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

some limit results for lag sums of independent non i i d random variables let x k be an independent but not necessarily i i d sequence of random variables suppose ex k 0 and that nk 2 e s n s n k 2 where s n x1 xn we obtain conditions under which mathop lim sup limits n to infty mathop max limits n alpha leqq k leqq n frac s n s n k sigma n k 2 log n k log log k 1 2 1 text a text s text and obtain other related results we assume either that p x k t mt r for all k and all t 0 or that e expx k b for all k and all t0 the results are like earlier results for the i i d case but the method of proof is entirely different

application of the inverse fast multipole method as a preconditioner in a 3d helmholtz boundary element method abstract we investigate an efficient preconditioning of iterative methods such as gmres for solving dense linear systems a x b that follow from a boundary element method bem for the 3d helmholtz equation focusing on the low frequency regime while matrix vector products in gmres can be accelerated through the low frequency fast multipole method lffmm the bem often remains computationally expensive due to the large number of gmres iterations we propose the application of the inverse fast multipole method ifmm as a preconditioner to accelerate the convergence of gmres the ifmm is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low rank approximations the proposed ifmm based preconditioning has a tunable parameter e that balances the cost to construct a preconditioner m which is an approximation of a 1 and the cost to perform the iterative process by means of m namely using a small respectively large value of e takes a long respectively short time to construct m while the number of iterations can be small respectively large a comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method we show that the ifmm preconditioner with a nearly optimal e of 10 2 clearly outperforms some common preconditioners for the bem achieving 1 2 10 8 times speed up of the computations in particular when the scale of the underlying scatterer is about five wavelengths or more in addition the ifmm preconditioner is capable of solving complicated problems in a reasonable amount of time that bd preconditioner can not

a comparison of techniques for overcoming non uniqueness of boundary integral equations for the collocation partition of unity method in two dimensional acoustic scattering the partition of unity method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements the non uniqueness of solution obtained from the conventional boundary integral equation cbie is well known the cbie derived through green s identities suffers from a problem of non uniqueness at certain characteristic frequencies two of the standard methods of overcoming this problem are the so called combined helmholtz integral equation formulation chief method and that of burton and miller the latter method introduces a hypersingular integral which may be treated in various ways in this paper we present the collocation partition of unity boundary element method pubem for the helmholtz problem and compare the performance of chief against a burton miller formulation regularised using the approach of li and huang

generalized finite element method for vector electromagnetic problems the generalized finite element method gfem as introduced by babuska is a partition of unity framework for solving scalar partial differential equations pdes to date they have been applied extensively to the solution of elliptic and parabolic pdes this technique can be interpreted as a unified prescription of a host of well known techniques such as classical finite element methods that are based on tessellation point cloud techniques and element free galerkin methods the focus of this paper is to extend this technique to solve vector electromagnetic problems however modification of gfem to address this problem poses a novel set of challenges i the vector nature of the problem and the different continuity requirements on each component imply that basis functions developed should share similar characteristics ii the basis functions have to be able to represent divergence free electromagnetic fields in a source free region when they themselves are not divergence free these have been the primary impediments to extension of this technique in this paper we propose a method that may be used to overcome these obstacles and will demonstrate h and p convergence characteristics of the proposed method for a range of problems finally we will demonstrate convergence characteristics of this technique for analyzing scattering from a sharp wedge when the basis functions include analytical local field representation

notes on the norm map between the hecke algebras of the gelfand graev representations of gl 2 q2 and u 2 q abstract let g be a connected reductive algebraic group defined over the field f q and let f and f be two frobenius maps such that f m f m for some integer m let g f g f and g f m g f m be the finite groups of fixed points in this article we consider the case where g gl 2 f q f is the usual frobenius map so that g f gl 2 q and f is the twisted frobenius map such that g f u n q in this case f 2 f 2 and g f 2 g f 2 gl 2 q 2 this article provides connections between the complex representation theory of these groups using the norm maps see c curtis t shoji a norm map for endomorphism algebras of gelfand graev representations in progr math vol 141 1997 pp 185 194 from the gelfand graev hecke algebra of gl 2 q 2 to the gelfand graev hecke algebras of both gl 2 q and u 2 q

