tangent cones to tensor train varieties abstract as already done for the matrix case in 1 p 256 2 thm 6 1 p 1872 and 3 thm 3 2 we give a parametrization of the bouligand tangent cone of the variety of tensors of bounded tensor train tt rank we discuss how the proof generalizes to any binary hierarchical format the parametrization can be rewritten as an orthogonal sum of tt tensors its retraction onto the variety is particularly easy to compose we also give an implicit description of the tangent cone as the solution of a system of polynomial equations

the treatment of lubrication forces in boundary integral equations despite their many advantages over other numerical methods boundary integral formulations still fail to provide accurate predictions of mesoscale motion in dense suspensions of rigid particles because the nearly singular flow between surfaces in close proximity cannot be resolved accurately a procedure for incorporating analytical solutions for the lubrication flow within a large scale boundary integral equation method is shown although the method is applied to the case of spherical particles in conjunction with the completed double layer boundary integral equation it can be developed further to treat more complex geometries and can be adapted to other numerical techniques in contrast to other apparently similar approaches the present method does not resort to effective medium approximations and in principle retains all the advantages typical of boundary integral approaches the framework also allows for forces other than those due to hydrodynamic lubrication between particles provided that they are a linear function of the relative velocity or at least can be linearized for example forces due to sub continuum fluid behaviour or forces resulting from surface chemistry it is shown using several benchmarks that the relative motion between two particles in various flows is captured accurately both statically and dynamically in situations where uncorrected simulations fail moreover the computational effort is reduced substantially by the application of the method

limits of tangent spaces to real surfaces we investigate the tangent semicone c and the nash space n the fiber of the nash blowup of an algebraic surface v with singular locus s in r 3 we prove a structure theorem for n there are finitely many exceptional rays in c so that n is the union of n c and the set of elements in n containing one of the exceptional rays the set of elements in n containing an exceptional ray is semialgebraic but can be disconnected and have discrete elements if the ray is tangent to s any ray not tangent to s but along which c is singular must be exceptional except in one case and the set of elements in n containing the exceptional ray is closed connected and 1 dimensional and we can give a lower bound on the size of this set

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

an efficient numerical method for studying interfacial motion in two dimensional creeping flows we present new methods for computing the motion of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal spacing in arclength of the marker points a small scale decomposition is performed to extract the dominant term in the evolution of the interface and we show that this dominant term leads to a cfl type stability constraint when in an equal arclength frame this term is linear and we show that implicit time integration schemes that are explicit in fourier space can be formulated we verify this analysis through several numerical examples

a practical determination strategy of optimal threshold parameter for matrix compression in wavelet bem a practical strategy is developed to determine the optimal threshold parameter for wavelet based boundary element be analysis the optimal parameter is determined so that the amount of storage and computational work is minimized without reducing the accuracy of the be solution in the present study the beylkin type truncation scheme is used in the matrix assembly to avoid unnecessary integration concerning the truncated entries of a coefficient matrix a priori estimation of the matrix entries is introduced and thus the truncated entries are determined twice before and after matrix assembly the optimal threshold parameter is set based on the equilibrium of the truncation and discretization errors these errors are estimated in the residual sense for laplace problems the discretization error is in particular indicated with the potential s contribution c to the residual norm r used in error estimation for mesh adaptation since the normalized residual norm c u u the potential components of be solution cannot be computed without main be analysis the discretization error is estimated by the approximate expression constructed through subsidiary be calculation with smaller degree of freedom dof the matrix compression using the proposed optimal threshold parameter enables us to generate a sparse matrix with o n 1 7 0 1 non zero entries although the quasi optimal memory requirements and complexity are not attained the compression rate of a few per cent can be achieved for n 1000

numerical methods for multiple inviscid interfaces in creeping flows we present new highly accurate and efficient methods for computing the motion of a large number of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal arclength spacing of the marker points we assume that the fluid on one side of the interface is inviscid and we discuss two different physical phenomena bubble dynamics and interfacial motion driven by surface tension viscous sintering applications from buoyancy driven bubble interactions the motion of polydispersed bubbles in an extensional flow and the removal of void spaces through viscous sintering are considered and we present large scale fully resolved simulations involving o 100 closed interfaces

boundary element and finite element coupling for aeroacoustics simulations we consider the scattering of acoustic perturbations in the presence of a flow we suppose that the space can be split into a zone where the flow is uniform and a zone where the flow is potential in the first zone we apply a prandtl glauert transformation to recover the helmholtz equation the well known setting of boundary element method for the helmholtz equation is available in the second zone the flow quantities are space dependent we have to consider a local resolution namely the finite element method herein we carry out the coupling of these two methods and present various applications and validation test cases the source term is given through the decomposition of an incident acoustic field on a section of the computational domain s boundary validations against analytic another numerical method and measurements on different test cases are presented

estimating functions and approximate conditional likelihood summary the approximate conditional likelihood method proposed by cox reid 1987 is applied to the estimation of a scalar parameter 0 in the presence of nuisance parameters the estimating function of 0 based on the approximate conditional likelihood is shown to be preferable to that based on the profile likelihood a sufficient condition for both approaches to be equivalent is given the role of parameter orthogonality is emphasized several examples including bivariate normal means with known coefficient of variation are presented

