moments of spectral functions monte carlo evaluation and verification the subject of the present study is the monte carlo path integral evaluation of the moments of spectral functions such moments can be computed by formal differentiation of certain estimating functionals that are infinitely differentiable against time whenever the potential function is arbitrarily smooth here i demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods e g exact differentiation of a chebyshev polynomial interpolant which utilize information from the entire interval ensuremath ensuremath beta ensuremath hbar 2 ensuremath beta ensuremath hbar 2 the algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated although the resulting approximation to the estimating functional is nonlinear the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane with the help of the cauchy integral formula e g by lyness method an interesting aspect of the present development is that hamburger s conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm finally the issue of appearance of the sign problem in the computation of moments albeit in a milder form than for other quantities is addressed

velocity induced numerical solutions of reaction diffusion systems on continuously growing domains reaction diffusion systems have been widely studied in developmental biology chemistry and more recently in financial mathematics most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions it therefore becomes convenient to look for numerical solutions finite difference finite element finite volume and spectral methods are typical examples of the numerical methods used most of these methods are locally based schemes we examine the implications of mesh structure on numerically computed solutions of a well studied reaction diffusion model system on two dimensional fixed and growing domains the incorporation of domain growth creates an additional parameter the grid point velocity and this greatly influences the selection of certain symmetric solutions for the adi finite difference scheme when a uniform square mesh structure is used domain growth coupled with grid point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure we compare our results to those obtained by use of finite elements on unstructured triangular elements

a general weighted median filter structure admitting negative weights weighted median smoothers which were introduced by edgemore in the context of least absolute regression over 100 years ago have received considerable attention in signal processing during the past two decades although weighted median smoothers offer advantages over traditional linear finite impulse response fir filters it is shown in this paper that they lack the flexibility to adequately address a number of signal processing problems in fact weighted median smoothers are analogous to normalized fir linear filters constrained to have only positive weights it is also shown that much like the mean is generalized to the rich class of linear fir filters the median can be generalized to a richer class of filters admitting positive and negative weights the generalization follows naturally and is surprisingly simple in order to analyze and design this class of filters a new threshold decomposition theory admitting real valued input signals is developed the new threshold decomposition framework is then used to develop fast adaptive algorithms to optimally design the real valued filter coefficients the new weighted median filter formulation leads to significantly more powerful estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior weighted median smoother structures

conformal maps to multiply slit domains and applications by exploiting conformal maps to vertically slit regions in the complex plane a recently developed rational spectral method t w tee and l n trefethen siam j sci comput 28 2006 pp 1798 1811 is able to solve pdes with interior layer like behavior using significantly fewer collocation points than traditional spectral methods the conformal maps are chosen to enlarge the region of analyticity in the solution an idea which can be extended to other numerical methods based upon global polynomial interpolation here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries and apply them to some challenging differential equations

robust frequency selective filtering using weighted myriad filters admitting real valued weights weighted myriad smoothers have been proposed as a class of nonlinear filters for robust non gaussian signal processing in impulsive noise environments however weighted myriad smoothers are severely limited since their weights are restricted to be non negative this constraint makes them unusable in bandpass or highpass filtering applications that require negative filter weights further they are incapable of amplifying selected frequency components of an input signal in this paper we generalize the weighted myriad smoother to a richer structure a weighted myriad filter admitting real valued weights this involves assigning a pair of filter weights one positive and the other negative to each of the input samples equivalently the filter can be described as a weighted myriad smoother applied to a transformed set of samples that includes the original input samples as well as their negatives the weighted myriad filter is analogous to a normalized linear fir filter with real valued weights whose absolute values sum to unity by suitably scaling the output of the weighted myriad filter we extend it to yield the so called scaled weighted myriad filter which includes but is more powerful than the traditional unconstrained linear fir filter finally we derive stochastic gradient based nonlinear adaptive algorithms for the optimization of these novel myriad filters under the mean square error criterion

the accuracy of the chebyshev differencing method for analytic functions the chebyshev spectral collocation method is one of the most powerful tools for numerical differentiation particularly when the function under consideration is smooth an upper bound on the error in the discrete maximum norm is derived here when the function is analytic two model functions are analyzed in detail

