fast directional multilevel summation for oscillatory kernels based on chebyshev interpolation many applications lead to large systems of linear equations with dense matrices direct matrix vector products become prohibitive since the computational cost increases quadratically with the size of the problem by exploiting specific kernel properties fast algorithms can be constructed a directional multilevel algorithm for translation invariant oscillatory kernels of the type k x y g x y e i k x y with g x y being any smooth kernel will be presented we will first present a general approach to build fast multipole methods fmms based on chebyshev interpolation and the adaptive cross approximation aca for smooth kernels the chebyshev interpolation is used to transfer information up and down the levels of the fmm the scheme is further accelerated by compressing the information stored at chebyshev interpolation points using aca and qr decompositions this leads to a nearly optimal computational cost with a small pre processing time due to the low computational cost of aca this approach is in particular faster than performing singular value decompositions this does not address the difficulties associated with the oscillatory nature of k for that purpose we consider the following modification of the kernel k u k x y e i k u x y where u is a unit vector see brandt 1 we proved that the kernel k u can be interpolated efficiently when x y lies in a cone of direction u this result is used to construct an fmm for the kernel k theoretical error bounds will be presented to control the error in the computation as well as the computational cost of the method the paper ends with the presentation of 2d and 3d numerical convergence studies and computational cost benchmarks

global structure and finite size effects in the f alpha of diffusion limited aggregates the multifractal spectrum f alpha characterizing the scaling properties of the growth probability on the boundary of radial diffusion limited aggregates is known to exhibit strong finite size effects we demonstrate that there exists a correlation between these finite size effects and those present in measurements of the angular width of the fjords which lie between the principal cluster arms we subsequently conclude that it is the evolution in the global structure of the clusters which is responsible for the slow convergence in f alpha and discuss how this global structure induces a phase transition in f alpha

on spherical convergence convexity and block iterative projection algorithms in hilbert space abstract we call a sequence x n in hilbert space spherical if there exists u such that lim x n u exists and is finite if moreover u is a weak accumulation point of the sequence we call the sequence spherically convergent we demonstrate that for large classes of nonexpansive possibly nonstationary discrete time processes in hilbert space the iterates are spherically convergent basic identities and orthogonality relations pertinent to this type of convergence are exhibited sufficient conditions for weak and spherical convergence in terms of the fullness of the set of fixed points of the process are established and compared spherical convergence of the general block iterative projection scheme in hilbert space is established

block iterative projection methods for parallel computation of solutions to convex feasibility problems abstract an iterative method is proposed for solving convex feasibility problems each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step it is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions special cases are block iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used in particular a block iterative version of the agmon motzkin schoenberg relaxation method for solving systems of linear inequalities is derived such processes lend themselves to parallel implementation and will be useful in various areas of applications including image reconstruction from projections image restoration and other fully discretized inversion problems

analytical study and numerical experiments for radiation and scattering problems using the chief method in this paper the well known failure in numerical computations of the exterior helmholtz integral equation at certain characteristic frequencies is investigated analytically and numerically the combined helmholtz integral equation formulation chief method is a very popular technique that can overcome the non unique problem but this method breaks down if the interior points are not properly chosen the chief method in conjunction with the singular value decomposition svd technique is an easy and efficient method to ensure a unique solution for the exterior problem based on the circulant properties and degenerate kernels an analytical scheme in discrete system of a cylinder is achieved the optimum numbers and proper positions for the collocation points in the interior domain are analytically studied and suggested in the numerical scheme numerical experiments are designed to verify the analytical results one analytical example for a circular cylinder and one numerical example for a square rod are demonstrated to show the validity of the chief method

strong convergence of almost simultaneous block iterative projection methods in hilbert spaces abstract we prove strong convergence of a class of block iterative projection methods for finding a common point of a finite family of closed convex subsets in a hilbert space

a weighted residual formulation for the chief method in acoustics in this article the combined helmholtz integral equation formulation chief proposed by schenck is greatly improved so that it will alleviate the annoying nodal surface problem the constraint provided by each chief equation and its first order derivatives is enforced in a weighted residual sense over a small interior region rather than only at a single point consequently chief blocks rather than chief points are used in the numerical implementation although the corresponding interior problem may have nodal surfaces at characteristic frequencies it has no such things as nodal blocks therefore the chief block chief b method is expected to work even if all of the chief blocks fall on the intersection of several nodal surfaces this new formulation is implemented in an isoparametric element environment numerical examples in spherical radiation and scattering at intermediate frequencies are given to test the new formulation