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

an efficient solution for cauchy likesystems of linear equations using the displacement approach and transformation of vectors the complexity ofsolving strongly nonsingular cauchy like systems of linear equations email protected email protected will be improved off a factor of log n from the known algorithms of the recursive triangular factorization where f is a field and email protected f n x 1 is a vector and c f n x n is a strongly nonsingular cauchy like matrix the efficient algorithms were presented by morf bitmead and anderson to solve strongly nonsingular toeplitz like equations of linear systems by using the recursive triangular factorization in 1980 recently this approach of the recursive triangular factorization was extended to solve cauchy like systems with the complexity of o n log 3n operations by pan et al this is the best known complexity bound by using the direct approach of recursive triangular factorization in cauchy like cases however these algorithms are still slower than the well known algorithms with the asymptotic bound of o n log 2n operations which have been proposed by the means of reducing cauchy like matrices into toeplitz like matrices in our present paper the algorithms of the recursive triangular factorization are modified so that the complexity bound of the direct recursive approach can be decreased to o n log 2n operations this matches the asymptotic bound without transforming to toeplitz like matrices our improvement of the direct recursive approach is by a factor of log n due to changing the original vectors expressed in the given cauchy like matrix into the special vectors where the entries are unit roots the applications of structured matrices include nevanlinna pick tangential interpolation problems

convergence results for projected line search methods on varieties of low rank matrices via łojasiewicz inequality the aim of this paper is to derive convergence results for projected line search methods on the real algebraic variety mathcal m le k of real m times n matrices of rank at most k such methods extend riemannian optimization methods which are successfully used on the smooth manifold mathcal m k of rank k matrices to its closure by taking steps along gradient related directions in the tangent cone and afterwards projecting back to mathcal m le k considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of mathcal m k the pointwise convergence is obtained for real analytic functions on the basis of a ojasiewicz inequality for the projection of the antigradient to the tangent cone if the derived limit point lies on the smooth part of mathcal m le k i e in mathcal m k this boils down to more or less known results but with the benefit that asymptotic convergence rate estimates for specific step sizes

low rank matrix completion by riemannian optimization the matrix completion problem consists of finding or approximating a low rank matrix based on a few samples of this matrix we propose a new algorithm for matrix completion that minimizes the least square distance on the sampling set over the riemannian manifold of fixed rank matrices the algorithm is an adaptation of classical nonlinear conjugate gradients developed within the framework of retraction based optimization on manifolds we describe all the necessary objects from differential geometry necessary to perform optimization over this low rank matrix manifold seen as a submanifold embedded in the space of matrices in particular we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions finally we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations the numerical experiments indicate that our approach

on expected volumes of multidimensional confidence sets associated with the usual and adjusted likelihoods we consider a general multiparameter set up where both the interest and the nuisance parameters are possibly vector valued we derive an explicit higher order asymptotic formula to compare the expected volumes of confidence sets given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof our general framework also allows us to include highest posterior density regions with approximate frequentist validity in the study the fact that our interest parameter is possibly vector valued complicates the derivation and warrants the development of special tools and techniques

robust exponential convergence of hp fem in balanced norms for singularly perturbed reaction diffusion problems corner domains abstract the h p version of the finite element method is applied to singularly perturbed reaction diffusion type equations on polygonal domains the solution exhibits boundary layers as well as corner layers on a class of meshes that are suitably refined near the boundary and the corners robust exponential convergence in the polynomial degree is shown in both a balanced norm and the maximum norm

data sparse approximation of non local operators by h2 matrices many of today s most efficient numerical methods are based on multilevel decompositions the multigrid algorithm is based on a hierarchy of grids wavelet techniques use a hierarchy of basis functions while fast panel clustering and multipole methods employ a hierarchy of clusters the high efficiency of these methods is due to the fact that the hierarchies are nested i e that the information present on a coarser level is also present on finer levels thus allowing efficient recursive algorithms h2 matrices employ nested local expansion systems in order to approximate matrices in optimal or for some problem classes at least optimal up to logarithmic factors order of complexity this paper presents a criterion for the approximability of general matrices in the h2 matrix format and an algorithm for finding good nested expansion systems and constructing the approximation efficiently

