a modified decomposition abstract the maclaurin series is quite limited in comparison to the adomian series obtained in the decomposition method by adding procedures from the decomposition method and the expansion of nonlinearities using the adomian polynomials as well as a recent result of adomian and rach on transformation of series using the above polynomials the maclaurin series can be made much more useful in its applicability however the convergence is still slower than for adomian s results using decomposition

application of the fast multipole method for the evaluation of magnetostatic fields in micromagnetic computations micromagnetic simulations are elaborated to describe the magnetic dynamics in ferromagnetic bodies in these simulations most of the time is spent on the evaluation of the magnetostatic field in the magnetic material this paper presents a new numerical finite difference scheme for the evaluation of the magnetostatic field based on the fast multipole method fmm the interactions between finite difference cells are described in terms of far and near field interactions the far field computations are conducted using the spherical harmonic expansion of the magnetostatic field while the near field computations are accelerated using fast fourier transforms fft the performance of the presented fmm scheme is studied by comparing the scheme with a pure fft scheme the fmm scheme is more memory efficient and more flexible then the fft scheme it is accurate and still has a good time efficiency

an innovative eigenvalue problem solver for free vibration of uniform timoshenko beams by using the adomian modified decomposition method this paper deals with free vibration problems of uniform timoshenko beams under various supporting boundary conditions the technique we have used is based on applying the adomian modified decomposition method amdm to our vibration problems doing some simple mathematical operations on the method we can obtain ith natural frequencies and mode shapes one at a time the computed results agree well with those analytical and numerical results given in the literature these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is simpler and more straightforward than the other modal analysis

an improved fast multipole algorithm for potential fields a new version of the fast multipole method fmm for potential fields is presented we introduce a new representation of potentials in which most translation operators are diagonal as a result for double precision calculations in two dimensions we obtain an improvement of a factor of two to four in speed compared to previously published algorithms the improvement is expected to be much greater in three dimensions the performance of the method is illustrated with several numerical examples

design of shaped piezoelectric modal sensor for beam with arbitrary boundary conditions by using adomian decomposition method the theory for designing distributed piezoelectric modal sensors is well established for beam structures however the current modal sensor theory is limited in scope in that it can only be applied in the case of classical boundary conditions i e either clamped free simply supported or sliding in this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a beam with arbitrary boundary conditions is proposed using the adomian decomposition method adm a general expression for designing the shape of a piezoelectric modal sensor is presented in which the output signal of the designed sensor is proportional to the response of the target mode other modes are filtered out the modal sensor shape is expressed as a function of the second spatial derivative of the structural mode shape function based on the adm and employing some simple mathematical operations the closed form series solution of the second spatial derivative of the mode shapes can be determined then the shapes of the designed modal sensors are obtained finally some numerical examples are given to demonstrate the feasibility of the proposed modal sensors it is shown that for classical boundary conditions the shapes of the modal sensors based on the adm agree well with analytical and numerical results given in the literature for general boundary conditions it is found that the shape of the modal sensors is influenced by the number of modes of interest because the second spatial derivatives of the mode shapes are not orthogonal to one another the modal sensors for general boundary conditions can be considered as modal filters within a limited frequency band

numerical experiments on free and forced vibrations of beams of non uniform cross section the objective of this paper is to obtain an approximate solution of the title problem by using rayleigh s optimization technique when the mechanical system and its excitation in the case where it is subject to a p 0 cos t type loading is symmetric it is shown that if one takes into account the symmetry of the system the calculated eigenvalues are in general more accurate than those which would result if the complete structural system were analyzed it is also assumed that the beam is subject to an axial force and results for the buckling problem are also presented the analytically determined eigenvalues are compared with values calculated by using the finite element method in several instances

a computational stochastic methodology for the design of random meta materials under geometric constraints we present a computational stochastic methodology for generating and optimizing random meta material mm configurations with nonoverlapping geometric constraints subject to various types of covari