recovering exponential accuracy from collocation point values of smooth functions with end point singularities gibbs phenomenon is the particular manner how a global spectral approximation of a piecewise analytic function behaves at the jump discontinuity the truncated spectral series has large oscillations near the jump and the overshoot does not decay as the number of terms in the truncated series increases there is therefore no convergence in the maximum norm and convergence in smooth regions away from the discontinuity is also slow in gottlieb and shu 1995 5 a methodology is proposed to completely overcome this difficulty in the context of spectral collocation methods resulting in the recovery of exponential accuracy from collocation point values of a piecewise analytic function in this paper we extend this methodology to handle spectral collocation methods for functions which are analytic in the open interval but have singularities at end points with this extension we are able to obtain exponential accuracy from collocation point values of such functions similar to gottlieb and shu 1995 5 the proof is constructive and uses the gegenbauer polynomials c n l x the result implies that the gibbs phenomenon can be overcome for smooth functions with endpoint singularities

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

numerical approximation of a free boundary problem arising in electromagnetic shaping this paper deals with the numerical approximation of the section of a column of liquid metal shaped by an external electromagnetic field the bidimensional model used here leads up to solve by a spectral method a nonlinear boundary value problem posed on the exterior of the unit disk to find a convenient conformal mapping we give accurate bounds on the errors in the approximation process some examples of shapes for both exterior and interior shaping are presented

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

adaptive weighted myriad filter algorithms for robust signal processing in spl alpha stable noise environments stochastic gradient based adaptive algorithms are developed for the optimization of weighted myriad filters wmyfs wmyfs form a class of nonlinear filters motivated by the properties of spl alpha stable distributions that have been proposed for robust non gaussian signal processing in impulsive noise environments the weighted myriad for an n long data window is described by a set of nonnegative weights w sub i sub i l sup n and the so called linearity parameter k 0 in the limit as k spl rarr spl infin the filter reduces to the familiar weighted mean filter which is a constrained linear fir filter necessary conditions are obtained for optimality of the filter weights under the mean absolute error criterion an implicit formulation of the filter output is used to find an expression for the gradient of the cost function using instantaneous gradient estimates an adaptive steepest descent algorithm is then derived to optimize the weights this algorithm involves a very simple update term that is computationally comparable to the update in the classical lms algorithm the robust performance of this adaptive algorithm is demonstrated through a computer simulation example involving lowpass filtering of a one dimensional chirp type signal in impulsive noise

exponential convergence of the hp version for the boundary element method on open surfaces we analyze the boundary element galerkin method for weakly singular and hypersingular integral equations of the first kind on open surfaces we show that the hp version of the galerkin method with geometrically refined meshes converges exponentially fast for both integral equations the proof of this fast convergence is based on the special structure of the solutions of the integral equations which possess specific singularities at the corners and the edges of the surface we show that these singularities can be efficiently approximated by piecewise tensor products of splines of different degrees on geometrically graded meshes numerical experiments supporting these results are presented

convergence of spectral methods for burgers equation in this paper a general framework is presented for analyzing numerical methods for the evolutionary equations that admit semigroup formulations this framework is then applied to spectral and pseudospectral methods for the burgers equation using trigonometric chebyshev and legendre polynomials optimal order of convergence is obtained which implies the spectral accuracy of these methods

optimal weighted median filtering under structural constraints a new expression for the output moments of weighted median filtered data is derived the noise attenuation capability of a weighted median filter can now be assessed using the l vector and m vector parameters in the new expression the second major contribution of the paper is the development of a new optimality theory for weighted median filters this theory is based on the new expression for the output moments and combines the noise attenuation and some structural constraints on the filter s behavior in certain special cases the optimal weighted median filter can be obtained by merely solving a set of linear inequalities this leads in some cases to closed form solutions for optimal weighted median filters some applications of the theory developed in this paper in 1 d signal processing and image processing are discussed throughout the analysis some striking similarities are pointed out between linear fir filters and weighted median filters

fourier spectral method on ensemble architectures the direct simulation of three dimensional unsteady fluid flows is performed on some of the most powerful parallel computers presently available connection machine 2 intel i860 and the ncube 2 a fourier pseudospectral method and second order timestepping scheme is used implementation performance analysis and experimental timings are given results of fluid flow simulations on meshes up to 2563 show sustained rates of 520 mflops on the 64 k node cm 2 484 mflops on the 1 k node ncube 2 and 103 mflops on the 32 node intel i860