short note moore s law and the saffman taylor instability ten years ago hou lowengrub and shelley 4 published a state of the art boundary integral simulation of a classical viscous fingering problem the saffman taylor instability 12 in terms of complexity and level of detail those computations 4 are still among the most ramified and accurately computed interfacial instability patterns that have appeared in the literature since 1994 the computational power of a standard workstation has increased a hundredfold as predicted by moore s law 7 the purpose of this note is to consider moore s law and its consequences in computational science and in particular its impact on studying the saffman taylor instability we illustrate moore s law and fast algorithms in action by presenting the worlds largest viscous fingering simulation to date viscous fingering is one of the fundamental interfacial instabilities in fluid dynamics perturbations to an expanding circular air bubble displacing a viscous fluid in a thin gap flow device become unstable resulting in intricate densely branched interfacial patterns the viscous fingering problem is governed by the three dimensional incompressible navier stokes equations with moving free boundaries at the fluid air interface without simplification this problem is computationally intractable and will remain so for the foreseeable future the simulations in this note build upon decades of advances in mathematical modeling and numerical methods i the fluid dynamics in a thin gap is reduced from the threedimensional navier stokes equations by asymptotic analysis to a darcy s law 12

a mixed form thin stratified medium fast multipole algorithm for both low and mid frequency problems a mixed form thin stratified medium fast multipole algorithm is proposed for fast simulation of general microstrip structures at both low and mid frequencies the newly developed matrix friendly formula of layered medium green s function is applied in this algorithm for well separated interactions the path deformation technique is implemented to achieve a smoother and exponentially convergent integral the two dimensional addition theorem is then incorporated into the integrand to expedite the matrix vector product in our approach multipole expansion low frequency fast multiple algorithm as well as the plane wave expansion mid frequency fast multipole algorithm of the translational addition theorem are combined into a single multilevel tree to capture quasi static physics and wave physics simultaneously the outgoing wave is represented first in terms of multipole expansion at leafy levels and then switched to plane wave expansion automatically at higher levels this seamless connection makes the algorithm applicable in simulations where subwavelength interaction and wave physics both exist

analysis of arbitrarily shaped dielectric radomes using adaptive integral method based on volume integral equation the adaptive integral method aim is employed to solve the volume integral equation vie for analyzing the radiation of the antenna with an arbitrarily shaped radome small dipoles are used as exciting sources modeling the radomes by tetrahedron cells the induced volume current is determined by the aim based on vie the application of aim significantly reduces cpu time and computer memory requirement hence the method presented in the paper can be applied to simulate electrically large sized radomes finally the radiation patterns of small dipole arrays in the presence of spherical and conical radomes are calculated

an efficient surface integral algorithm applied to unsteady gravity waves a computationally fast method for calculating the unsteady motion of a surface on a two dimensional fluid is described cauchy s integral theorem is used iteratively to solve laplace s equation for successive time derivatives of the surface motion and time stepping is performed using truncated taylor series this allows fairly large time steps to be made for a given accuracy while the required number of spatial points is minimised by using high order differencing formulae this reduces the overall number of required calculations the numerical implementation of the method is found to be accurate and efficient a fairly thorough examination of this implementation is carried out revealing that high accuracies are often achievable using surprisingly few numerical surface points extensive calculations are also performed using modest computing resources some numerical instabilities are identified although these would not usually be significant in practical calculations a model analysis reveals that two of these instabilities can be eliminated by using suitable methods of time stepping should the third steep wave instability become significant it is shown that it can be completely controlled by using high order smoothing techniques at little cost to accuracy using a routine to ensure asymptotic conservation of energy this is confirmed by time stepping a very steep but stable wave over thousands of wave periods 0 1992 academx press inc since all of the interior properties of a body of fluid undergoing inviscid incompressible irrotational motion are fully determined by properties at its boundary it is possible to reduce the calculation of the motion of such a fluid to the evaluation of the motion of its surface alone in a numerical scheme the entire motion can thus be modelled using only a point discretisation of the surface such an approach was first suggested by svendsen in 1971 as a means of calculating the motion of two dimensional gravity waves on water 13 since then the idea has been implemented using a variety of boundary integral or conformal mapping techniques by longuet higgins and cokelet 2 vinje and brevig 3 baker meiron and orszag 4 roberts s fornberg 6 and based on his original formulation by svendsen and co workers 7 although this paper presents yet another numerical method the approach is aimed specifically at producing a computationally fast and efficient numerical scheme in order to be able to study problems of increased complexity in numerically representing complicated surface motions ranging from the breaking of waves to the evolution of instabilities in a train of travelling waves it typically becomes necessary to use a large number of surface points if running times tend to increase too rapidly with the number of points then this can destroy any apparent advantage in being able to treat the whole fluid only in terms of its moving surface the penalty is particularly severe for algorithms that require matrix inversion or factorisation techniques yielding running times per time step that increase like n3 the cube of the number of surface points by using iterative techniques the penalty can be reduced to running times per time step that increase in proportion to n2 while being less severe this still has the effect that increased numbers of surface points become increasingly expensive to calculate the use of a conformal mapping technique 6 can lead to running times per timestep that vary like n log n however this method is severely limited in its ability to resolve steep wave surfaces as are found in breaking waves requiring disproportionately many surface points to achieve a given accuracy it is therefore vital to ensure computational elliciency if it is to be possible accurately to describe wave processes for which large numbers of surface points may be required the most direct way of achieving this is to employ the lowest number of points that can adequately describe a fluid surface in this paper a further improvement is introduced by showing that higher time derivatives of the motion can be calculated directly using exactly the same technique that is used to determine the velocity of the surface at each timestep this yields significantly more information about the motion for relatively little extra computation in turn this allows larger time steps to be performed for a given accuracy so that the overall number of calculations can be reduced the method presented here combines these two ideas into producing a practical numerical algorithm the properties of which are examined in the following pages in appendix a the formulae are derived that make it

