multilevel fast multipole algorithm with global and local interpolators in this paper a new version of the multilevel fast multipole algorithm mlfma based on an interpolator that utilizes trigonometric polynomial expansion is presented this novel version allows in levels where the cube size is large the use of the local interpolator that utilizes the lagrange interpolating polynomial the accuracy of aggregation and disaggregation is improved by choosing the sample rates for each cube according to the distribution of the basis functions inside the cube in question additionally an algorithm for an efficient evaluation of the translator is presented further some optimizations and good compromises both in terms of memory and cpu time are suggested hierarchical matrices are employed to reduce the memory requirements of the sparse block system matrix the provided numerical results demonstrate that the presented version is able to maintain good accuracy provided that the local interpolator is only utilized in levels where cube size is large enough

an efficient hybrid mlfma fft solver for the volume integral equation in case of sparse 3d inhomogeneous dielectric scatterers electromagnetic scattering problems involving inhomogeneous objects can be numerically solved by applying a method of moments discretization to the volume integral equation for electrically large problems the iterative solution of the resulting linear system is expensive both computationally and in memory use in this paper a hybrid mlfma fft method is presented which combines the fast fourier transform fft method and the high frequency multilevel fast multipole algorithm mlfma in order to reduce the cost of the matrix vector multiplications needed in the iterative solver the method represents the scatterers within a set of possibly disjoint identical cubic subdomains which are meshed using a uniform cubic grid this specific mesh allows for the application of ffts to calculate the near interactions in the mlfma and reduces the memory cost considerably since the aggregation and disaggregation matrices of the mlfma can be reused additional improvements to the general mlfma framework such as an extention of the fft interpolation scheme of sarvas et al from the scalar to the vectorial case in combination with a more economical representation of the radiation patterns on the lowest level in vector spherical harmonics are proposed and the choice of the subdomain size is discussed the hybrid method performs better in terms of speed and memory use on large sparse configurations than both the fft method and the hf mlfma separately and it has lower memory requirements on general large problems this is illustrated on a number of representative numerical test cases

relative perturbation bound for invariant subspaces of graded indefinite hermitian matrices abstract we give a bound for the perturbations of invariant subspaces of graded indefinite hermitian matrix h d ad which is perturbed into h h d a a d such relative perturbations include an important case where h is given with an element wise relative error application of our bounds requires only the knowledge of the size of relative perturbation a and not the perturbation a itself this typically occurs when data are given with relative uncertainties when the matrix is being stored into computer memory and when analyzing some numerical algorithms subspace perturbations are measured in terms of perturbations of angles between subspaces and our bound is therefore a relative variant of the well known davis kahan sin theorem our bounds generalize some of the recent relative perturbation results

broadband muller mlfma for electromagnetic scattering by dielectric objects the multilevel fast multipole algorithm mlfma is very efficient for solving large scale electromagnetic scattering problems however at low frequencies or when the discretization is small compared with the wavelength both the mlfma and the underlying integral equation formulation typically suffer from a subwavelength breakdown for the electromagnetic scattering from a homogeneous dielectric object we obtain a stable and well conditioned surface integral formulation using a variant of the classical muumlller formulation and linear basis functions to overcome the subwavelength breakdown of the mlfma we use both propagating and evanescent plane waves to represent the fields the implementation is based on a combination of the spectral representation of the green s function and rokhlin s translation formula we also present a new interpolation scheme for the evanescent part which significantly improves the error controllability of the mlfma implementation several numerical results verify both the error controllability and scalability of the proposed algorithm

relative perturbation theory for hyperbolic eigenvalue problem we give relative perturbation bounds for eigenvalues and perturbation bounds for eigenspaces of a hyperbolic eigenvalue problem hx jx where h is a positive definite matrix and j is a diagonal matrix of signs we consider two types of perturbations when a graded matrix h d ad is perturbed in a graded sense to h h d a a d and the multiplicative perturbations of the form h h i e h i e our bounds are simple to compute compare well to the classical results and can be used when analyzing numerical algorithms