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

numerical proof of stability of roll waves in the small amplitude limit for inclined thin film flow abstract we present a rigorous numerical proof based on interval arithmetic computations categorizing the linearized and nonlinear stability of periodic viscous roll waves of the kdv ks equation modeling weakly unstable flow of a thin fluid film on an incline in the small amplitude kdv limit the argument proceeds by verification of a stability condition derived by bar nepomnyashchy and johnson noble rodrigues zumbrun involving inner products of various elliptic functions arising through the kdv equation one key point in the analysis is a bootstrap argument balancing the extremely poor sup norm bounds for these functions against the extremely good convergence properties for analytic interpolation in order to obtain a feasible computation time another is the way of handling analytic interpolation in several variables by a two step process carving up the parameter space into manageable pieces for rigorous evaluation these and other general aspects of the analysis should serve as blueprints for more general analyses of spectral stability

nonparametric estimation with recurrent event data the problem of nonparametric estimation for the distribution function governing the time to occurrence of a recurrent event in the presence of censoring is considered we derive nelson aalen and kaplan meier type estimators for the distribution function and establish their respective finite sample and asymptotic properties we allow for random observation periods for each subject under study and explicitly account for the informative sum quota nature of the data accrual scheme these allowances complicate technical matters considerably and in particular invalidate the direct use of martingale methods consistency and weak convergence of our estimators are obtained by extending an approach due to sellke who considered a single renewal process i e recurrent events on a single subject observed over an infinite time period a useful feature of the present analysis is that strong parallels are drawn to the usual single event setting providing a natural route toward developing extensions that involve

the stability of pseudospectral chebyshev methods abstract the stability of pseudospectral chebyshev methods is demonstrated for parabolic and hyperbolic problems with variable coefficients the choice of collocation points is discussed numerical examples are given for the case of variable coefficient hyperbolic equations author

analysis of spectral projectors in one dimensional domains in this paper we analyze a class of projection operators with values in a subspace of polynomials these projection operators are related to the hubert spaces involved in the numerical analysis of spectral methods they are in the first part of the paper the standard sobolev spaces and in the second part some weighted sobolev spaces the weight of which is related to the orthogonality relation satisfied by the chebyshev polynomials these results are used to study the approximation of a model fourth order problem

filon clenshaw curtis rules for a class of highly oscillatory integrals with logarithmic singularities in this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities for these quadrature rules we derive error estimates in terms of n the number of nodes k the rate of oscillations and a sobolev like regularity of the function we prove that the method is not only robust but the error even decreases for fixed n as k increases practical issues about the implementation of the rule are also covered in this paper by a writing down ready to implement algorithms b analysing the numerical stability of the computations and c estimating the overall computational cost we finish by showing some numerical experiments which illustrate the theoretical results presented in this paper

error bounds for approximate eigenvalues of periodic coefficient linear delay differential equations we describe a new chebyshev spectral collocation method for systems of variable coefficient linear delay differential equations with a single fixed delay computable uniform a posteriori bounds are given for this method when the coefficients are periodic the system has a unique compact nonnormal monodromy operator whose spectrum determines the stability of the system the spectral method approximates this operator by a dense matrix of modest size in cases where the coefficients are smooth we observe spectral convergence of the eigenvalues of that matrix to those of the operator our main result is a computable a posteriori bound on the eigenvalue approximation error in the case that the coefficients are analytic

parametric estimation for window censored recurrence data many applications in particular the failure repair and replacement of industrial components or physical infrastructure involve recurrent events frequently the available data are window censored only events that occurred during a particular interval are recorded window censoring presents a challenge for recurrence data analysis for statistical inference from window censored recurrence data we derive the likelihood function for a model in which the distributions of inter recurrence intervals in a single path need not be identical and may be associated with covariate information we assume independence among different sample paths we propose a distribution to model the effect of external interventions on recurrence processes this distribution can represent a phenomenon frequently observed in practice that the probability of process regeneration increases with the number of historical interventions for example an item that had a given number of repairs is generally more likely to be replaced i