adjusted versions of profile likelihood and directed likelihood and extended likelihood several adjusted and invariant versions of profile likelihood and of directed likelihood short for signed log likelihood ratio are discussed these adjusted versions which fall into two groups are closely related to modified profile likelihood and modified directed likelihood the first group has the advantage of not requiring an explicit specification of an ancillary but partly at a cost of a loss of asymptotic accuracy in comparison with the modified versions the second group consists of some asymptotically sharpened versions of modified profile likelihood furthermore a concept of extended likelihood is proposed which associates a likelihood like function to certain combinants

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

accelerating boundary integral equation method using a special purpose computer we present a new solution to accelerate the boundary integral equation method biem the calculation time of the biem is dominated by the evaluation of the layer potential in the boundary integral equation we performed this task using mdgrape 2 a special purpose computer designed for molecular dynamics simulations mdgrape 2 calculates pairwise interactions among particles e g atoms and ions using hardwired pipeline processors we combined this hardware with an iterative solver during the iteration process mdgrape 2 evaluates the layer potential the rest of the calculation is performed on a conventional pc connected to mdgrape 2 we applied this solution to the laplace and helmholtz equations in three dimensions numerical tests showed that biem is accelerated by a factor of 10 100 our rather naive solution has a calculation cost of o n2 niter where n is the number of unknowns and niter is the number of iterations copyright 2005 john wiley sons ltd

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

numerical p version refinement studies for the regularized stress bem in the development of the boundary element method bem and the finite element method fem researchers have typically selected similar basis functions that is both methods typically employ low order interpolations such as piece wise linear or piece wise quadratic and rely on h version refinement to increase accuracy as required in the case of the fem the decision to use low order elements is made for computational efficiency as an attractive compromise between local modeling accuracy and sparseness of the resulting linear system however in many bem formulations low order elements may be the only practical choice given the complexity of using analytic integration formulae in conjunction with special integral interpretations unlike their efficient use in the fem fine meshes of low order elements in the bem are highly inefficient from a computational standpoint given the dense nature of bem systems moreover owing to singularities in the kernel functions the bem should be expected to benefit more so than the fem from very high levels of local accuracy through the use of regularized algorithms which only require numerical integration p version refinement in the bem is easily extended to include any set of basis functions with no significant increase in programming complexity numerical results show that by using interpolations as high as 12th and 16th order one can expect reductions in error by as many as five orders of magnitude over comparable algorithms based on similar system size for two dimensional problems it is also shown that for a given level of error one can expect reductions in system size by an order of magnitude thus leading to a reduction in computational expense for conventional algorithms by three orders of magnitude copyright 2003 john wiley sons ltd

why finite element discretizations can be factored by triangular hierarchical matrices although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second order elliptic partial differential operators is far from being optimal these methods are often preferred over modern iterative methods this is mainly due to their robustness in this article it is shown that an approximate lu decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical lu decomposition low precision approximations may be used for preconditioning iterative solvers as a byproduct we prove that schur complements of stiffness matrices can be approximated with almost linear complexity

integral equation methods for stokes flow and isotropic elasticity in the plane we present a class of integral equation methods for the solution of biharmonic boundary value problems with applications to two dimensional stokes flow and isotropic elasticity the domains may be multiply connected and finite infinite or semi infinite in extent our analytic formulation is based on complex variables and our fast multipole based iterative solution procedure requireso n operations wherenis the number of nodes in the discretization of the boundary the performance of the methods is illustrated with several large scale numerical examples

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

adaptive spatial decomposition in fast multipole method this work presents a new adaptive node cluster algorithm for fast multipole method in the algorithm we use rectangular boxes instead of cubes subdivide a box based on its shape and tighten the child boxes at each subdivision step more importantly we determine the number of expansion terms in multipole to local translations according to the distance between the two interaction boxes our method is tested using benchmark examples for three dimensional potential problems the results obtained show that the new algorithm can solve a problem with 100 thousands nodes in about 20 min and runs nearly three times faster than the standard algorithm the proposed algorithm is especially suitable for treating slender and shell like structures

a second order adjustment to the profile likelihood in the case of a multidimensional parameter of interest summary inference in the presence of nuisance parameters is often carried out by using the x2approximation to the profile likelihood ratio statistic however in small samples the accuracy of such procedures may be poor in part because the profile likelihood does not behave as a true likelihood in particular having a profile score bias and information bias which do not vanish to account better for nuisance parameters various researchers have suggested that inference be based on an additively adjusted version of the profile likelihood function each of these adjustments to the profile likelihood generally has the effect of reducing the bias of the associated profile score statistic however these adjustments are not applicable outside the specific parametric framework for which they were developed in particular it is often difficult or even impossible to apply them where the parameter about which inference is desired is multidimensional in this paper we propose a new adjustment function which leads to an adjusted profile likelihood having reduced score and information biases and is readily applicable to a general parametric framework including the case of vector valued parameters of interest examples are given to examine the performance of the new adjusted profile likelihood in small samples and also to compare its performance with other adjusted profile likelihoods