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

computation of scattering from clusters of spheres using the fast multipole method a t matrix based method of solution of the multiple scattering problem was presented by the authors j acoust soc am 112 2688 2701 2002 this method can be applied to the computation of relatively small problems since the number of operations required grows with the number of spheres n as o n3 and with the sixth power of the wave number the use of iterative techniques accelerated using the fast multipole method fmm can accelerate this solution as presented by koc and chew j acoust soc am 103 721 734 1998 originally in this study we present a method that combines preconditioned krylov subspace iterative techniques fmm accelerated matrix vector products a novel fmm based preconditioner and fast translation techniques that enable us to achieve an overall algorithm in which the cost of the matrix vector multiplication grows with n as o n logn and with the third power of the wave number we discuss the convergence of the iterative techniques selection of the truncation number errors in the solution and other issues the results of the solution of test problems obtained with the method for n approximately 10 2 10 4 for different wave numbers are presented

recursions for the computation of multipole translation and rotation coefficients for the 3 d helmholtz equation we develop exact expressions for the coefficients of series representations of translations and rotations of local and multipole fundamental solutions of the helmholtz equation in spherical coordinates these expressions are based on the derivation of recurrence relations some of which to our knowledge are presented here for the first time the symmetry and other properties of the coefficients are also examined and based on these efficient procedures for calculating them are presented our expressions are direct and do not use the clebsch gordan coefficients or the wigner 3 j symbols although we compare our results with methods that use these to prove their accuracy for evaluating an nt term truncation of the translated series involving o n t 2 multipoles our expressions require o n t 3 evaluations compared to previous exact expressions that require o n t 5 operations

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

a fast and stable solver for acoustic scattering problems based on the nonuniform grid approach a fast and stable boundary element method bem algorithm for solving external problems of acoustic scattering by impenetrable bodies is developed the method employs the burton miller integral equation which provides stable convergence of iterative solvers and a generalized multilevel nonuniform grid mlng algorithm for fast evaluation of field integrals the mlng approach is used here for the removal of computational bottlenecks involved with repeated matrix vector multiplications as well as for the low order basis function regularization of the hyper singular integral kernel the method is used for calculating the fields scattered by large acoustic scatterers including nonconvex bodies with piece wise smooth surfaces as a result the algorithm is capable of accurately incorporating high frequency effects such as creeping waves and multiple edges diffractions in all cases stable convergence of the method is observed high accuracy of the method is demonstrated by comparison with the traditional b

some results on increments of the wiener process with applications to lag sums of i i d random variables obtained for the wiener process but with n replacing t and sn replacing w t using the work of koml6s major and tusnady on the invariance principle we obtain the desired results for i i d sequences as immediate corollaries to our work for the wiener process 1 introduction the motivation for the work done here came from a specific statistical problem suppose xk is a sequence of random variables whose distribution functions approach in some appropriate sense a fixed distribution with mean it if the sequence is independent or is appropriately dependent then under mild regularity conditions sn n t almost surely there will usually be bias or non random error associated with the earlier xk s one might hope to reduce this bias by throwing away

the fast multipole method i error analysis and asymptotic complexity this paper is concerned with the application of the fast multipole method fmm to the maxwell equations this application differs in many aspects from other applications such as the n body problem laplace equation and quantum chemistry etc the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products this leads to faster resolution of scattering of harmonic plane waves from perfectly conducting obstacles emphasis here is on a rigorous mathematical approach to the problem we focus on the estimation of the error introduced by the fmm and a rigorous analysis of the complexity o n log n of the algorithm we show that error estimates reported previously are not entirely satisfactory and provide sharper and more precise estimates

efficient fast multipole method for low frequency scattering the solution of the helmholtz and maxwell equations using integral formulations requires to solve large complex linear systems a direct solution of those problems using a gauss elimination is practical only for very small systems with few unknowns the use of an iterative method such as gmres can reduce the computational expense most of the expense is then computing large complex matrix vector products the cost can be further reduced by using the fast multipole method which accelerates the matrix vector product for a linear system of size n the use of an iterative method combined with the fast multipole method reduces the total expense of the computation to n log n there exist two versions of the fast multipole method one which is based on a multipole expansion of the interaction kernel exp ikr r and which was first proposed by v rokhlin and another based on a plane wave expansion of the kernel first proposed by w c chew in this paper we propose a third approach the stable plane wave expansion spw fmm which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion the computational complexity is n log n as with the other methods

generalized method of moments a boundary integral framework for adaptive analysis of acoustic scattering boundary integral equations bies find applications in problems ranging from sonar to medical diagnostics the two ingredients of the bie solution technique are 1 representation of the domain and 2 design of approximation spaces to represent physical quantities on the domain these in concert affect accuracy and convergence of the simulation this paper presents a framework that permits the development of a scheme for refinement of size and order in both geometry and function representations further this permits flexibility in the types of basis functions that can be used capabilities of the proposed framework are shown via a number of numerical examples

a wideband fast multipole method for the helmholtz kernel this work presents a new fast multipole method fmm based on plane wave expansions pwfmm combining the advantages of the low and high frequency formulations we revisit the method of greengard etal 1998 devoted to the low frequency regime and based on the splitting of the greens function into a propagative and an evanescent part more precisely we give an explicit formula of the filtered translation function for the propagative part we derive a new formula for the evanescent part and we provide a new interpolation algorithm at all steps we check the accuracy of the method by providing error estimates these theoretical developments are used to propose a wideband fmm based entirely on plane wave expansions the numerical efficiency and accuracy of this broadband pwfmm are illustrated with a numerical example