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

regularity of the solutions for elliptic problems on nonsmooth domains in r3 part 2 regularity in neighborhoods of edges this paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in 3 the present paper concentrates on the regularity of solutions of the poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted sobolev spaces and countably normed spaces these results can be generalised to elliptic problems arising from mechanics and engineering for instance the elasticity problem on polyhedral domains hence the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems

bounded influence propagation tau estimation a new robust method for arma model estimation a new robust and statistically efficient estimator for arma models called the bounded influence propagation estimator is proposed the estimator incorporates an auxiliary model which prevents the propagation of outliers strong consistency and asymptotic normality of the estimator for arma models that are driven by independently and identically distributed iid innovations with symmetric distributions are established to analyze the infinitesimal effect of outliers on the estimator the influence function is derived and computed explicitly for an ar 1 model with additive outliers to obtain estimates for the ar p model a robust durbin levinson type and a forward backward algorithm are proposed an iterative algorithm to robustly obtain arma p q parameter estimates is also presented the problem of finding a robust initialization is addressed which for orders p q 2 is a nontrivial matter numerical experiments are conducted to compare the finite sample performance of the proposed estimator to existing robust methodologies for different types of outliers both in terms of average and of worst case performance as measured by the maximum bias curve to illustrate the practical applicability of the proposed estimator a real data example of outlier cleaning for r r interval plots derived from electrocardiographic data is considered the proposed estimator is not limited to biomedical applications but is also useful in any real world problem whose observations can be modeled as an arma process disturbed by outliers or impulsive noise

on the robustness of the class of stack filters a widely held view in the nonlinear signal processing community is that the class of stack filters is robust although this is supported by extensive experimental evidence no systematic theoretical justification exists despite the availability of analytical tools for studying robustness of individual stack filters we focus on rank selection probabilities rsps as measures of robustness as it is well known that other statistical characterizations of stack filters such as output distributions breakdown probabilities and output distributional influence functions can be represented in terms of rsps we show in a very general sense that the class of stack filters is highly robust it is also shown that almost all stack filters have very similar output distributions for independent and identically distributed i i d input signals and thus very similar statistical behavior

conservative numerical methods for solitary wave interactions the purpose of this paper is to show the advantages that represent the use of numerical methods that preserve invariant quantities in the study of solitary wave interactions for the regularized long wave equation it is shown that the so called conservative methods are more appropriate to study the phenomenon and provide a dynamic point of view that allows us to estimate the changes in the parameters of the solitary waves after the collision

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

stability analysis of finite difference pseudospectral and fourier galerkin approximations for time dependent problems we consider finite difference pseudospectral and fourier galerkin methods for the approximate solution of time dependent problems the paper provides a unified framework for the stability analysis of all three discrete methods in particular the problem of stability for highly accurate stencils is studied in some detail

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

boundary concentrated finite element methods a method with optimal up to logarithmic terms complexity for solving elliptic problems is proposed the method relies on interior regularity but the solution may have globally low regularity due to rough boundary data or geometries elliptic regularity results high order approximation results and an efficient preconditioner are presented the method is utilized to realize with linear logarithmic complexity an accurate and data sparse approximation to the associated elliptic poincare steklov operators further applications include the treatment of exterior boundary value problems and the solution of problems in the framework of domain decomposition methods

adaptive galerkin boundary element methods with panel clustering in this paper we will propose a boundary element method for solving classical boundary integral equations on complicated surfaces which possibly contain a large number of geometric details or even uncertainties in the given data the small size of such details is characterised by a small parameter varepsilon and the regularity of the solution is expected to be low in such zones on the surface which we call the wire basket zones we will propose the construction of an initial discretisation for such type of problems afterwards standard strategies for boundary element discretisations can be applied such as the h p and the adaptive hp version in a straightforward way for the classical boundary integral equations we will prove the optimal approximation results of our so called wire basket boundary element method and discuss the stability aspects then we construct the panel clustering and mathcal h matrix approximations to the corresponding galerkin bem stiffness matrix the method is shown to have an almost linear complexity with respect to the number of degrees of freedom located on the wire basket

an enhanced pseudospectral chebyshev method for dissipative partial differential equations we present an enhanced pseudospectral chebyshev method based on a postprocess previously introduced for the galerkin method de frutos et al applied mathematics and computation reports july 1996 when the regularity of the solution is limited the new method is shown to possess a superior rate of convergence and improved efficiency than the classical pseudospectral scheme in the particular case of smooth solutions the reduction of the error obtained by postprocessing is not so clear however the use of a filter in the numerical solution before applying the postprocess improves again the efficiency of the method the numerical analysis of the new method and some numerical experiments showing its superior efficiency are presented