a fast algorithm for the multiplication of generalized hilbert matrices with vectors on presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de hilbert generalisees par des vecteurs

rcs computation of large inhomogeneous objects using a fast integral equation solver in this paper we apply a fast fourier transform fft accelerated volume integral equation solver to compute the radar cross section of large scale inhomogeneous objects this method is related to bojarski s k space method and the subsequent conjugate cg and biconjugate gradient bcg fft algorithms the method developed here combines the weak form discretization with the bcg and stabilized bcg bcgs solvers we show that this method has marked improvements over related algorithms the numerical method has been validated by mie series for multilayer spheres and applied to some practical problems with this method we are currently able to solve a three dimensional problem with a volume of size 3648 spl lambda sup 3 21 23 million unknowns on a single workstation

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

accelerating t sne using tree based algorithms the paper investigates the acceleration of t sne an embedding technique that is commonly used for the visualization of high dimensional data in scatter plots using two tree based algorithms in particular the paper develops variants of the barnes hut algorithm and of the dual tree algorithm that approximate the gradient used for learning t sne embeddings in o n log n our experiments show that the resulting algorithms substantially accelerate t sne and that they make it possible to learn embeddings of data sets with millions of objects somewhat counterintuitively the barnes hut variant of t sne appears to outperform the dual tree variant

adjusted versus conditional likelihood power properties and bartlett type adjustment summary under contiguous alternatives the adjusted likelihood ratio alr test is seen to have the same power properties as the conditional likelihood ratio test up to the third order of comparison in particular an optimum property of the alr test in terms of second order local maximinity follows it is also seen that the alr statistic admits a bartlett type adjustment

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

improving error bounds for multipole based treecodes rapid evaluation of potentials in particle systems is an important time consuming step in many physical simulations over the past decade the development of treecodes such as the fast multipole method and barnes hut method has enabled large scale simulations in astrophysics molecular dynamics material science etc these methods use fixed degree polynomial approximations of the potential at a point due to a set of particles in a hierarchical representation of the particle system in this paper we present analysis and experiments to illustrate that fixed degree multipole approximations can lead to large aggregate errors we describe an alternate strategy based on careful selection of the multipole degree that leads to asymptotically lower errors while incurring minimal computation overhead first we estimate the error associated with each particle cluster interaction and the aggregate error for each particle then we describe a technique for computing the multipole degree of each interaction with a view to reducing aggregate error and establish the computational complexity of the new method following this we discuss numerical experiments that demonstrate favorable error properties of the new method at the expense of marginal increase in computation finally we report a parallel implementation on the sgi origin 2000 that achieves excellent speedup for problems from domains such as astrophysics and boundary element solvers

iterative methods for low rank approximation of graph similarity matrices in this paper we analyze an algorithm to compute a low rank approximation of the similarity matrix s introduced by blondel et al in 1 this problem can be reformulated as an optimization problem of a continuous function s tr stm2 s where s is constrained to have unit frobenius norm and m2 is a non negative linear map we restrict the feasible set to the set of matrices of unit frobenius norm with either k nonzero identical singular values or at most k nonzero not necessarily identical singular values we first characterize the stationary points of the associated optimization problems and further consider iterative algorithms to find one of them we analyze the convergence properties of our algorithm and prove that accumulation points are stationary points of s we finally compare our method in terms of speed and accuracy to the full rank algorithm proposed in 1

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

on bias reduction and incidental parameters summary firth 1993 introduced a method for reducing the bias of the maximum likelihood estimator here it is shown that the approach is also effective in reducing the sensitivity of inferential procedures to incidental parameters

a multi level boundary element method for stokes flows in irregular two dimensional domains recently we have developed multi level boundary element methods mlbem for the solution of the laplace and helmholtz equations that involve asymptotically decaying non oscillatory and oscillatory singular kernels respectively the accuracy and efficiency of the fast boundary element methods for steady state heat diffusion and acoustics problems have been investigated for square domains the current work extends the mlbem methodology to the solution of stokes equation in more complex two dimensional domains the performance of the fast boundary element method for the stokes flows is first investigated for a model problem in a unit square then we consider an example problem possessing an analytical solution in a rectangular domain with 5 1 aspect ratio and finally we study the performance of the mlbem algorithm in a c shaped domain

bartlett s bias correction to the profile score function is a saddlepoint correction summary bartlett s bias correction to the profile score function is proved equal to first order to the leading correction term of the double saddlepoint approximation to the conditional score function in canonical exponential families gart 1985 1987 has pointed out the effectiveness of bartlett s bias correction to the profile score function in approximate conditional binomial likelihood analysis while the leading term in bartlett s 1953 1955 expansion of the bias function ignores terms of order o n 2 the first term correction works well for conditional analysis because as shown in this note in canonical exponential families it agrees with the first correction term of the double saddlepoint approximation to the conditional score function and that approximation is accurate up to terms of order o n