calculation of acoustical scattering from a cluster of scatterers the problem of determining the field scattered by a cluster of scatterers when they are insonified by a known acoustical field is addressed the problem is formulated by using the t matrix method and the resulting system of linear equations is solved by using the multilevel fast multipole algorithm mlfma and the fast multipole method fast fourier transform fmmfft method and the efficiency of the two methods is compared it was observed that in general the mlfma performs better than the fmmfft algorithm however when the scatterers are distributed uniformly on a rectangular grid the fmmfft algorithm performs as good as the mlfma the accuracy of the methods is evaluated by modeling a spherical scatterer as composed of many small spheres

a new relative perturbation theorem for singular subspaces abstract this note addresses the sensitivity of singular subspaces of a matrix under relative perturbations it employs a new technique of separating a multiplicative perturbation d into two components one is the distance of a scalar multiple of d to the nearest unitary matrix q and the other is the distance of q to the identity consequently the new bounds reflect the intrinsic differences in how left and right multiplicative perturbations affect left and right singular subspaces

an analysis of the rayleigh ritz method for approximating eigenspaces this paper concerns the rayleigh ritz method for computing an approximation to an eigenspace x of a general matrix a from a subspace w that contains an approximation to x the method produces a pair n x that purports to approximate a pair l x where x is a basis for x and ax xl in this paper we consider the convergence of n x as the sine e of the angle between x and w approaches zero it is shown that under a natural hypothesis called the uniform separation condition the ritz pairs n x converge to the eigenpair l x when one is concerned with eigenvalues and eigenvectors one can compute certain refined ritz vectors whose convergence is guaranteed even when the uniform separation condition is not satisfied an attractive feature of the analysis is that it does not assume that a has distinct eigenvalues or is diagonalizable

efficient evaluation of the rokhlin translator in multilevel fast multipole algorithm in multilevel fast multipole algorithm mlfma the direct evaluation of the rokhlin translator is computationally expensive and usually the cost is lowered by using local lagrange interpolation in the evaluation which requires oversampling of the translator in this paper we improve the interpolation procedure by introducing a new accurate and fast oversampling technique based on the fast fourier transform fft in addition to speeding up the oversampling this also allows the use of lower number of points in the interpolation stencils improving the efficiency of the evaluation of the rokhlin translator we have optimized the interpolation parameters i e the number of the stencil points and the oversampling factor by using as the error criterion the accuracy in the translated incoming field rather than the usually used interpolation error this choice leads to better optimized parameter pairs which further lowers the interpolation cost we have computed and tabulated the optimized pairs for a wide range of target accuracies and the mlfma division levels these tables can be used for a good error control and maximal speed up in practical computations

optimal perturbation bounds for the hermitian eigenvalue problem there is now a large literature on structured perturbation bounds for eigenvalue problems of the formvbox hx mx where h and m are hermitian these results give relative error bounds on the ith eigenvalue i of the form i i i and bound the error in the ith eigenvector in terms of the relative gap minj i i j i j 1 2 in general this theory usually restricts h to be nonsingular and m to be positive definite we relax this restriction by allowing h to be singular for our results on eigenvalues weallow m to be positive semi definite and for a few results we allow it to be more general for these problems for eigenvalues that are not zero or infinity under perturbation it is possible to obtain local relative error bounds thus a wider class of problems may be characterized by this theory although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully

fourier based fast multipole method for the helmholtz equation the fast multipole method fmm has had great success in reducing the computational complexity of solving the boundary integral form of the helmholtz equation we present a formulation of the helmholtz fmm that uses fourier basis functions rather than spherical harmonics by modifying the transfer function in the precomputation stage of the fmm time critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast fourier transforms retaining the diagonality of the transfer function and providing a simplified error analysis using fourier analysis constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance sharp error bounds are derived and verified numerically various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function

relative perturbation theory for hyperbolic singular value problem we give relative perturbation bounds for singular values and perturbation bounds for singular subspaces of a hyperbolic singular value problem for the pair g j where g is a full rank matrix and j is a diagonal matrix of signs we consider two types of relative perturbations g g b b d and g g d b b depending whether g has full column or full row rank respectively in both cases we also consider relative element wise perturbations of g which typically occur in numerical computations