optimality of the myriad filter in practical impulsive noise environments linear filtering theory has been largely motivated by the characteristics of gaussian signals in the same manner the proposed myriad filtering methods are motivated by the need for a flexible filter class with high statistical efficiency in non gaussian impulsive environments that can appear in practice we introduce several important properties of the myriad filter and prove its optimality in the family of spl alpha stable distributions

confidence limits for recurrence data applied to cost or number of product repairs this article presents a plot and new confidence limits for censored recurrence data they are applied here to data on product repairs they also apply to sociological demographic production business disease and other recurrence data for concreteness reliability terminology is used

analysis of the fully discrete fat boundary method the fat boundary method is a method of the fictitious domain class which was proposed to solve elliptic problems in complex geometries with non conforming meshes it has been designed to recover optimal convergence at any order despite of the non conformity of the mesh and without any change in the discrete laplace operator on the simple shape domain we propose here a detailed proof of this high order convergence and propose some numerical tests to illustrate the actual behaviour of the method

adaptive l_p norm spectrum sensing for cognitive radio networks in cognitive radio cr systems reliable spectrum sensing techniques are required in order to avoid interference to the primary users of the spectrum whereas most of the existing literature on spectrum sensing considers impairment by additive white gaussian noise awgn only in practice crs also have to cope with various types of non gaussian noise such as man made impulsive noise co channel interference and ultra wideband interference in this paper we propose an adaptive lp norm detector which does not require any a priori knowledge about the primary user signal and performs well for a wide range of circularly symmetric non gaussian noises with finite moments we analyze the probabilities of false alarm and missed detection of the proposed detector in rayleigh fading in the low signal to noise ratio regime and investigate its asymptotic performance if the number of samples available for spectrum sensing is large furthermore we consider the deflection coefficient for optimization of the lp norm parameters and discuss its connection to the probabilities of false alarm and missed detection based on the deflection coefficient an adaptive algorithm for online optimization of the lp norm parameters is developed analytical and simulation results show that the proposed lp norm detector yields significant performance gains compared to conventional energy detection in non gaussian noise and approaches the performance of the locally optimal detector which requires knowledge of the noise distribution

meridian filtering for robust signal processing a broad range of statistical processes is characterized by the generalized gaussian statistics for instance the gaussian and laplacian probability density functions are special cases of generalized gaussian statistics moreover the linear and median filtering structures are statistically related to the maximum likelihood estimates of location under gaussian and laplacian statistics respectively in this paper we investigate the well established statistical relationship between gaussian and cauchy distributions showing that the random variable formed as the ratio of two independent gaussian distributed random variables is cauchy distributed we also note that the cauchy distribution is a member of the generalized cauchy distribution family recently proposed myriad filtering is based on the maximum likelihood estimate of location under cauchy statistics an analogous relationship is formed here for the laplacian statistics as the ratio of laplacian statistics yields the distribution referred here to as the meridian interestingly the meridian distribution is also a member of the generalized cauchy family the maximum likelihood estimate under the obtained statistics is analyzed motivated by the maximum likelihood estimate under meridian statistics meridian filtering is proposed the analysis presented here indicates that the proposed filtering structure exhibits characteristics more robust than that of median and myriad filtering structures the statistical and deterministic properties essential to signal processing applications of the meridian filter are given the meridian filtering structure is extended to admit real valued weights utilizing the sign coupling approach finally simulations are performed to evaluate and compare the proposed meridian filtering structure performance to those of linear median and myriad filtering

harmonic density interpolation methods for high order evaluation of laplace layer potentials in 2d and 3d abstract we present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular laplace boundary integral operators and layer potentials in two and three spatial dimensions the method relies on the use of green s third identity and local taylor like interpolations of density functions in terms of harmonic polynomials the proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials and recasts the latter in terms of integrands that are bounded or even more regular depending on the order of the density interpolation the resulting boundary integrals can then be easily accurately and inexpensively evaluated by means of standard quadrature rules a variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule to integrate over smooth curves in two dimensions and with a chebyshev type quadrature rule to integrate over surfaces given as unions of non overlapping quadrilateral patches in three dimensions