high performance blas formulation of the multipole to local operator in the fast multipole method the multipole to local m2l operator is the most time consuming part of the far field computation in the fast multipole method for laplace equation its natural expression though commonly used does not respect a sharp error bound we here first prove the correctness of a second expression we then propose a matrix formulation implemented with basic linear algebra subprograms blas routines in order to speed up its computation for these two expressions we also introduce special data storages in memory to gain greater computational efficiency this blas scheme is finally compared for uniform distributions to other m2l improvements such as block fft fft with polynomial scaling rotations and plane wave expansions when considering runtime extra memory storage numerical stability and common precisions for laplace equation the blas version appears as the best one

fast monte carlo algorithms for matrices iii computing a compressed approximate matrix decomposition in many applications the data consist of or may be naturally formulated as an m times n matrix a which may be stored on disk but which is too large to be read into random access memory ram or to practically perform superlinear polynomial time computations on it two algorithms are presented which when given an m times n matrix a compute approximations to a which are the product of three smaller matrices c u and r each of which may be computed rapidly let a cur be the computed approximate decomposition both algorithms have provable bounds for the error matrix a a in the first algorithm c columns of a and r rows of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling and the r times n matrix r consists of those r rows of a also after appropriate rescaling then the c times r matrix u may be calculated from c and r for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively it is proven that left a a right xi le min d mathrm rank d le k left a d right xi poly k 1 c left a right f holds in expectation and with high probability for both xi 2 f and for all k 1 ldots mbox rank a thus by appropriate choice of k left a a right 2 le epsilon left a right f also holds in expectation and with high probability this algorithm may be implemented without storing the matrix a in ram provided it can make two passes over the matrix stored in external memory and use o m n additional ram assuming that c and r are constants independent of the size of the input the second algorithm is similar except that it approximates the matrix c by randomly sampling a constant number of rows of c thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a f both algorithms take time which is a low degree polynomial in k 1 epsilon and 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n the proofs for the error bounds make important use of matrix perturbation theory and previous work on approximating matrix multiplication and computing low rank approximations to a matrix the probability distribution over columns and rows and the rescaling are crucial features of the algorithms and must be chosen judiciously

an efficient algorithm for hydrodynamical interaction of many deformable drops an efficient and accurate 3d algorithm for dynamical simulations of many deformable drops with strong hydrodynamical interactions at zero reynolds numbers is developed the drop to medium viscosity ratio and the bond number are arbitrary and the drops are subject to gravity with stationary triply periodic boundary conditions the algorithm at each step is a hybrid of boundary integral and economical multipole techniques with extensive use of rotational transformations and economical truncation of multipole expansions to optimize near field interactions a significant part of the code is the new best paraboloid spline technique for calculating the normal vectors and curvatures on drop surfaces which greatly improves the quality of long time simulations examples show the phenomenon of clustering in a concentrated sedimenting emulsion for 0 25 and 1 which leads to an increase in the average sedimentation velocity with time a high efficiency of the method is demonstrated with two orders of magnitude gains over the standard o n2n2 boundary integral technique for n 102 drops in a periodic cell with n 103 triangular boundary elements per drop so that typical long time dynamical simulations can be performed in a few days or weeks on a standard workstation as compared to the several years which would be required for the same simulations using standard boundary integral techniques the effects of drop triangulation and truncation of multipole expansions on dynamical simulations are assessed

an improved fast multipole algorithm for potential fields on the line a new version of the fast multipole method fmm for the evaluation of potential fields on the line is introduced it uses a new representation of potential fields based on generalized gaussian quadratures j ma v rokhlin and s wandzura siam j numer anal 33 1996 pp 971 996 n yarvin and v rokhlin siam j sci comput 20 1999 pp 699 718 in this representation most translation operators are diagonal to efficiently incorporate this representation into the fmm an apparatus is introduced for transforming between different types of expansions this apparatus is highly general and is based on formulae for the least squares approximation of linear operators the performance of the method is illustrated with several numerical examples it is roughly twice as fast as previously published algorithms