stokes flow around cylinders in a bounded two dimensional domain using multipole accelerated boundary element methods the multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from o n2 iterative solvers or o n3 gaussian elimination to o n log n or o n where n is the number of nodes in the discretized system such a reduction in the growth of the calculation time and data storage is crucial in applications where n is large such as when modelling the macroscopic behaviour of suspensions of particles in such cases a minimum of 1000 particles is needed to obtain statistically meaningful results leading to systems with n of the order of 10 000 for the smallest problems when only boundary velocities are known the indirect boundary element formulation for stokes flow results in fredholm equations of the second kind which generally produce a well posed set of equations when discretized a necessary requirement for iterative solution methods the direct boundary element formulation on the other hand results in fredholm equations of the first kind which upon discretization produce ill conditioned systems of equations the model system here is a two dimensional wide gap couette viscometer where particles are suspended in the fluid between the cylinders this is a typical system that is efficiently modelled using boundary element method simulations the multipolar technique is applied to both direct and indirect formulations it is found that the indirect approach is sufficiently well conditioned to allow the use of fast multipole methods the direct approach results in severe ill conditioning to a point where application of the multipole method leads to non convergence of the solution iteration copyright 1999 john wiley sons ltd

median bias reduction of maximum likelihood estimates for regular parametric problems we show how median centring of the maximum likelihood estimate can be achieved by a simple modification of the score equation for a scalar parameter of interest the estimator is equivariant under interest respecting reparameterizations and is third order median unbiased with a vector parameter of interest componentwise equivariance and third order median centring are obtained like the implicit method of firth 1993 for bias reduction the new method does not require finiteness of the maximum likelihood estimate and is effective in preventing infinite estimates simulation results for continuous and discrete models including binary and beta regression confirm that the method succeeds in achieving componentwise median centring and in solving the boundary estimate problem while keeping comparable dispersion and the same approximate distribution as its main competitors

fast adaptive summation of point forces in the two dimensional poisson equation direct summation of the velocity field introduced by point vortices tends to be time consuming since the velocity of each vortex is found as a sum over all other vortices the resulting number of numerical operations is proportional to the square of the number of vortices here a relatively simple procedure is outlined which significantly reduces the number of operations by replacing selected partial sums by asymptotic series the resulting number of operations appears to vary roughly in proportion to the number of unknowns corresponding to a fast solver

a comparison of domain integral evaluation techniques for boundary element methods in many cases boundary integral equations contain a domain integral this can be evaluated by discretization of the domain into domain elements historically this was seen as going against the spirit of boundary element methods and several methods were developed to avoid this discretization notably dual and multiple reciprocity methods and particular solution methods these involved the representation of the interior function with a set of basis functions generally of the radial type in this study meshless methods dual reciprocity and particular solution are compared to the direct domain integration methods the domain integrals are evaluated using traditional methods and also with multipole acceleration it is found that the direct integration always results in better accuracy as well as smaller computation times in addition the multipole method further improves on the computation times in particular where multiple evaluations of the integral are required as when iterative solvers are used the additional error produced by the multipole acceleration is negligible copyright 2001 john wiley sons ltd

efficient numerical methods for multiple surfactant coated bubbles in a two dimensional stokes flow we present efficient and highly accurate numerical methods to compute the deformation of surfactant coated two dimensional bubbles in a slow viscous flow surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion in this paper we restrict our attention to the case of a dilute insoluble surfactant the convection diffusion equation for the surfactant concentration on the interface is coupled with the stokes equations in the fluid domain through a boundary condition based on the laplace young condition the stokes equations are first recast as an integral equation and then solved using a fast multipole accelerated iterative procedure the computational cost per time step is only o n log n operations with n being the number of discretization points on the interface the bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points this equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient implicit treatment of the stiffest terms in the dynamics several phenomenologically different examples are presented

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

approximate likelihood ratios for general estimating functions summary the method of estimating functions godambe 1991 is commonly used when one desires to conduct inference about some parameters of interest but the full distribution of the observations is unknown however this approach may have limited utility due to multiple roots for the estimating function a poorly behaved wald test or lack of a goodness of fit test this paper presents approximate likelihood ratios that can be used along with estimating functions when any of these three problems occurs we show that the approximate likelihood ratio provides correct large sample inference under very general circumstances including clustered data and misspecified weights in the estimating function two methods of constructing the approximate likelihood ratio one based on the quasi likelihood approach and the other based on the linear projection approach are compared and shown to be closely related in particular we show that quasi likelihood is the limit of the projection approach we illustrate the technique with two applications