accurate high order representation of complex three dimensional surfaces via fourier continuation analysis we present a new method for construction of high order parametrizations of surfaces starting from point clouds the method we propose can be used to produce full surface parametrizations by sets of local charts each one representing a large surface patch which typically contains thousands of the points in the original point cloud for complex surfaces of scientific and engineering relevance the proposed approach accurately renders both smooth and non smooth portions of a surface it yields super algebraically convergent fourier series approximations to a given surface up to and including all points of geometric singularity such as corners edges conical points etc in view of their c smoothness except at true geometric singularities and their properties of high order approximation the surfaces produced by this method are suitable for use in conjunction with high order numerical methods for boundary value problems in domains with complex boundaries including pde solvers integral equation solvers etc our approach is based on a very simple concept use of fourier analysis to continue smooth portions of a piecewise smooth function into new functions which defined on larger domains are both smooth and periodic the continuation functions arising from a function f converge super algebraically to f in its domain of definition as discretizations are refined we demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance

an interior penalty method for the generalized method of moments the generalized method of moments gmm is a technique to discretize integral equations that permits integration of different types of basis functions as well as different geometric descriptions using a partition of unity framework while accuracy and efficacy of the method have been demonstrated the integration quadratures required to compute the inner products are often high as they have to respect spatial variation of the integrand to overcome this problem we introduce an interior penalty function method within the gmm framework the penalty formulation yields solutions that are smooth and accurate both in surface currents and fields with significantly lower numerical quadrature orders than would be required for the uncompensated operator to demonstrate and analyze the method we conduct an analytical and numerical investigation of the properties of the penalty method applied to the 2 d text te z electric field integral equation efie operator

further development of vector generalized finite element method and its hybridization with boundary integrals recently vector generalized finite element method vgfem was introduced for the solution of the vector helmholtz equation and its applicability was validated for canonical problems vgfem uses a local helmholtz decomposition to construct basis functions in overlapping local domains of some canonical shape while using a canonical shape for local domains adds flexibility to the method one needs to provide information regarding boundaries of domains inhomogeneities the need for surface information proves to be a bottleneck in using the method for a larger class of problems this paper is targeted towards overcoming these deficiencies here we will introduce the modifications to this method that permit interfacing with arbitrarily shaped local domains to facilitate interfacing with existing meshing software integrate this method with boundary integrals and provide a framework for studying dispersion as will be apparent the hybridization of the method with boundary integrals is not a simple adaptation of existing methods onto the vgfem framework likewise dispersion analysis is nontrivial due to the overlapping nature of vgfem basis functions a range of practical problems has been analyzed within the presented framework and results are compared either against measurements or existing fem data to validate the presented methodology

a broadband fast multipole accelerated boundary element method for the three dimensional helmholtz equation the development of a fast multipole method fmm accelerated iterative solution of the boundary element method bem for the helmholtz equations in three dimensions is described the fmm for the helmholtz equation is significantly different for problems with low and high kd where k is the wavenumber and d the domain size and for large problems the method must be switched between levels of the hierarchy the bem requires several approximate computations numerical quadrature approximations of the boundary shapes using elements and these errors must be balanced against approximations introduced by the fmm and the convergence criterion for iterative solution these different errors must all be chosen in a way that on the one hand excess work is not done and on the other that the error achieved by the overall computation is acceptable details of translation operators for low and high kd choice of representations and bem quadrature schemes all consistent with these approximations are described

use of simple solutions for boundary integral methods in elasticity and fracture analysis the simple solutions or indirect method of analysing cauchy and hypersingular integrands in the gradient flux or traction boundary integral equation bie is applied to linear elastic fracture analysis because of the geometric singularity of the crack surface application of the simple solutions formulas on the crack face requires integration over a temporary closure surface rather than the remainder of the body closure surface constructions are exhibited for crack surfaces allowing the gradient bie to be applied as a constraint equation on a crack surface where the primary bie is degenerate computational results are given for two benchmark fracture problems

a burton miller inverse boundary element method for near field acoustic holography an inverse boundary element method based on the burton miller integral equation is proposed for reconstructing the neumann boundary data from pressure values on a conformal surface in the near field of an arbitrary radiating object the accuracy of the reconstruction is compared with that of a method based on the more commonly used helmholtz integral equation in particular the behavior at characteristic frequencies which are known to be problematic in the helmholtz integral equation for the forward problem is examined the effect of regularization is considered including the l curve parameter selection method numerical computations are given for noisy data generated from an internal point source