves for the currents crossing the edges of the triangular patch basis functions by use of a matrix transformation the currents source terms and impedance matrix can be arranged in the form of different scales from one scale to another scale the initial guess for the solution can be predicted according to the properties of the multiscale technique ammm can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional mm the basic difference between this methodology and the other wavelet based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown two numerical results are presented which demonstrate that the ammm is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3d perfectly conducting bodies

approximation by complex genuine hybrid operators abstract in the recent years several complex operators of summation integral type have been proposed and their approximation properties have been discussed the operators which have baskakov or szasz basis functions have very complicated analysis in the present article we propose a complex hybrid operator the special case of which provides the genuine baskakov szasz operators here we establish an upper estimate voronovskaja type asymptotic formula and exact order of approximation

scattering from planar structures containing small features using the adaptive integral method aim fast integral equation algorithms such as the adaptive integral method aim have been demonstrated to reduce memory and execution time associated with moment method solutions for arbitrarily shaped three dimensional 3 d geometries in this paper we examine the efficiency of aim in modeling planar structures that contain small and intricate details as is the case with spirals and slot antennas such geometries require high tessellation due to the inclusion of very small features resulting in a large number of unknowns aim with its capability to translate the original grid to an equivalent sparser uniform grid is uniquely suited for the analysis of such geometries in the latter part of the paper we demonstrate the application of aim in connection with a finite element boundary integral formulation for cavity backed antennas

on the numerical evaluation of the singular integrals of scattering theory in a previous work the authors introduced a scheme for the numerical evaluation of the singular integrals which arise in the discretization of certain weakly singular integral operators of acoustic and electromagnetic scattering that scheme is designed to achieve high order algebraic convergence and high accuracy when applied to operators given on smoothly parameterized surfaces this paper generalizes the approach to a wider class of integral operators including many defined via the cauchy principal value operators of this type frequently occur in the course of solving scattering problems involving boundary conditions on tangential derivatives the resulting scheme achieves high order algebraic convergence and approximately 12 digits of accuracy

numerical accuracy of a pade type non reflecting boundary condition for the finite element solution of acoustic scattering problems at high frequency the present text deals with the numerical solution of two dimensional high frequency acoustic scattering problems using a new high order and asymptotic pade type artificial boundary condition the pade type condition is easy to implement in a galerkin least squares iterative finite element solver for arbitrarily convex shaped boundaries the method accuracy is investigated for different model problems and for the scattering problem by a submarine shaped scatterer as a result relatively small computational domains optimized according to the shape of the scatterer can be considered while yielding accurate computations for high frequencies copyright 2005 john wiley sons ltd

a mixed form fast multipole algorithm the fast multipole algorithm manifests in two very different forms at low frequencies and at mid frequencies each can operate in their respective regimes but are not tenable in the other regimes the paper reports on a way to factorize the green s function for fast algorithm using a mixed form the low frequency fast multipole algorithm lf fma will be used at low frequencies or the long wavelength regime and the multilevel fast multipole algorithm mlfma will be used for the mid frequencies or the shorter wavelength regime for object modeling where both long wavelength and wave physics are important we propose a mixed form fast multipole algorithm mf fma this algorithm has no low frequency break down and it can work seamlessly from static where circuit physics is important to dynamic where wave physics is important

truncation error analysis of multipole expansion the multilevel fast multipole algorithm is based on the multipole expansion which has numerical error sources such as truncation of the addition theorem numerical integration and interpolation anterpolation of these we focus on the truncation error and discuss its control precisely the conventional selection rule fails when the buffer size is small compared to the desired numerical accuracy we propose a new approach and show that the truncation error can be controlled and predicted regardless of the number of buffer sizes

quantitative estimates for a new complex durrmeyer operator in compact disks abstract in the recent years the extension of linear positive operators from real to complex domain is one of the interesting area of research in this context we present the exact order of simultaneous approximation and voronovskaja kind results with quantitative estimate for a new type of complex durrmeyer operator different from those previously studied attached to analytic functions in compact disks in this way we put in evidence the overconvergence phenomenon for this kind of durrmeyer operator namely the extensions of approximation properties with exact quantitative estimates from the real interval 1 3 2 3 to compact disks in the complex plane

stabilizing block diagonal preconditioners for complex dense matrices in electromagnetics abstract preconditioning techniques are widely used to speed up the convergence of iterative methods for solving large linear systems with sparse or dense coefficient matrices for certain application problems however the standard block diagonal preconditioner makes the krylov iterative methods converge more slowly or even diverge to handle this problem we apply diagonal shifting and stabilized singular value decomposition svd to each diagonal block which is generated from the multilevel fast multiple algorithm mlfma to improve the stability and efficiency of the block diagonal preconditioner our experimental results show that the improved block diagonal preconditioner maintains the computational complexity of mlfma converges faster and also reduces the cpu cost

finding numerical solutions of diophantine equations using ant colony optimization the paper attempts to find numerical solutions of diophantine equations a challenging problem as there are no general methods to find solutions of such equations it uses the metaphor of foraging habits of real ants the ant colony optimization based procedure starts with randomly assigned locations to a fixed number of artificial ants depending upon the quality of these positions ants deposit pheromone at the nodes a successor node is selected from the topological neighbourhood of each of the nodes based on this stochastic pheromone deposit if an ant bumps into an already encountered node the pheromone is updated correspondingly a suitably defined pheromone evaporation strategy guarantees that premature convergence does not take place the experimental results which compares with those of other machine intelligence techniques validate the effectiveness of the proposed method

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

application of volume integrals to the solution of partial differential equations abstract a group of algorithms for the numerical solution of elliptic partial differential equations is presented the differential equation is reduced to a fredholm equation of the second kind which in turn is discretized by means of the nystrom algorithm leading to a dense large scale system of linear algebraic equations this linear system is solved by means of the generalized conjugate residual algorithm which requires that the matrix of the system be applied to a sequence of recursively generated vectors since applying a dense matrix to a vector is a process of order n 2 where n is the dimension of the system the resulting algorithms are usually not competitive with finite differences and finite elements in terms of cpu time requirements however it turns out that the matrices of linear systems resulting from many partial differential equations can be applied to vectors in a fast manner i e for the cost proportional to n in some cases and to n log n or n log n 2 in others resulting in extremely efficient algorithms for the solution of certain elliptic partial differential equations the performances of several such algorithms are illustrated by numerical examples

fast inverse scattering solutions using the distorted born iterative method and the multilevel fast multipole algorithm the distorted born iterative method dbim computes iterative solutions to nonlinear inverse scattering problems through successive linear approximations by decomposing the scattered field into a superposition of scattering by an inhomogeneous background and by a material perturbation large or high contrast variations in medium properties can be imaged through iterations that are each subject to the distorted born approximation however the need to repeatedly compute forward solutions still imposes a very heavy computational burden to ameliorate this problem the multilevel fast multipole algorithm mlfma has been applied as a forward solver within the dbim the mlfma computes forward solutions in linear time for volumetric scatterers the typically regular distribution and shape of scattering elements in the inverse scattering problem allow the method to take advantage of data redundancy and reduce the computational demands of the normally expensive mlfma setup additional benefits are gained by employing kaczmarz like iterations where partial measurements are used to accelerate convergence numerical results demonstrate both the efficiency of the forward solver and the successful application of the inverse method to imaging problems with dimensions in the neighborhood of ten wavelengths