diffusion limited aggregation a paradigm of disorderly cluster growth the purpose of this talk is to present a brief overview of our group s recent research into dynamic mechanisms of disorderly growth an exciting new branch of condensed matter physics in which the methods and concepts of modern statistical mechanics are proving to be useful our strategy has been to focus on attempting to understand a single model system diffusion limited aggregation dla this philosophy was the guiding principle for years of research in phase transitions and critical phenomena for example by focusing on the ising model steady progress was made over a period of six decades and eventually led to understanding a wide range of critical point phenomena since even systems for which the ising model was not appropriate turned out to be described by variants of the ising model such as the xy and heisenberg models so also we are optimistic that whatever we may learn in trying to understand dla will lead to generic information helpful in understanding general aspects of dynamic mechanisms underlying disorderly growth

acoustic analogy and alternative theories for jet noise prediction several methods for the prediction of jet noise are described all but one of the noise prediction schemes are based on lighthill s or lilley s acoustic analogy whereas the other is the jet noise generation model recently proposed by tam and auriault in all of the approaches some assumptions must be made concerning the statistical properties of the turbulent sources in each case the characteristic scales of the turbulence are obtained from a solution of the reynolds averaged navier stokes equation using a kappa sigma turbulence model it is shown that for the same level of empiricism tam and auriault s model yields better agreement with experimental noise measurements than the acoustic analogy it is then shown that this result is not because of some fundamental flaw in the acoustic analogy approach but instead is associated with the assumptions made in the approximation of the turbulent source statistics if consistent assumptions are made both the acoustic analogy and tam and auriault s model yield identical noise predictions in conclusion a proposal is presented for an acoustic analogy that provides a clearer identification of the equivalent source mechanisms as is a discussion of noise prediction issues that remain to be resolved

a method to overcome computational difficulties in the exterior acoustics problem when the boundary integral equation method is applied to exterior acoustics problems singularities occur in the resulting algebraic equations at various frequencies associated with the eigenvalues of an interior problem these frequencies are referred to as forbidden and various methods have been devised to overcome the computational difficulties presented at these frequencies the work presented here is an extension to the chief method in that higher derivatives in addition to the function itself are constrained to be zero at selected points in the interior domain whereas the relative success of either method depends on the quantity and selection of interior points the superchief method requires fewer interior points and is less sensitive to point selection resulting in improved robustness without a significant increase in computational complexity

self similarity of harmonic measure on dla abstract the right hand side of the curve of the harmonic measure on dla is undefined this does not necessarily imply that the harmonic measure and the dla geometry are not self similar we show for off lattice dla that the right hand tail satisfies a different rescaling rule this cauchy rescaling is compatible with self similarity the analysis is done on off off lattice dla in which both the brownian motion and the laplace equation are off lattice the cluster sizes range between 32 and 50 000 atoms the square lattice used to numerically estimate the laplacian potential introduces a lower cutoff on the spatial resolution of this potential we find a dependence of the right tail of the distribution of holders on this ultraviolet cutoff whereas the shape of the tail does depend on this ultraviolet lattice cutoff the applicability of the collapse rules do not

harmonic measure around a linearly self similar tree the authors use the concept of random multiplicative processes to help describe and understand the distribution of the harmonic measure on growing fractal boundaries the laplacian potential around a linearly self similar square koch tree is studied in detail the multiplicative nature of this potential and the consequent multifractality of the harmonic measure are discussed on prefractal stages the density d mu of the harmonic measure and the corresponding holder alpha ln d mu are well defined along the boundary except in the folds where the tangent is undefined a regularization scheme is introduced to eliminate these local effects they then consider the probability distributions p alpha d alpha of successive stages and discuss their collapse into an f alpha curve both the left and right hand sides of this curve show good convergence other studies indicate that for dla the right hand tail does not converge a brief comparison is made between the multifractality of these two cases

new anomalous multiplicative multifractals left sided ƒ α and the modelling of dla abstract a sharp distinction is drawn between general multiplicative multifractals as originally introduced by the author and the more familiar but more restricted class defined by frisch and parisi and by halsey et al all the general multiplicative multifractals are exactly renormalizable by design however this is important and perhaps suprising it is shown that the scaling relations that serve to define the restricted class fail to extend to certain multiplicative multifractals as a result in addition to the familiar restricted class the general multifractals are shown to allow for a wide variety of anomalous behaviors several of which are described we believe that suitable examples of these anomalies are of the most important concrete applications to fully developed turbulence and to diffusion limited aggregates dla this paper concerns several new classes of multifractals we have recently proposed as tools to model the anomalies of dla they are left sided that is characterized by max with max d 0 related multifractals characterized by max 0 and even by max and min 0 are encountered along the way

an estimate for lipschitz constants of metric projections abstract the main purpose of the paper is to give a global estimate for lipschitz constants of metric projections in a p uniformly convex and q uniformly smooth banach space