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

a parallel multipolar indirect boundary element method for the neumann interior stokes flow problem a multipolar expansion technique is applied to the indirect formulation of the boundary element method in order to solve the two dimensional internal stokes flow second kind boundary value problems the algorithm is based on a multipolar expansion for the far field and numerical evaluation for the near field due to the nature of the algorithm it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations a parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvements copyright 2000 john wiley sons ltd

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

a note on the difference between profile and modified profile likelihood the difference between profile likelihood and modified profile likelihood depends primarily on the expected value of a certain third order derivative of the log likelihood it is shown that in exponential family problems this derivative vanishes if the parameter of interest is a mean parameter but in general not when it is a canonical parameter as examples some matched pair problems for exponential family models are considered

fast multipole boundary element method for the laplace equation in a locally perturbed half plane with a robin boundary condition a fast multipole boundary element method fm bem for solving large scale potential problems ruled by the laplace equation in a locally perturbed 2 d half plane with a robin boundary condition is developed in this paper these problems arise in a wide gamut of applications being the most relevant one the scattering of water waves by floating and submerged bodies in water of infinite depth the method is based on a multipole expansion of an explicit representation of the associated green s function which depends on a combination of complex valued exponential integrals and elementary functions the resulting method exhibits a computational performance and memory requirements similar to the classic fm bem for full plane potential problems numerical examples demonstrate the accuracy and efficiency of the method

information bias and adjusted profile likelihoods summary the bias and information bias of the ordinary profile score statistic are both typically of order 0 1 several additive adjustments to the profile score statistic that reduce its bias to order o n have been proposed in certain situations the information bias of these adjusted profile score statistics is also reduced to order o n q specifically the modified profile likelihood of barndorff nielsen yields an adjusted profile score statistic having both reduced bias and reduced information bias in general however a bias reducing adjustment to the profile score statistic will not automatically reduce the order of the information bias as well in this paper an analytical formula is obtained for the information bias of a general bias adjusted profile score statistic this formula is used to compare bias reducing adjustments to the profile score statistic as well as to construct further adjustments that reduce the information bias to o n 1 several examples are presented to illustrate use of the formula for information bias in particular the information bias formula may be utilized in a criterion for choosing between orthogonal parameterizations for the conditional profile likelihood of cox and reid

a fast bie iteration method for an arbitrary body in a flow of incompressible inviscid fluid the paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics the stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid with a discrete numerical treatment we approximate the exact matrix by a certain toeplitz one and then apply a fast algorithm for this matrix on each iteration step we illustrate the convergence of this iteration scheme by a number of numerical examples both for hard and soft boundary conditions it appears that the method is highly efficient for hard boundaries being much less efficient for soft boundaries

an adaptive multigrid technique for evaluating long range forces in biomolecular simulations we propose an adaptive multigrid method for evaluating the nonlocal long range interactions in biomolecular models the method provides o h 3 accuracy for the forces where h is the mesh size of the finest cubic grid even though the stencil size is minimal two new algorithmic components are developed an adaptive technique for spreading a three dimensional charge distribution onto a space mesh and a procedure for performing coarse to fine grid interpolation of the potential force components and their derivatives grid to particle interpolation at the finest level in a consistent fashion the adaptive mesh used in the spreading procedure contains two types of nodes regular positioned at nodes cubic grid and flexible one per cubic cell thus the charges located in each grid cell are spread to these nine nodes the method conserves charge dipole and quadrupole moments of cells the spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils implementation is also straightforward our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique

a fast dual boundary element method for 3d anisotropic crack problems in the present paper a fast solver for dual boundary element analysis of 3d anisotropic crack problems is formulated implemented and tested the fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix the admissible low rank blocks are computed by adaptive cross approximation aca the performance of aca against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated focusing on the balance between the kernel representation accuracy and the accuracy required for aca the system solution is computed by a preconditioned gmres and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format the effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated highlighting the accuracy as well as the significant reduction in memory storage and analysis time in particular it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom obtaining up to solution speedups of order 10 for systems of order 104 moreover the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed copyright 2009 john wiley sons ltd

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

on the equivalence of marginal and approximate conditional likelihoods for correlation parameters under a normal model summary marginal and approximate conditional likelihoods are obtained for the correlation parameters in a normal linear regression model with correlated errors these likelihoods may be evaluated using the kalman filter it is shown that the marginal and conditional likelihoods are equivalent for any correlation matrix whose entries are continuous and differentiable functions of its parameters the results are illustrated by the application of marginal and conditional likelihood methods to estimation of the correlation parameters in time series and correlated spatial processes the concept of a marginal likelihood was introduced originally in the structural inference context by fraser 1967 and later developed in the classical framework by kalbfleisch sprott 1970 as a general method for eliminating nuisance parameters from the likelihood function more recently cox reid 1987 introduced approximate conditional likelihoods which also address this problem they argued that this conditional likelihood was preferable to the profile likelihood obtained by replacing the nuisance parameters in the likelihood by their maximum likelihood estimates when the parameters of interest are given following on the work of cox reid 1987 cruddas reid cox 1989 obtained an approximate conditional likelihood for the correlation parameter in several short series of autoregressive processes of order one with common variance and autocorrelation parameter based on a simulation study cruddas et al 1989 showed that the estimate based on the approximate conditional likelihood had a much smaller bias and better coverage properties of the confidence interval than the maximum likelihood estimate from the profile likelihood they also showed that the approximate conditional and the marginal likelihood were the same in this situation the approximate conditional likelihood for the correlation parameters in a general normal regression model with correlated errors is obtained here and is shown to be equivalent to the marginal likelihood for the same parameters

