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

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

fast bem fem mortar coupling for acoustic structure interaction a coupling algorithm based on lagrange multipliers is proposed for the simulation of structure acoustic field interaction finite plate elements are coupled to a galerkin boundary element formulation of the acoustic domain the interface pressure is interpolated as a lagrange multiplier thus allowing the coupling of non matching grids the resulting saddle point problem is solved by an approximate uzawa type scheme in which the matrix vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method the algorithm is demonstrated on the example of a cavity backed elastic panel

nonuniform spectrum on banach spaces abstract for a sequence of bounded linear operators acting on a banach space we consider the notion of nonuniform spectrum this is defined in terms of the existence of nonuniform exponential dichotomies with an arbitrarily small nonuniform part and can be seen as a nonuniform version of the spectrum introduced by sacker and sell in the case of a single trajectory although now in the infinite dimensional setting we give a complete characterization of all possible forms of the nonuniform spectrum for sequences of compact linear operators and more generally for sequences of bounded linear operators satisfying a certain asymptotic compactness moreover we provide explicit examples of sequences of compact linear operators acting on the l 2 space of sequences of real numbers for all the possible forms of the nonuniform spectrum as nontrivial applications we show that the nonuniform spectrum of a lyapunov regular sequence is the set of lyapunov exponents and that the asymptotic behavior persists under sufficiently small nonlinear perturbations in the sense that the lower and upper lyapunov exponents of the perturbed dynamics belong to a connected component of the nonuniform spectrum finally we obtain appropriate versions of the results for nonuniformly hyperbolic cocycles

the sampling window corresp the concept of the sampling window is introduced for the central interpolation of finite energy band limited functions the sampling window does not increase the rate of convergence of the truncation error series as do various convergence factors but does significantly reduce the truncation error bound

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

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

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

sternberg theorems for random dynamical systems in this paper we prove the smooth conjugacy theorems of sternberg type for random dynamical systems based on their lyapunov exponents we also present a stable and unstable manifold theorem with tempered estimates that are used to construct conjugacy 2005 wiley periodicals inc

optimal interpolation of translation operator in multilevel fast multipole algorithm lagrange interpolation of the translation operator in the three dimensional multilevel fast multipole algorithm mlfma is revisited parameters of the interpolation namely the number of interpolation points and the oversampling factor are optimized for controllable error via optimization it becomes possible to obtain the desired level of accuracy with the minimum processing time

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

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

normal forms via nonuniform hyperbolicity abstract we develop a normal form theory for a nonautonomous dynamics x m 1 a m x m f m x m with discrete time based on the nonuniform spectrum of the sequence of matrices a m in particular we show that any nonresonant terms of the perturbations f m can be eliminated through an appropriate coordinate change with the resonances expressed in terms of the connected components of the nonuniform spectrum the latter is defined in terms of the notion of a nonuniform exponential dichotomy with a small nonuniform part which is ubiquitous in the context of ergodic theory we first make a preparation of the linear part of the dynamics that is of independent interest we show that any sequence of matrices with a bounded nonuniform spectrum can be reduced to a sequence of matrices in block form via a lyapunov coordinate change this allows maintaining the lyapunov exponents as well as the nonuniform spectrum as further developments we describe normal form theories in two additional contexts we consider nonuniformly hyperbolic cocycles over a diffeomorphism of a compact manifold as well as perturbations f m of a sequence of compact linear operators a m on a banach space the latter includes the particular case of a sequence of matrices that need not be invertible

a geometric inequality in the heisenberg group and its applications to stable solutions of semilinear problems in the heisenberg group framework we obtain a geometric inequality for stable solutions of delta mathbb h u f u in a domain omega subseteq mathbb h more precisely if we denote the horizontal intrinsic hessian by hu the mean curvature of a level set by h its imaginary curvature by p the intrinsic normal by and the unit tangent by we have that int limits omega nabla mathbb h phi 2 nabla mathbb h u 2 quad geq int limits omega cap nabla mathbb h u ne0 left hu 2 langle hu 2 nu nu rangle mathbb h 2 left tyuxu txuyu right right phi 2 quad int limits omega cap nabla mathbb h u ne0 nabla mathbb h u 2 big h 2 big p frac langle huv nu rangle mathbb h nabla mathbb h u big 2 2 langle t nu v rangle mathbb h big phi 2 for any phi in c infty 0 omega stable solutions in the entire mathbb h satisfying a suitably weighted energy growth and such that langle t nu v rangle mathbb h ge0 are then shown to have level sets with vanishing mean curvature

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

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

optimal interpolation of radiated fields over a sphere an optimal sampling interpolation algorithm is developed that allows the accurate recovery of scattered or radiated fields over a sphere from a minimum number of samples using the concept of the field equivalent spatial bandwidth a central interpolation scheme is developed to compute the field in theta phi coordinates starting from its samples the maximum allowable sample spacing and error upper bounds are also rigorously derived several simulated examples of pattern reconstruction are presented for both the cases of field and power pattern interpolation the interpolation error as a function of the retained sample number has been also evaluated and compared with the theoretical upper bounds the algorithm stability versus randomly distributed errors added to the exact samples is demonstrated

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

a diagonalized multilevel fast multipole method with spherical harmonics expansion of the k space integrals diagonalization of the fast multipole method fmm for the helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves the resulting k space integral over the ewald sphere is numerically evaluated storing the k space quadrature samples of the method of moments mom basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems in this paper it is proposed to expand the k space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules aggregations plane wave translations and disaggregations in the realized multilevel fast multipole method mlfmm are carried out using the k space samples of a numerical quadrature rule however the incoming plane waves on the finest mlfmm level are expanded in spherical harmonics again thus due to the orthonormality of spherical harmonics the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients overall the resulting mlfmm can save a considerable amount of memory without compromising accuracy and numerical speed

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