a global interpolator with low sample rate for multilevel fast multipole algorithm a new improved version of a global interpolator utilizing trigonometric polynomials is presented for the high frequency multilevel fast multipole algorithm the number of required points to sample the outgoing and incoming field patterns is low almost half in some levels compared with the earlier published versions compared with local interpolators based on lagrange interpolating polynomials the proposed technique performs even more favorably and reduces the number of sample points by a factor of eight the numerical examples demonstrate that the interpolator allows full numerical accuracy control during the aggregation and disaggregation phases regardless of the number of the levels in the octree

floating point perturbations of hermitian matrices abstract we consider the perturbation properties of the eigensolution of hermitian matrices for the matrix entries and the eigenvalues we use the realistic floating point error measure a a recently demmel and veselic considered the same problem for a positive definite matrix h showing that the floating point perturbation theory holds with constants depending on the condition number of the matrix a dhd where aii 1 and d is a diagonal scaling we study the general hermitian case along the same lines thus obtaining new classes of well behaved matrices and matrix pairs our theory is applicable to the already known class of scaled diagonally dominant matrices as well as to matrices given by factors like those in symmetric indefinite decompositions we also obtain norm estimates for the perturbations of the eigenprojections and show that some of our techniques extend to non hermitian matrices however unlike in the positive definite case we are still unable to describe simply the set of all well behaved hermitian matrices

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

an overview of relative sin t theorems for invariant subspaces of complex matrices abstract relative perturbation bounds for invariant subspaces of complex matrices are reviewed with emphasis on bounding the sines of the largest principal angle between two subspaces i e sin theorems the goal is to provide intuition as well as an idea for why the bounds hold and why they look the way they do relative bounds have the advantage of being better at exploiting structure in a perturbation than absolute bounds therefore the reaction of subspaces to relative perturbations can be different than to absolute perturbations in particular there are certain classes of relative perturbations to which subspaces of indefinite hermitian matrices can be more sensitive than subspaces of definite matrices

a note on unifying absolute and relative perturbation bounds abstract perturbation bounds for invariant subspaces and eigenvalues of complex matrices are presented that lead to absolute as well as a large class of relative bounds in particular it is shown that absolute bounds such as those by davis and kahan bauer and fike and hoffman and wielandt and some relative bounds are special cases of universal bounds as a consequence we obtain a new relative bound for subspaces of normal matrices which contains a deviation of the matrix from positive definiteness we also investigate how row scaling affects eigenvalues and their sensitivity to perturbations and we illustrate how the departure from normality can affect the condition number with respect to inversion of the scaled eigenvectors

a nondirective plane wave mlfma stable at low frequencies a novel method called the nondirective stable plane wave multilevel fast multipole algorithm nspwmlfma is presented to evaluate the low frequency lf interactions that cannot be handled by the multilevel fast multipole algorithm mlfma it is well known that the mlfma cannot be used for lf interactions since it suffers from numerical instability contrary to current techniques the proposed technique is not based on the spectral representation of the green function instead the addition theorem of the mlfma is manipulated into a form that allows numerically stable translations along the z axis the translation operator for these translations is derived in closed form a qr based method is devised to allow stable translations in all the other directions interpolations and anterpolations are also provided allowing a full multilevel algorithm since the nspwmlfma is based on the same mathematical foundations as the mlfma it requires limited adaptations to existing mlfma codes the fact that a qr is needed limits this algorithm to lf interactions however a coupling with the mlfma is straightforward allowing the easy construction of a broadband algorithm the dc limit of the algorithm is also presented and it is shown that the algorithm remains valid for static problems finally it is shown that the error introduced in the different steps of the algorithm is controllable and a single level vectorial version of the algorithm is applied to a generic scattering application to demonstrate its validity

embedding calderon multiplicative preconditioners in multilevel fast multipole algorithms calderon preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation efie for perfect electric conductors at lower frequencies previous authors have shown that by using a dense matrix preconditioner based on the calderon identities the low frequency instability is removed while still maintaining the inherent accuracy of the efie it was also demonstrated that the spectral properties of the caldero n preconditioner are conserved during discretization if the efie operator is discretized with rao wilton glisson expansion functions and the preconditioner with buffa christiansen expansion functions in this article we will show how the calderon multiplicative preconditioner cmp can be combined with fast multipole methods to accelerate the numerical solution leading to an overall complexity of o n logn for the entire iterative solution at low frequencies where the cmp is most useful the traditional multilevel fast multipole algorithm mlfma is unstable and we apply the nondirectional stable plane wave mlfma nspwmlfma that resolves the low frequency breakdown of the mlfma the combined algorithm will be called the cmp nspwmlfma applying the cmp nspwmlfma at open surfaces or very low frequencies leads to certain problems which will be discussed in this article