a measure of similarity between graph vertices applications to synonym extraction and web searching we introduce a concept of similarity between vertices of directed graphs let ga and gb be two directed graphs with respectively na and nb vertices we define an nb times na similarity matrix s whose real entry sij expresses how similar vertex j in ga is to vertex i in gb we say that sij is their similarity score the similarity matrix can be obtained as the limit of the normalized even iterates of sk 1 bskat btska where a and b are adjacency matrices of the graphs and s0 is a matrix whose entries are all equal to 1 in the special case where ga gb g the matrix s is square and the score sij is the similarity score between the vertices i and j of g we point out that kleinberg s hub and authority method to identify web pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them in analogy to kleinberg we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix potential applications of our similarity concept are numerous we illustrate an application for the automatic extraction of synonyms in a monolingual dictionary

combining kronecker product approximation with discrete wavelet transforms to solve dense function related linear systems a new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory this technique combines kronecker product approximation and wavelet sparsification for the kronecker product factors the user is only required to supply a procedure for computation of each entry of the given matrix the main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic skeleton method of the second author and data sparse preconditioners the incomplete lu decomposition with dynamic choice of the fill in structure with a prescribed threshold and the inverse kronecker product preconditioner constructed for the sum of kronecker products of sparsified finger like matrices computed by the discrete wavelet transform in some model but quite representative examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 gbyte operative memory

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

numerical computation of an analytic singular value decomposition of a matrix valued function this paper extends the singular value decomposition to a path of matricese t an analytic singular value decomposition of a path of matricese t is an analytic path of factorizationse t x t s t y t t wherex t andy t are orthogonal ands t is diagonal to maintain differentiability the diagonal entries ofs t are allowed to be either positive or negative and to appear in any order this paper investigates existence and uniqueness of analytic svd s and develops an algorithm for computing them we show that a real analytic pathe t always admits a real analytic svd a full rank smooth pathe t with distinct singular values admits a smooth svd we derive a differential equation for the left factor develop euler like and extrapolated euler like numerical methods for approximating an analytic svd and prove that the euler like method converges

boundary algebraic equations for lattice problems boundary algebraic equations corresponding to dirichlet boundary value problems on lattices are introduced these equations are based on the lattice green s function from which discrete single and double layer potentials are derived structurally the boundary algebraic equations are similar to the boundary integral equations of classical potential theory numerical experiments indicate that boundary algebraic equations possess excellent spectral properties

a time series illustration of approximate conditional likelihood summary we consider the calculation of confidence intervals for the correlation parameter based on observations of a large number of short gaussian first order autoregressive processes with different means but common correlation and variance ordinary profile likelihood gives very misleading answers the approximate conditional likelihood of cox reid 1987 is calculated and shown to lead to consistent estimation of the correlation simulations verify that the usual asymptotic properties of the statistics derived from the conditional likelihood are quite satisfactory in this example the approximate conditional likelihood is the same as the marginal likelihood for the standardized residuals although this is not true in general

a boundary element method for steady convective heat diffusion in three dimensions a higher order boundary element method recently developed by the current authors numer heat trans part b fundamentals 45 2004 109 for two dimensional steady convective heat diffusion is generalized to three dimensions in order to facilitate an accurate and efficient boundary element formulation we introduce an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence the localization of the kernels becomes more prominent as the peclet number of the flow increases this in turn leads to increasing sparsity and improved conditioning of the global matrix consequently iterative solvers for sparse matrices become the primary choice in this paper we consider an example problem with an exact solution and investigate the accuracy and efficiency of the boundary element formulation for peclet numbers in the range from 10 to 1000 the bi quartic boundary elements included in this study are shown to provide acceptable levels of resolution

the hybrid boundary node method accelerated by fast multipole expansion technique for 3d potential problems this paper presents a fast formulation of the hybrid boundary node method hybrid bnm for solving problems governed by laplace s equation in 3d the preconditioned gmres is employed for solving the resulting system of equations at each iteration step of the gmres the matrix vector multiplication is accelerated by the fast multipole method green s kernel function is expanded in terms of spherical harmonic series an oct tree data structure is used to hierarchically subdivide the computational domain into well separated cells and to invoke the multipole expansion approximation formulations for the local and multipole expansions and also conversion of multipole to local expansion are given and a binary tree data structure is applied to accelerate the moving least square approximation on surfaces all the formulations are implemented in a computer code written in c numerical examples demonstrate the accuracy and efficiency of the proposed approach copyright 2005 john wiley sons ltd