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

self similarity of the loop structure of diffusion limited aggregates the structure of fjords in diffusion limited aggregation dla clusters can be described in terms of the loop size distribution nr x which is the normalised number of loops with a neck to depth ratio x within a circle of radius r centred at the origin of the cluster the authors find from the numerical study of very large off lattice aggregates that nr x converges quickly to a limiting distribution with a well defined smallest ratio xmin larger than zero indicating the self similarity of the loop structure if the fjords are self similar i e they do not have long tube like structures one does not expect a phase transition in the multifractal spectrum of growth probabilities of typical dla clusters generated on the plain their study is essentially statistical and they cannot rule out the possibility of such rare events e g the occurrence of a few loops with anomalously small x which may result in a qualitatively different behaviour concerning the multifractal spectrum

a multifractal wavelet model with application to network traffic we develop a new multiscale modeling framework for characterizing positive valued data with long range dependent correlations 1 f noise using the haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results the model provides a rapid o n cascade algorithm for synthesizing n point data sets we study both the second order and multifractal properties of the model the latter after a tutorial overview of multifractal analysis we derive a scheme for matching the model to real data observations and to demonstrate its effectiveness apply the model to network traffic synthesis the flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics variance time plots and moment scaling and queuing behavior although for illustrative purposes we focus on applications in network traffic modeling the multifractal wavelet model could be useful in a number of other areas involving positive data including image processing finance and geophysics

a direct boundary integral formulation for acoustic radiation in a subsonic uniform flow abstract in this paper a new boundary integral formulation is proposed for acoustic radiation in a subsonic flow this new boundary integral formulation uses the green s function derived from the adjoint operator of the original governing differential equation unlike the conventional formulation that uses the helmholtz integral equation in a transformed domain the new boundary integral formulation requires no co ordinate transformation the variables in the integral equation can be expressed explicitly in terms of the velocity potential as well as its normal and tangential derivatives this will facilitate a direct coupling of the boundary element method with other numerical techniques such as the finite element method for radiation and scattering in a non uniform flow region

multifractal formalism for self similar bridges we derive the thermodynamics of self similar paths or bridges joining the two points and of the plane these paths may be constituted with both macroscopic and microscopic fragments each deserving its specific statistics while remaining continuous such discontinuous paths are also studied with some information related to the statistics of their jumps if the bridges under study are bound to be non decreasing paths this study coincides with the one of multifractal measures on the unit interval relaxing this condition leads to an extension of the multifractal formalism whose main lines are derived here

noise from fine scale turbulence of nonaxisymmetric jets the noise from the fine scale turbulence of high speed nonaxisymmetric jets is considered a prediction method is developed by extending the work of tam and auriault a set of improved numerical boundary conditions for use in nonaxisymmetricjet mean flow and turbulence calculation is developed these new boundary conditions allow the computation to be carried out in a smaller computation domain it is known that nonaxisymmetric mean flow has a significant impact on the radiated noise spectrum and directivity through refraction in the tam and auriault theory this effect is accounted for by means of the adjoint green s function here the adjoint green s function method is extended to nonaxisymmetric mean flows the adjoint green s function is first recast into the solution of a sound scattering problem the sound scattering problem is then solved computationally by computational aeroacoustics methods extensive comparisons between calculated and experimentally measured jet noise spectra are presented they include both rectangular and elliptic jets at supersonic and subsonic mach numbers

behaviour of the harmonic measure at the bottom of fjords diffusion limited aggregates are among many important fractal shapes that involve deep indentations usually called fjords to estimate the harmonic measure at the bottom of a fjord seems a prohibitive task but the authors find that a new mathematical equality due to beurling carleson and jones makes it easy they find that the harmonic measure at the bottom of a fjord as a function of its euclidean depth can exhibit a wide range of behaviours they introduce an infinite family of model fjords for which the equality takes a very simple form in this family the decay of the harmonic measure at their bottoms can be for example power law semi exponential stretched exponential and exponentially stretched exponential they show that self affinity or randomness can lead to faster than power law decays of the minimal growth probability on boundaries

convergence results for an accelerated nonlinear cimmino algorithm we present an accelerated version of cimmino s algorithm for solving the convex feasibility problem in finite dimension the algorithm is similar to that given by censor and elfving for linear inequalities we show that the nonlinear version converges locally to a weighted least squares solution in the general case and globally to a feasible solution in the consistent case applications to the linear problem are suggested

multifractality of the harmonic measure on fractal aggregates and extended self similarity abstract we show that dla follows a surprising new scaling rule it expresses that the screened region in which the harmonic measure is tiny increases more than proportionately as the cluster grows this scaling rule also gives indirect evidence that the harmonic measure of lattice dla follows a hyperbolic probability distribution of exponent equal to 1 this distribution predicts that sample moments behave erratically hence explains why the common restricted multifractal formalism fails to apply to dla