a note on the perturbation bounds of eigenspaces for hermitian matrices let h be a hermitian matrix and h be its perturbed matrix in this paper both additive and multiplicative perturbations of eigenspace are studied and some absolute and relative bounds are presented which are better than the existing bounds in some sense

canonical angles of unitary spaces and perturbations of direct complements abstract let n plane1d 4b2 be a given direct sum decomposition we determine the largest number r i such that all subspaces for which the gap between and plane1d 4b2 satisfies have the property dim plane1d 4b2 the problem involves angles between subspaces i e singular values of products of projections

on expansion of search subspaces for large non hermitian eigenproblems abstract how to expand a given subspace efficiently is an important task for large sparse eigenvalue problems in this paper we are interested in the following question which vector from a given subspace after multiplied by the matrix a will give a better expanded search subspace in 2008 33 ye investigated this problem and determined how to choose the optimal expansion vector theoretically unfortunately the result is not computable since it involves some unknown information on the desired eigenvector when a is symmetric it was suggested that the ritz vector may lead to a good candidate for subspace expansion however when a is non hermitian the ritz vector may not be a satisfactory choice the contribution of this paper is twofold first we suggest to use the refined ritz vector to expand the search subspace and propose a residual expansion arnoldi method for subspace expansion theoretical results justify the use of the refined ritz vector second we prove that the elements of the primitive refined ritz vector have a decreasing pattern going to zero we then show that the decreasing pattern exists in an arbitrary primitive approximate eigenvector and derive an inexact residual expansion arnoldi method for subspace expansion numerical examples show the effectiveness of our theoretical results and illustrate the numerical behavior of the new method for subspace expansion

stable periodic orbits for a predator prey model with delay abstract in this paper we consider a predator prey model with time lag which improves upon that proposed by cavani and farkas 1994 acta math hungar 63 3 213 229 we show that when the model has exactly one non trivial unstable and hyperbolic equilibrium there exists a stable periodic orbit

broadband mlfma with plane wave expansions and optimal memory demand a broadband multilevel fast multipole algorithm mlfma in 3d is presented based on plane wave expansions and diagonal translations on all division levels the radiation and incoming wave patterns on all levels are presented by trigonometric polynomials and optimal sampling rates and translator degrees are found and tabulated on the super wavelength levels the memory need and the computational cost are lowered more than by half compared to the traditional approach on the sub wavelength levels also a novel saving method for the field patterns is presented which is as efficient as saving the multipole series coefficients while all translations with the proposed method can still be carried out in the fast diagonal form also the direction dependence of the radiation and incoming wave patterns is lowered by half the proposed broadband mlfma has a very good error control on all division levels which we demonstrate by numerical testing

absolute and relative perturbation bounds for invariant subspaces of matrices absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices in the two norm and in the frobenius norm the absolute bounds can be considered extensions of davis and kahan s sin theorem to general matrices and invariant subspaces of any dimension the relative bounds are the most general relative bounds for invariant subspaces because they place no restrictions on the matrix or the perturbation when the perturbed subspace has dimension one the relative bound is implied by the absolute bound

fast solution methods in electromagnetics various methods for efficiently solving electromagnetic problems are presented electromagnetic scattering problems can be roughly classified into surface and volume problems while fast methods are either differential or integral equation based the resultant systems of linear equations are either solved directly or iteratively a review of various differential equation solvers their complexities and memory requirements is given the issues of grid dispersion and hybridization with integral equation solvers are discussed several fast integral equation solvers for surface and volume scatterers are presented these solvers have reduced computational complexities and memory requirements