multilevel methods for the h p and hp versions of the boundary element method in this paper we give an overview on the definition of finite element spaces for the h p and hp version of the bem along with preconditioners of additive schwarz type we consider screen problems with a hypersingular or a weakly singular integral equation of first kind on an open surface as model problems for the hypersingular integral equation and the h version with piecewise bilinear functions on a coarse and a fine grid we analyze a preconditioner by iterative substructuring based on a non overlapping decomposition of we prove that the condition number of the preconditioned linear system behaves polylogarithmically in h h here h is the size of the subdomains and h is the size of the elements for the hp version and the hypersingular integral equation we comment in detail on an additive schwarz preconditioner which uses piecewise polynomials of high degree on the fine grid and yields also a polylogarithmically growing condition number for the weakly singular integral equation where no continuity of test and trial functions across the element boundaries has to been enforced the method works for nonuniform degree distributions as well numerical results supporting our theory are reported

analysis of window observation recurrence data many systems experience recurrent events recurrence data are collected to analyze quantities of interest such as the mean cumulative number of events methods of analysis are available for recurrence data with left and or right censoring due to practical constraints however recurrence data are sometimes recorded only in windows between the windows are gaps over which the process cannot be observed this article extends existing statistical methods both nonparametric and parametric to window observation recurrence data the nonparametric estimator requires minimum assumptions but will be inconsistent if the size of the risk set is not positive over the entire period of interest no such difficulty arises when using a parametric model for the recurrence data for cases in which the size of the risk set is zero for some period of time we propose and compare two alternative hybrid estimators we illustrate the methods with two example applications

on exponential convergence of gegenbauer interpolation and spectral differentiation this paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the gegenbauer gauss and gegenbauer gauss lobatto points when the underlying function is analytic on and within an ellipse sharp error estimates in the maximum norm are derived

alpha stable signals and adaptive filtering let d sub k x sub k be jointly stationary symmetric spl alpha stable processes with index 1 spl alpha 2 we first consider the problem of optimal fir linear and regression filters we then consider the signed error and sign sign adaptive filtering algorithms for the estimation of the desired signal d sub j on the basis of the input vector x sub j various convergence results are established for the signal and tap weights estimation errors

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

on the gibbs phenomenon v recovering exponential accuracy from collocation point values of a piecewise analytic function this paper presents a method to recover exponential accuracy at all points including at the discontinuities themselves from the knowledge of an approximation to the interpolation polynomial or trigonometrical polynomial we show that if we are given the collocation point values or a highly accurate approximation at the gauss or gauss lobatto points we can reconstruct an uniform exponentially convergent approximation to the function f x in any sub interval of analyticity the proof covers the cases of fourier chebyshev legendre and more general gegenbauer collocation methods a numerical example is also provided

matched median filtering a class of median type filters called matched median filters is defined for the estimation of the information which is carried by the amplitude of a noisy signal these filters form the natural counterpart of linear matched filters in the class of median type filters and are maximum likelihood estimators if the noise is biexponential matched median filters are defined both for baseband and passband systems their statistical properties are analyzed and simulation results presented

recursive weighted myriad based filters and their optimizations this paper proposes two new recursive filtering structures based on the nonlinear myriad operator first we develop the general class of recursive weighted myriad rwmy filters as a robust filtering structure against impulsive noise that includes as particular cases a normalized version of the linear infinite impulse response iir filter the recursive mode type filter and the non recursive weighted myriad filter secondly considering the fact that the additive noise that contaminates the previous filter s outputs is no longer impulsive we introduce a novel class of recursive hybrid myriad rhmy filters whose structure gathers the advantages of both the weighted myriad and the weighted mean in a single cost function to be minimized some properties of the rhmy filter are derived and a fast algorithm to compute the rhmy filter output is proposed furthermore adaptive algorithms for designing the proposed recursive structures under the equation error formulation framework are developed finally extensive numerical simulations are shown to evaluate both the iterative update of the adaptive algorithms and the performance of the proposed recursive filters against impulsive noise

some simple robust methods for the analysis of recurrent events nelson discussed a method of estimating the cumulative mean function for identically distributed processes of recurrent events we show that a similar approach can be used with more general models including regression the key idea is to use point estimates based on poisson models and to develop robust variance estimates that are valid more generally the methods are illustrated on reliability and warranty data