a bem fmm approach applied to the combined convected helmholtz integral formulation for the solution of aeroacoustic problems abstract exterior and interior acoustic calculations often exploit the advantages of boundary element method bem numerical scheme due to the large dimension of dense matrices arising in real case applications direct matrix vector products become prohibitive since the computational cost of standard methods increases quadratically in this work the black box fast multipole method bbfmm based on the chebyshev interpolation is extended to the integral solution of the convective helmholtz equation stabilized using the combined helmholtz integral equation formulation chief the chief approach relies on the solution of an over determined linear system achieved by adding a set of points internal to the scattering body in order to handle the distribution of the chief points within the body volume a robust stochastic approach has been also developed moreover an adaptive bbfmm abbfmm is presented for handling the oscillating kernels such as those arising in the solution of the convective helmholtz equation the proposed approach which has a complexity of o n l o g n is interesting for industrial applications especially in the aeroacoustic field

power law tail in the radial growth probability distribution for dla using both analytic and numerical methods we study the radial growth probability distribution p r m for large scale off lattice diffusion limited aggregation dla clusters if the form of p r m is a gaussian we show analytically that the width xi m of the distribution cannot scale as the radius of gyration rg of the cluster we generate about 1750 clusters of masses m up to 500000 particles and calculate the distribution by sending 106 further random walkers for each cluster we give strong support that the calculated distribution has a power law tail in the interior r approximately 0 of the cluster and can be described by a scaling ansatz p r m varies as ralpha xi g r r0 xi where g x denotes some scaling function which is centred around zero and has a width of order unity the exponent alpha is determined to be approximately 2 which is now substantially smaller than values measured earlier we show by including the power law tail that the width can scale as rg if alpha df 1

growth zone scaling properties and fjord structure of aggregates grown by particle cluster interaction abstract aggregation by particle cluster interaction of type r leads to tenuous growth products with continuously tunable fractal dimension by using 2 7 aggregation clusters are generated with scaling behavior mass scaling capacity dimension and fjord structure similar to the one observed in dla aggregates it is shown that a simple backward iteration scheme provides an accurate estimation of the interaction induced off lattice growth probabilities a phase transition in the multifractal measure is found

decomposition through formalization in a product space when an optimization problem is posed in a product space it is classical to decompose this problem the goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space by associating systematically to this problem an equivalent one posed in ann fold cartesian product space we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former applications to three rather classical mathematical programming problems are given

structural five fold symmetry in the fractal morphology of diffusion limited aggregates abstract the statistical self similarity of the geometry of diffusion limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform this study reveals the existence of a predominant structural five fold symmetry in the internal frozen region as well as in the active outer region of the interface this observation is corroborated by a statistical analysis of the screening effects that govern diffusion limited aggregation dla growth in linear and sector shaped cells the existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering we report on the discovery of fibonacci sequences in the inner extinct region of large mass off lattice dla clusters with a branching ratio which converges asymptotically to the golden mean we suggest an interpretation of the dla morphology as a quasifractal counterpart of the well ordered snowflake fractal architecture

acoustic isogeometric boundary element analysis an isogeometric boundary element method based on t splines is used to simulate acoustic phenomena we restrict our developments to low frequency problems to establish the fundamental properties of the proposed approach using t splines the computer aided design cad and boundary element analysis are integrated without recourse to geometry clean up or mesh generation a regularized burton miller formulation is used resulting in integrals which are at most weakly singular we employ a collocation based approach to generate the linear system of equations the method is verified against closed form solutions and direct comparisons are made with conventional lagrangian discretizations it is demonstrated that the superior accuracy of the isogeometric approach emanates from the exact geometric description encapsulated in the t spline the method is then applied to a real world application to illustrate the potential for integrated engineering design and analysis

aggregation by attractive particle cluster interaction a quasideterministic irreversible growth model is studied where aggregates are generated by attractive cluster particle forces proportional ro r alpha it is shown that fractal dimensions and growth site probability measures of the resulting fractal structures strongly depend on the parameter alpha comparison with experimental results indicates that this model represents a further step towards realistic simulations of growth processes

reply by the authors to c k w tam the prediction of noise generation and radiation by turbulence has been the subject of continuous research for over fifty years the essential problem is how to model the noise sources when one s knowledge of the detailed space time properties of the turbulence is limited we attempted to provide a comparison of models based on acoustic analogies and recent alternative models our goal was to demonstrate that the predictive capabilities of any model are based on the choice of the turbulence property that is modeled as a source of noise our general definition of an acoustic analogy is a rearrangement of the equations of motion into the form l u q where l is a linear operator that reduces to an acoustic propagation operator outside a region upsilon u is a variable that reduces to acoustic pressure or a related linear acoustic variable outside upsilon and q is a source term that can be meaningfully estimated without knowing u and tends to zero outside upsilon