an iterative block arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems a modified m step arnoldi algorithm proposed by jia and elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version in the new method we use new modified approximate eigenvectors obtained from a linear combination of ritz vectors and the wasted m 1 th block basis vector whose residual norms are satisfied with some p 1 dimensional minimization problems the resulting block arnoldi algorithm is not only better than the standard m step one both in theory and in practice but also cheaper than the standard m 1 step one the relationships among the residual norms of ritz pairs and those of new approximate eigenpairs are analyzed theoretical results show that the new method can overcome the drawback of nonconvergence which exists in the conventional one in some extent we carry out some numerical experiments which exhibit that the new algorithm is more efficient and works better than its counterparts

performing interpolation and anterpolation entirely by fast fourier transform in the 3 d multilevel fast multipole algorithm the fast multipole methods are used for solving a scalar acoustic or vector electromagnetic wave equation by integral equation methods with a large number of unknowns in this paper a new method is presented for performing interpolation and anterpolation in both spherical coordinates theta and phi by fft in the three dimensional 3 d multilevel fast multipole algorithm mlfma the key idea is to approximate functions on the unit sphere by truncated fourier series in two variables rather than by the usual spherical harmonics the proposed method is exact in interpolating and anterpolating and has the high numerical efficiency of fft the method is numerically compared to the method of performing interpolation and anterpolation using lagrangian interpolation which presently is probably the fastest method for those operations and the results suggest that the proposed new method is equally or more efficient depending on the desired accuracy

error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem the numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximate but fast methods such as the fast multipole method however since these methods are only approximate it is important to have an estimate for the error introduced in such calculations an analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtained the error caused by polynomial interpolation in a multilevel fast multipole algorithm is also analyzed the total error introduced in a multilevel implementation is also investigated numerically

monte carlo simulation of electromagnetic scattering from two dimensional random rough surfaces the fast multipole method fast fourier transform fmm fft method is developed to compute the scattering of an electromagnetic wave from a two dimensional 2 d rough surface the resulting algorithm computes a matrix vector multiply in o n log n operations this algorithm is shown to be more efficient than another o n log n algorithm the multilevel fast multipole algorithm mlfma for surfaces of small height for surfaces with larger roughness the mlfma is found to be more efficient using the mlfma monte carlo simulations are carried out to compute the statistical properties of the electromagnetic scattering from 2 d random rough surfaces using a workstation for the rougher surface backscattering enhancement is clearly observable as a pronounced peak in the backscattering direction of the computed bistatic scattering coefficient for the smoother surface the monte carlo results compare well with the results of the approximate kirchhoff theory

positive periodic solutions for a class of delay differential equations abstract in this paper we employ fixed point theorem and functional equation theory to study the existence of positive periodic solutions of the delay differential equation x t t x t t x 2 t t x t t x t moreover new explicit conditions for the related population growth models are also given

broadband multilevel fast multipole algorithm for electric magnetic current volume integral equation a volume integral equation method given in terms of the equivalent electric and magnetic volume currents and discretized using non conforming element wise constant approximations is applied to electromagnetic scattering analysis of inhomogeneous and anisotropic objects a broadband version of the multilevel fast multipole algorithm mlfma combining low frequency stable planewave expansion technique with the high frequency mlfma and utilizing global interpolators based on trigonometric polynomials is used to accelerate the computations

perturbations of the eigenprojections of a factorized hermitian matrix abstract we give the perturbation bounds for the eigenprojections of a hermitian matrix h gjg where g has full column rank and j is nonsingular under the perturbations of the factor g our bounds hold for example when g is given with elementwise relative error our bounds contain relative gaps between the eigenvalues and may thus be much less pessimistic than the standard norm estimates

a low frequency stable plane wave addition theorem the multilevel fast multipole algorithm mlfma is a well known and very successful method for accelerating the matrix vector products required for the iterative solution of helmholtz problems the mlfma has an important drawback namely its inability to handle scattering problems with a lot of subwavelength detail due to the low frequency lf breakdown of the mlfma there is a need to extend the mlfma to lf since alternative methods are less efficient multipole methods or hard to implement spectral methods in this paper a new addition theorem will be developed that does not suffer from an lf breakdown instead it suffers from a high frequency hf breakdown the new method relies on a novel set of distributions the so called pseudospherical harmonics closely related to the spherical harmonics these allow the discretization points and translation operators to be calculated in closed form hence the method presented in this paper allows the easy implementation of a method that is stable at lf furthermore a combination of the traditional mlfma and the new method allows for the construction of a broadband mlfma