optimal pivot selection in fast weighted median search weighted median filters are increasingly being used in signal processing applications and thus fast implementations are of importance this paper introduces a fast algorithm to compute the weighted median wm of n samples which has linear time and space complexity as opposed to o n log n which is the time complexity of traditional sorting algorithms a popular selection algorithm often used to find the wm in large data sets is quickselect whose performance is highly dependent on how the pivots are chosen we introduce an optimization based pivot selection strategy which results in significantly improved performance as well as a more consistent runtime compared to traditional approaches the selected pivots are order statistics of subsets in order to find the optimal order statistics as well as the optimal subset sizes a set of cost functions are derived which when minimized lead to optimal design parameters we compare the complexity to floyd and rivest s algorithm select which to date has been the fastest median finding algorithm and we show that the proposed algorithm compared with select requires close to 30 fewer comparisons it is also shown that the proposed selection algorithm is asymptotically optimal for large n

approximation of integral operators by variable order interpolation we employ a data sparse recursive matrix representation so called matrices for the efficient treatment of discretized integral operators we obtain this format using local tensor product interpolants of the kernel function and replacing high order approximations with piecewise lower order ones the scheme has optimal i e linear complexity in the memory requirement and time for the matrix vector multiplication we present an error analysis for integral operators of order zero in particular we show that the optimal convergence h is retained for the classical double layer potential discretized with piecewise constant functions

error estimates for spectral and pseudospectral approximations of hyperbolic equations the linear one dimensional advection equation with variable coefficients and nonperiodic boundary conditions is considered we investigate some spectral and pseudospectral collocation approximations of jacobi and legendre type in the latter case the boundary condition is implicitly treated stability and convergence results in weighted norms are given some precise estimates of the rate of convergence in terms of the regularity of the continuous solution are derived

polynomial weighted median filtering this paper extends weighted median wm filters to the class of polynomial weighted median pwm filters traditional polynomial filtering theory based on linear combinations of polynomial terms is able to approximate important classes of nonlinear systems the linear combination of polynomial terms however yields poor performance in environments characterized by heavy tailed distributions weighted median filters in contrast are well known for their outlier suppression and detail preservation properties the weighted median sample selection methodology is naturally extended to the polynomial sample case yielding a filter structure that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers moreover the pwm filter class is well motivated by an analysis of cross and square term statistics a presented probability density function analysis shows that these terms have heavier tails than the observed samples indicating that robust combination methods should be utilized to avoid undue influence of outliers further analysis shows weighted median processing of polynomial terms is justified from a maximum likelihood perspective the established pwm filter class is statistically analyzed through the determination of the filter output distribution and breakdown probability filter parameter optimization procedures are also presented finally the effectiveness of pwm filters is demonstrated through simulations that include temporal spectrum and bispectrum analysis

on the performance of single threshold detectors for binary communications in the presence of gaussian mixture noise in this paper probability of error performance of single threshold detectors is studied for binary communications systems in the presence of gaussian mixture noise first sufficient conditions are proposed to specify when the sign detector is not an optimal detector among all the single threshold detectors then a monotonicity property of the error probability is derived for the optimal single threshold detector in addition a theoretical limit is obtained on the maximum ratio between the average probabilities of error for the sign detector and the optimal single threshold detector finally numerical examples are presented to investigate the theoretical results

primal and dual interface concentrated iterative substructuring methods this paper is devoted to the fast solution of interface concentrated finite element equations the interface concentrated finite element schemes are constructed on the basis of a nonoverlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain similar to boundary element domain decomposition methods the total number of unknowns per subdomain behaves like o h h d 1 where h h and d denote the usual scaling parameter of the subdomains the average discretization parameter of the subdomain boundaries and the spatial dimension respectively we propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns in particular the so called all floating finite element tearing and interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

compressed absorbing boundary conditions via matrix probing absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the helmholtz equation in a heterogeneous medium it is always possible to reduce an absorbing layer to an operator at the boundary by layer stripping elimination of the exterior unknowns but the linear algebra involved is costly we propose bypassing the elimination procedure and directly fitting the surface to surface operator in compressed form from a few exterior helmholtz solves with random dirichlet data the result is a concise description of the absorbing boundary condition with a complexity that grows slowly often logarithmically in the frequency parameter