accurate and efficient nystrom volume integral equation method for electromagnetic scattering of 3 d metamaterials in layered media in this paper we develop an accurate and efficient nystrom volume integral equation vie method for the maxwell equations to compute the electromagnetic scattering of three dimensional metamaterials in layered media the vie for meta atom scatterers in a layered medium is derived using dyadic green s functions for layered media where cauchy principal values are evaluated accurately using a finite size exclusion volume with the help of some correction integrals of removable singularities several desingularization techniques are also introduced for the vie including interpolated quadrature formulas with tensor product quadrature nodes for self integration terms of typical meta atoms a regularization scheme for closely packed meta atoms and adaptive integration approaches for high gradient components of the dyadic green s functions the resulting nystrom method demonstrates fast numerical convergence with high accuracy and efficiency with only a small number of collocation points in calculating the sca

min max boundary domain decomposition domain decomposition is one of the most effective and popular parallel computing techniques for solving large scale numerical systems in the special case when the amount of computation in a subdomain is proportional to the volume of the subdomain domain decomposition amounts to minimizing the surface area of each subdomain while dividing the volume evenly motivated by this fact we study the following min max boundary multi way partitioning problem given a graph g and an integer k 1 we would like to divide g into k subgraphs g1 gk by removing edges such that i gi g k for all i 1 k and ii the maximum boundary size of any subgraph the set of edges connecting it with other subgraphs is minimized we provide an algorithm that given g a well shaped mesh in d dimensions finds a partition of g into k subgraphs g1 gk such that for all i gi has g k vertices and the number of edges connecting gi with the other subgraphs is o g k 1 1 d our algorithm can find such a partition in o g logk time finally we extend our results to vertex weighted and vertex based graph decomposition our results can be used to simultaneously balance the computational and memory loads on a distributed memory parallel computer without incurring large communication overhead

free vibration analysis of a type of tapered beams by using adomian decomposition method abstract the adomian decomposition method adm is employed in this paper to investigate the free vibrations of tapered euler bernoulli beams with a continuously exponential variation of width and a constant thickness along the length under various boundary conditions based on adm the governing differential equation for the tapered beam becomes a recursive algebraic equation by using the boundary condition equations the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously the computed results for different boundary conditions and non uniformity ratios are presented the accuracy is assured from the convergence and comparison published results

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

application of fast multipole galerkin boundary integral equation method to elastostatic crack problems in 3d fast multipole method fmm has been developed as a technique to reduce the computational cost and memory requirements in solving large scale problems this paper discusses an application of fmm to three dimensional boundary integral equation method for elastostatic crack problems the boundary integral equation for many crack problems is discretized with fmm and galerkin s method the resulting algebraic equation is solved with generalized minimum residual method gmres the numerical results show that fmm is more efficient than conventional methods when the number of unknowns is more than about 1200 and therefore can be useful in large scale analyses of fracture mechanics copyright 2001 john wiley sons ltd

vibration of non uniform rods and beams we show that for some non uniform rods and beams the equation of motion can be transformed into the equation of motion for a uniform rod or beam then when the ends are completely fixed the eigenvalues of the non uniform continuum are the same as those of uniform rods or beams for other end support conditions exact solutions are obtained an efficient procedure is used to analyze the free vibration of non uniform beams with general shape and arbitrary boundary conditions simple formulas are presented for predicting the fundamental natural frequency of non uniform beams with various end support conditions

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

a parallel version of the fast multipole method abstract this paper presents a parallel version of the fast multipole method fmm the fmm is a recently developed scheme for the evaluation of the potential and force fields in systems of particles whose interactions are coulombic or gravitational in nature the sequential method requires o n operations to obtain the fields due to n charges rather than the o n2 operations required by the direct calculation here we describe the modifications necessary for implementation of the method on parallel architectures and show that the expected time requirements grow as log n when using n processors numerical results are given for a shared memory machine the encore multimax 320

free vibration analysis of stepped beams by using adomian decomposition method abstract the adomian decomposition method adm is employed in this paper to investigate the free vibrations of a stepped euler bernoulli beam consisting of two uniform sections each section is considered a substructure which can be modeled using adm by using boundary condition and continuity condition equations the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously the computed results for different boundary conditions step ratios and step locations are presented comparing the results using adm to those given in the literature excellent agreement is achieved

provably good partitioning and load balancing algorithms for parallel adaptive n body simulation we present an efficient and provably good partitioning and load balancing algorithm for parallel adaptive n body simulation the main ingredient of our method is a novel geometric characterization of a class of communication graphs that can be used to support hierarchical n body methods such as the fast multipole method fmm and the barnes hut method bh we show that communication graphs of these methods have a good partition that can be found efficiently sequentially and in parallel in particular we show that an n body communication graph either for bh or for fmm can be partitioned into two subgraphs with equal computation load by removing only o sqrt n log n and o n2 3 log n 1 3 number of nodes respectively for two and three dimensions these bounds on node partition imply bounds on edge partition of o sqrt n log n 3 2 and o n2 3 log n 4 3 respectively for two and three dimensions to the best of our knowledge this is the first theoretical result on the quality of partitioning n body communication graphs for nonuniformly distributed particles our results imply that parallel adaptive n body simulation can be made as scalable as computation on regular grids and as efficient as parallel n body simulation on uniformly distributed particles

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

the black box fast multipole method a new o n fast multipole formulation is proposed for non oscillatory kernels this algorithm is applicable to kernels k x y which are only known numerically that is their numerical value can be obtained for any x y this is quite different from many fast multipole methods which depend on analytical expansions of the far field behavior of k for x y large other black box or kernel independent fast multipole methods have been devised our approach has the advantage of requiring a small pre computation time even for very large systems and uses the minimal number of coefficients to represent the far field for a given l 2 tolerance error in the approximation this technique can be very useful for problems where the kernel is known analytically but is quite complicated or for kernels which are defined purely numerically

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

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

communications overlapping in fast multipole particle dynamics methods in molecular dynamics the fast multipole method fmm is an attractive alternative to ewald summation for calculating electrostatic interactions due to the operation counts however when applied to small particle systems and taken to many processors it has a high demand for interprocessor communication in a distributed memory environment this demand severely limits applicability of the fmm to systems with o 10 k atoms we present an algorithm that allows for fine grained overlap of communication and computation while not sacrificing synchronization and determinism in the equations of motion the method avoids contention in the communication subsystem making it feasible to use the fmm for smaller systems on larger numbers of processors our algorithm also facilitates application of multiple time stepping techniques within the fmm we present scaling at a reasonably high level of accuracy compared with optimized ewald methods

approximate complex polynomial evaluation in near constant work per point given the n complex coefficients of a degree n 1 complex polynomial we wish to evaluate the polynomial at a large number m ge n of points on the complex plane this problem is required by many algebraic computations and so is considered in most basic algorithm texts e g a v aho j e hopcroft and j d ullman the design and analysis of computer algorithms addison wesley 1974 we assume an arithmetic model of computation where on each step we can execute an arithmetic operation which is computed exactly all previous exact algorithms c m fiduccia proceeding 4th annual acm symposium on theory of computing 1972 pp 88 93 h t kung fast evaluation and interpolation carnegie mellon 1973 a b borodin and i munro the computational complexity of algebraic and numerical problems american elsevier 1975 v pan a sadikou e landowne and o tiga comput math appl 25 1993 pp 25 30 cost at least work omega log 2 n per point and previously there were no known approximation algorithms for complex polynomial evaluation within the unit circle with work bounds better than the fastest known exact algorithms there are known approximation algorithms v rokhlin j complexity 4 1988 pp 12 32 v y pan j h reif and s r tate in proceedings 32nd annual ieee symposium on foundations of computer science 1992 pp 703 713 for polynomial evaluation at real points but these do not extend to evaluation at general points on the complex plane we provide approximation algorithms for complex polynomial evaluation that cost in many cases near constant amortized work per point let k log p epsilon where p is the sum of the moduli of the coefficients of the input polynomial p z let it tilde p z j be an epsilon approx of p z if epsilon upper bounds the modulus of the error of the approximation tilde p z j at each evaluation point zj that is p z j tilde p z j le epsilon note that epsilon is an absolute error bound rather than a relative error bound in many applications particularly in signal processing the evaluation points zj are fixed and require only polylogarithmic k log p epsilon o log o 1 n for these cases we get a surprising reduction in work by use of approximation algorithms as compared to the fastest known exact algorithms we epsilon approx complex degree n 1 polynomial evaluation at m ge n log n log 2 k fixed points on or within the unit disk in the complex plane in amortized work o log2 k per point which is o log2 log n for polylogarithmic k if the m points are not fixed then we have increased amortized work o log2 k log m per point which is o log m for polylogarithmic k and m ge n log n log k and is still substantially below the previous bound of omega log 2 m for known exact algorithms we further reduce our amortized bounds for special sets of evaluation points widely used in signal processing applications the chirp transform is equivalent to evaluating a complex degree n 1 polynomial at the chirp points which are zeta j j 0 dots m 1 for some fixed complex number zeta we epsilon approx complex degree n 1 polynomial evaluation at these m chirp points where m ge n log n log 2 k and zeta le 1 or ii m ge n and zeta le a function that limits to 1 for k o n and large n in amortized work o log k per point whereas the previous best bounds for exact evaluation via the chirp transform were omega log m per point a v aho k steiglitz and j d ullman siam j comput 4 1975 pp 533 539 using instead a reduction to approximate real polynomial evaluation by interpolation at the chebyshev points in total work o n log k we epsilon approx the evaluation of a degree n polynomial at the first n powers of the n th root of unity where n ge omega n 2 k and epsilon approx the n point dft for certain inputs with descending coefficient magnitude all of our results require polylogarithmic that is logo 1 n depth with the same work bounds we also provide a lower bound for a wide class of schemes for approximate evaluation of a degree n 1 polynomial on the unit circle namely we prove that if a scheme uses an approximation polynomial of degree k 1 then it can be convergent only over a small fraction o k n of the unit circle we believe this is the first lower bound of this sort proved and the proof uses an interesting reduction to the approximation of a matrix product by a matrix of reduced rank

vibration of an euler bernoulli beam of constant depth and with linearly varying breadth abstract the transverse vibration of a homogeneous euler bernoulli beam of constant depth and linearly varying breadth is investigated in this paper a direct analytical solution is presented for the mode shape equation the constraints imposed by the boundary conditions yield the frequency equation the four ideal boundary conditions considered are clamped pinned sliding and free for a complete beam sharp ended it is a necessary condition that the sharp end remain free and as a result the frequency equations for clamped pinned sliding or free boundary conditions at the large end are the determinant of a 2 2 matrix equated to zero a truncated beam is considered as part of a hypothetical complete beam the frequency equation for truncated beam is the determinant of a 4 4 matrix equated to zero the roots of the frequency equations are determined by an iterative procedure the first four natural frequencies of complete beam and the first three frequencies of truncated beams for 16 combinations of the ideal boundary conditions for various truncation factors are presented the results are benchmarks and are therefore presented in tabular form to preserve the accuracy to six figures

the inverse fast multipole method using a fast approximate direct solver as a preconditioner for dense linear systems although some preconditioners are available for solving dense linear systems there are still many matrices for which preconditioners are lacking particularly in cases where the size of the matrix n becomes very large examples of preconditioners include incomplete lu ilu preconditioners that sparsify the matrix based on some threshold algebraic multigrid preconditioners and specialized preconditioners e g calderon and other analytical approximation methods when available despite these methods there remains a great need to develop general purpose preconditioners whose cost scales well with the matrix size n in this paper we propose a preconditioner with broad applicability and with cost mathcal o n for dense matrices when the matrix is given by a smooth as opposed to a highly oscillatory kernel extending the method using the same framework as general mathcal h 2 matrices i e algebraic instead of defined in terms of an analytical kernel is relatively straightforward but

a hierarchical o n force calculation algorithm a novel code for the approximate computation of long range forces between n mutually interacting bodies is presented the code is based on a hierarchical tree of cubic cells and features mutual cell cell interactions which are calculated via a cartesian taylor expansion in a symmetric way such that total momentum is conserved the code benefits from an improved and simple multipole acceptance criterion that reduces the force error and the computational effort for n 104 the computational costs are found empirically to rise sublinearly with n for applications in stellar dynamics this is the first competitive code with complexity o n it is faster than the standard tree code by a factor of 10 or more

askit approximate skeletonization kernel independent treecode in high dimensions we present a fast algorithm for kernel summation problems in high dimensions such problems appear in computational physics numerical approximation nonparametric statistics and machine learning in our context the sums depend on a kernel function that is a pair potential defined on a dataset of points in a high dimensional euclidean space a direct evaluation of the sum scales quadratically with the number of points fast kernel summation methods can reduce this cost to linear complexity but the constants involved do not scale well with the dimensionality of the dataset the main algorithmic components of fast kernel summation algorithms are the separation of the kernel sum between near and far field which is the basis for pruning and the efficient and accurate approximation of the far field we introduce novel methods for pruning and for approximating the far field our far field approximation requires only kernel evaluations and does not use analytic expansions pruning is not done using bounding

a matrix version of the fast multipole method we present a matrix interpretation of the three dimensional fast multipole method fmm the fmm is for efficient computation of gravitational electrostatic potentials and fields it has found various applications and inspired the design of many efficient algorithms the one dimensional fmm is well interpreted in terms of matrix computations the three dimensional matrix version reveals the underlying matrix structures and computational techniques used in fmm it also provides a unified view of algorithm variants as well as existing and emerging implementations of the fmm

an approximate method for one dimensional structures with strong nonlinear and nonhomogenous boundary conditions abstract there is a lack of analytic methods for vibration analysis of structures with strong nonlinear and non homogenous boundary conditions this paper proposes such a method a distinct feature of the method is to introduce the generalized coordinates for nonlinear and nonhomogeneous boundary conditions the total solution of the structure consists of a sum of a spatial particular solution for the boundary conditions and a field solution satisfying linear and homogeneous boundary conditions the field solution can be obtained in terms of the modal expansion the modal coefficients together with the generalized coordinates for the boundary conditions satisfy a set of nonlinear ordinary differential equations which are solved with the harmonic balance method two examples are presented to study the convergence and accuracy of the solution the first example is the longitudinal vibration of a bar subject to a base excitation on the left and restrained by a damper on the right the wave method and the differential quadrature method dqm are compared with the proposed method and are found in good agreement the second example is a beam with nonlinear torsional boundary absorbers the method shows a good accuracy to deal with the complex boundary conditions as compared with the differential quadrature element method dqem the proposed method is also compared with the multi scale method msm and shows an advantage to deal with strong nonlinearities the proposed method is highly promising for vibration analysis of complex structures with nonlinear and nonhomogeneous boundary conditions

a fast algorithm for three dimensional potential fields calculation fast fourier transform on multipoles in this paper we present a fast algorithm for rapid calculation of the potential fields in three dimensions this method arises from an observation that potential evaluation using the multipole to local expansion translation operator can be expressed as a series discrete convolutions of the multipole moments with their associated spherical harmonics functions the high efficiency of the algorithm is primarily due to the use of fft algorithms to evaluate the numerous discrete convolutions we refer to it as the fast fourier transform on multipoles fftm method it is demonstrated that fftm is all accurate method it is significantly more accurate than fmm for a given order of expansion it is also shown that the algorithm has computational complexity of o na where a ranges from 1 0 to 1 3

free vibration analysis of elastically connected multiple beams by using the adomian modified decomposition method abstract the adomian modified decomposition method amdm is employed in this paper to investigate the free vibrations of n elastically connected parallel euler bernoulli beams which are continuously joined by a winkler type elastic layer the proposed amdm method can be used to analyze the vibration of beam system consisting of an arbitrary number of beams by using boundary conditions the natural frequencies and corresponding mode shapes can be easily obtained simultaneously the numerical results for different boundary conditions beam numbers and the stiffness of the winkler type elastic layer are presented it is shown that the amdm offers an accurate and effective method of free vibration analysis of multiple connected beams with arbitrary boundary conditions

a kernel independent adaptive fast multipole algorithm in two and three dimensions we present a new fast multipole method for particle simulations the main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel and it is based only on kernel evaluations instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical fmm tree we use a continuous distribution of an equivalent density on a surface enclosing the box to find this equivalent density we match its potential to the potential of the original sources at a surface in the far field by solving local dirichlet type boundary value problems the far field evaluations are sparsified with singular value decomposition in 2d or fast fourier transforms in 3d we have tested the new method on the single and double layer operators for the laplacian the modified laplacian the stokes the modified stokes the navier and the modified navier operators in two and three dimensions our numerical results indicate that our method compares very well with the best known implementations of the analytic fmm method for both the laplacian and modified laplacian kernels its advantage is the relative simplicity of the implementation and its immediate extension to more general kernels

a treecode accelerated boundary integral poisson boltzmann solver for electrostatics of solvated biomolecules abstract we present a treecode accelerated boundary integral tabi solver for electrostatics of solvated biomolecules described by the linear poisson boltzmann equation the method employs a well conditioned boundary integral formulation for the electrostatic potential and its normal derivative on the molecular surface the surface is triangulated and the integral equations are discretized by centroid collocation the linear system is solved by gmres iteration and the matrix vector product is carried out by a cartesian treecode which reduces the cost from o n 2 to o n log n where n is the number of faces in the triangulation the tabi solver is applied to compute the electrostatic solvation energy in two cases the kirkwood sphere and a solvated protein we present the error cpu time and memory usage and compare results for the poisson boltzmann and poisson equations we show that the treecode approximation error can be made smaller than the discretization error and we compare two versions of the treecode one with uniform clusters and one with non uniform clusters adapted to the molecular surface for the protein test case we compare tabi results with those obtained using the grid based apbs code and we also present parallel tabi simulations using up to eight processors we find that the tabi solver exhibits good serial and parallel performance combined with relatively simple implementation efficient memory usage and geometric adaptability

a direct solution for the transverse vibration of euler bernoulli wedge and cone beams abstract a number of publications have appeared on the transverse vibration of complete and truncated wedge and cone beams the mode shape equations are linear differential equations with regular singularity the natural frequencies of complete beams are found in several publications for truncated beams several numerical solutions have been proposed and the natural frequencies documented in most cases for cantilever wedge and cone beams clamped at the large end analytical solutions of the mode shape equations have been derived based on bessel functions only in one publication are the natural frequencies tabulated for nine combinations of clamped pinned and free boundary conditions but the accuracy of the frequencies is limited to two figures after the decimal point for the fundamental frequency and the higher mode frequencies are rounded to integers the results are not suitable for judging the quality of the various approximate methods in this paper a direct solution of the mode shape equation is presented the sliding boundary condition is also considered and the first three dimensionless natural frequencies computed for 16 combinations of the boundary conditions are tabulated for values of truncation factors from 0 05 to 0 7 the tabulated results are benchmarks and frequencies for truncation factors not in the tables may be interpolated to within 0 1 accuracy

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

fast electrostatic force calculation on parallel computer clusters the fast multipole method fmm and smooth particle mesh ewald spme are well known fast algorithms to evaluate long range electrostatic interactions in molecular dynamics and other fields fmm is a multi scale method which reduces the computation cost by approximating the potential due to a group of particles at a large distance using few multipole functions this algorithm scales like o n for n particles spme algorithm is an o nlnn method which is based on an interpolation of the fourier space part of the ewald sum and evaluating the resulting convolutions using fast fourier transform fft those algorithms suffer from relatively poor efficiency on large parallel machines especially for mid size problems around hundreds of thousands of atoms a variation of the fmm called pwa based on plane wave expansions is presented in this paper a new parallelization strategy for pwa which takes advantage of the specific form of this expansion is described its parallel efficiency is compared with spme through detail time measurements on two different computer clusters

bottom up construction and 2 1 balance refinement of linear octrees in parallel in this article we propose new parallel algorithms for the construction and 2 1 balance refinement of large linear octrees on distributed memory machines such octrees are used in many problems in computational science and engineering e g object representation image analysis unstructured meshing finite elements adaptive mesh refinement and n body simulations fixed size scalability and isogranular analysis of the algorithms using an mpi based parallel implementation was performed on a variety of input data and demonstrated good scalability for different processor counts 1 to 1024 processors on the pittsburgh supercomputing center s tcs 1 alphaserver the results are consistent for different data distributions octrees with over a billion octants were constructed and balanced in less than a minute on 1024 processors like other existing algorithms for constructing and balancing octrees our algorithms have mathcal o n log n work and mathcal o n storage complexity under reasonable assumptions on the distribution of octants and the work per octant the parallel time complexity is mathcal o frac n n p log frac n n p n p log n p where n is the size of the final linear octree and n p is the number of processors

accurate and efficient nystrom volume integral equation method for the maxwell equations for multiple 3 d scatterers in this paper we develop an accurate and efficient nystrom volume integral equation vie method for the maxwell equations for a large number of 3 d scatterers the cauchy principal values that arise from the vie are computed accurately using a finite size exclusion volume together with explicit correction integrals consisting of removable singularities also the hyper singular integrals are computed using interpolated quadrature formulae with tensor product quadrature nodes for cubes spheres and cylinders that are frequently encountered in the design of meta materials the resulting nystrom vie method is shown to have high accuracy with a small number of collocation points and demonstrates p convergence for computing the electromagnetic scattering of these objects numerical calculations of multiple scatterers of cubic spherical and cylindrical shapes validate the efficiency and accuracy of the proposed method

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

free vibration of non uniform euler bernoulli beams with general elastically end constraints using adomian modified decomposition method abstract this paper deals with free vibration problems of non uniform euler bernoulli beam under various supporting conditions the technique we have used is based on applying the adomian modified decomposition method amdm to our vibration problems doing some simple mathematical operations on the method we can obtain ith natural frequencies and mode shapes one at a time the computed results agree well with those analytical and numerical results given in the literatures these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is simple and more straightforward than the other modal analysis

yet another fast multipole method without multipoles pseudoparticle multipole method in this paper we describe a new approach to implement theo n fast multipole method ando nlogn tree method which uses pseudoparticles to express the potential field the new method is similar to anderson s method which uses the values of potential at discrete points to represent the potential field however for the same expansion order the new method is more accurate

fast fourier transform accelerated fast multipole algorithm this paper describes an cal o p 2 log 2 p n implementation of the fast multipole algorithm fma for n body simulations this method of computing the fma is faster than the original which is cal o p 4n where p is the number of terms retained in the truncated multipole expansion representation of the potential field of a collection of charged particles the p term determines the accuracy of the calculation the limiting cal o p 4 computation in the original fma is a convolution like operation on a matrix of multipole coefficients this paper describes the implementation details of a conversion of this limiting computation to linear convolution which is then computed in the fourier domain using the fast fourier transform fft based on a method originally outlined by greengard and rokhlin in addition this paper describes a new block decomposition of the multipole expansion data that provides numerical stability and efficient computation the resulting cal o p 2 log 2 p subroutine has a speedup of 2 on a sequential processor over the original method for p 8 and a speedup of 4 for p 16 the new subroutine vectorizes well and has a speedup of 3 on a vector processor at p 8 and a speedup of 6 at p 16

an o nlogn hierarchical random compression method for kernel matrices by sampling partial matrix entries abstract in this paper we propose an o n log n hierarchical random compression method hrcm for kernel matrix compressing which only requires sampling o n log n entries of a matrix the hrcm combines the hierarchical framework of the h matrix and a randomized sampling technique of column and row spaces for far field interaction kernel matrix we show that a uniform column row sampling of a far field kernel matrix thus without the need and associated cost to pre compute a costly sampling distribution will give a low rank compression of such low rank matrix independent of the matrix size and only dependent on the separation of the source and target locations this far field random compression technique is then implemented at each level of the hierarchical decomposition for general kernel matrices resulting in an o n log n random compression method error and complexity analysis for the hrcm are included numerical results for electrostatic and low frequency helmholtz wave kernels have validated the efficiency and accuracy of the proposed method in comparison of direct o n 2 summations