nonparametric analysis of recurrent events and death this article is concerned with the analysis of recurrent events in the presence of a terminal event such as death we consider the mean frequency function defined as the marginal mean of the cumulative number of recurrent events over time a simple nonparametric estimator for this quantity is presented it is shown that the estimator properly normalized converges weakly to a zero mean gaussian process with an easily estimable covariance function nonparametric statistics for comparing two mean frequency functions and for combining data on recurrent events and death are also developed the asymptotic null distributions of these statistics together with consistent variance estimators are derived the small sample properties of the proposed estimators and test statistics are examined through simulation studies an application to a cancer clinical trial is provided

superconvergence points of polynomial spectral interpolation in this work we study superconvergence properties for some high order orthogonal polynomial interpolations at the superconvergent points the error is smaller by a factor of n alpha with alpha 0 where n is the polynomial degree the results are two fold when interpolating function values we identify those points where the first and second derivatives of the interpolant converge faster when interpolating the first derivative we locate those points where the function value of the interpolant superconverges for the earlier case we use various chebyshev polynomials and for the later case we also include the counterpart legendre polynomials

second order heavy tailed distributions and tail analysis this correspondence studies the second order distributions of heavy tail distributed random variables rvs two models for the heavy tailed distributions are considered power law and epsi contaminated distributions special cases of the models considered include 1 rvs formed by the product of two independent but not necessarily identically distributed heavy tailed rvs x and y such that z xy and 2 rvs formed through squaring a heavy tail distributed rv x such that w x2 tail analysis of the rvs their cross terms and square values shows the ordering of their tail heaviness the following results hold strictly for power law distributions and under mild conditions for epsi contaminated distributions the tail of f z x is heavier than that of fx x and the tail of fw x is heavier than that of fz x where f x denotes the probability density distribution of the corresponding random variable the heaviness of the tails indicates that robust methods of sample combination and output determination should be utilized to avoid undue influence of outliers and degradation in performance as examples the denoising and frequency selective filtering problems under the derived cross and square statistics for hyperbolic tailed and epsi contaminated models are considered simulation results indicate that the weighted myriad wmy filter outperforms the weighted median wm filter and the wm filter outperforms the weighted sum finite impulse response fir filter the results may be exploited in higher order applications of heavy tailed distributions in networking such as data traffic modeling and in nonlinear signal and image processing such as polynomial and volterra filtering

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

on the direct fourier method for computer tomography considers the direct fourier methods dfm s for reconstructing an image from its given x ray projections the main purpose here is to use concepts from numerical analysis to estimate the errors in the methods the authors also suggest an alternative to the interpolations involved in the dfm s and estimate the number of terms involved realizing that one of the main reasons for the degradation of the quality of the reconstruction is the gibbs phenomenon the authors suggest a way to overcome this difficulty and give numerical evidence of its efficacy

efficient automatic quadrature in 3 d galerkin bem we present cubature methods approximating the surface integrals arising by galerkin discretization of boundary integral equations on surfaces in r3 this numerical integrator does not depend on the explicit form of the kernel function the trial and test space or the surface parametrization thus it is possible to generate the system matrix for a broad class of integral equations just by replacing the sub routine for evaluating the kernel function we will present formulae to determine the minimal order of the cubature methods for a required accuracy emphasis is laid on numerical experiments confirming the theoretical results

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions this paper provides a rigorous and delicate analysis for exponential decay of jacobi polynomial expansions of analytic functions associated with the bernstein ellipse using an argument that can recover the best estimate for the chebyshev expansion we derive various new and sharp bounds of the expansion coefficients which are featured with explicit dependence of all related parameters and valid for degree n ge 1 we demonstrate the sharpness of the estimates by comparing with existing ones in particular the very recent results in siam j numer anal 50 2012 pp 1240 1263 we also extend this argument to estimate the gegenbauer gauss quadrature remainder of analytic functions which leads to some new tight bounds for quadrature errors

an adaptive algorithm for spectral computations on unbounded domains abstract we consider spectral methods for solving differential equations on unbounded domains in particular we consider a spectral rational function method and a method based on domain truncation combined with fourier series both methods contain a free parameter which determines the accuracy of the spectral method when solving evolution equations the optimal parameter may change with time resulting in a significant loss of accuracy if the parameter is kept fixed we develop an automatic algorithm which adapts the parameter at regular time levels to maintain optimum accuracy in numerical tests we found that the proposed method is robust and efficient

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows