guiding functions for generalized periodic problems and applications abstract in this paper developing the method of guiding functions for differential inclusions with a generalized periodic condition we obtain the existence theorem for such problems it is shown how the abstract result can be applied to the study of differential games and anti periodic problems

liapunov schmidt reduction and continuation for nonlinear schro dinger equations we study the bifurcation scenario of nonlinear schro dinger equations nls the liapunov schmidt reduction is applied to show that the simple bifurcations of a single nls are pitchfork the pitchfork bifurcation can be subcritical or supercritical depending on the coefficient of the cubic term we choose we also describe numerical methods so that the liapunov schmidt reduction can effectively handle a corank 2 bifurcation point next we apply numerical continuation methods to trace solution curves and surfaces of the nls where the system is discretized by the centered difference approximations numerical results on two and three dimensional m coupled nls are reported where the physical properties such as the effect of trapping potentials isotropic and nonisotropic trapping potentials mass conservation constraints and strong and weak repulsive interactions are considered in our numerical experiments

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

the complex cepstrum of higher order cumulants and nonminimum phase system identification a computationally efficient identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase system the method is based on the idea of computing the complex cepstrum of higher order cumulants of the system output in particular the differential cepstrum parameters of the nonminimum phase impulse response are estimated directly from higher order cumulants by least squares solution or two dimensional fft operations the method reconstructs the minimum phase and maximum phase impulse response components separately it is flexible enough to be applied on autoregressive ar moving average ma or arma system without a priori knowledge of the type of the system benchmark simulation examples demonstrate the effectiveness of the method even with short length data records

approachability and fixed points for non convex set valued maps this paper studies the approachability of upper semicontinuous nonconvex set valued maps by single valued continuous functions and its applications to fixed point theory the use of approximate selections was initiated by j von neumann 41 in the proof of his well known minimax principle this remarkable property was extended to the normed space setting by cellina s and cellina and lasota 9 in the context of a degree theory for upper semicontinuous convex compact valued maps it also holds for maps with compact contractible values defined on finite polyhedra of iw ma cole11 40 and more generally defined on compact anrs mclennan 39 it was recently taken up by gbrniewicz granas and kryszewski 22 241 in the context of an index theory for non convex maps defined on compact anrs the content of this paper is divided into three parts the first part is devoted to background material concerning the types of spaces and set valued maps studied herein in the second part we define the abstract class d of approachable set valued maps that is the class of maps a x y

harmonic retrieval using higher order statistics a deterministic formulation given a single record the authors consider the problem of estimating the parameters of a harmonic signal buried in noise the observed data are modeled as a sinusoidal signal plus additive gaussian noise of unknown covariance the authors define novel higher order statistics referred to as mixed cumulants that can be consistently estimated using a single record and are insensitive to colored gaussian noise employing fourth order mixed cumulants they estimate the sinusoid parameters using a consistent nonlinear matching approach the algorithm requires an initial estimate that is obtained from a consistent linear estimator finally the authors examine the performance of the proposed method via simulations

parameter estimation of exponentially damped sinusoids using higher order statistics a new approach for the estimation of the parameters of exponentially damped sinusoids is introduced based on third or fourth order statistics of the observation signal the method may be seen as an extension of the minimum norm principal eigenvectors method to higher order statistics domains the strong points and limitations of the method are discussed as well as sufficient conditions for the existence of the solution the utilization of the method in the case of finite length signals in the presence of additive gaussian noise white or colored is addressed monte carlo simulations demonstrate the effectiveness of the new method when the additive noise is colored gaussian with unknown autocorrelation sequence for different signal to noise ratios and a single data record the case of an ensemble of data records is studied when the exponentially damped sinusoids are assumed to have random phase

a fast two class classifier for 2d data using complex moment preserving principle a new moment preserving classifier for two class clustering is suggested based on preserving the complex moments of two dimensional 2d input data an analytic non iterative and unsupervised classifier is proposed this new classifier is suitable for applications requiring fast automatic two class clustering of 2d data or fast automatic hierarchical clustering furthermore the computation time is of order of data size and hence much faster than the well known iterative k means algorithm experimental results show that the proposed classifier can acquire acceptable clustering results

an efficient algorithm for the schrodinger poisson eigenvalue problem we present a new implementation of the two grid method for computing extremum eigenpairs of self adjoint partial differential operators with periodic boundary conditions a novel two grid centered difference method is proposed for the numerical solutions of the nonlinear schrodinger poisson sp eigenvalue problem we solve the poisson equation to obtain the nonlinear potential for the nonlinear schrodinger eigenvalue problem and use the block lanczos method to compute the first k eigenpairs of the schrodinger eigenvalue problem until they converge on the coarse grid then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid the rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid our numerical results show how the first few eigenpairs of the schrodinger eigenvalue problem are affected by the dopant in the schrodinger poisson sp system moreover the convergence rate of eigenvalue computations on the fine grid is o h 3

asymptotic boundary value problems in banach spaces abstract a continuation principle is given for solving boundary value problems on arbitrary possibly infinite intervals to caratheodory differential inclusions in banach spaces for this aim the appropriate fixed point index is defined to condensing decomposable multivalued operators in frechet spaces this index extends and unifies the one for compact maps in andres et al trans amer math soc 351 1999 4861 4903 as well as the one for operators in banach spaces in bader ph d thesis university of munich 1995 as an application we prove the existence of an entirely bounded solution of a semilinear evolution inclusion

stationary solutions of quasi linear schrodinger poisson systems abstract we show that a high field version of the periodic schrodinger poisson system including nonlinear terms in the poisson equation corresponding to a field dependent dielectric constant and effective potentials in the schrodinger equation has an infinite number of different stationary states which correspond to solutions of a nonlinear schrodinger poisson eigenvalue problem

a spectral galerkin continuation method using chebyshev polynomials for the numerical solutions of the gross pitaevskii equation we study an efficient spectral galerkin continuation method sgcm and two grid centered difference approximations for the numerical solutions of the gross pitaevskii equation gpe where the second kind chebyshev polynomials are used as the basis functions for the trial function space some basic formulae for the sgcm are derived so that the eigenvalues of the associated linear eigenvalue problems can be easily computed the sgcm is implemented to investigate the ground and first excited state solutions of the gpe both the parabolic and quadruple well trapping potentials are considered we also study bose einstein condensates bec in optical lattices where the periodic potential described by the sine or cosine functions is imposed on the gpe of particular interest here is the investigation of symmetry breaking solutions sample numerical results are reported

estimation of close sinusoids in colored noise and model discrimination this paper considers the estimation of the two close dominant frequencies in the signal when it is known a priori that the observation is a sum of two close sinusoids and an additive colored noise whose spectral density is unknown earlier attempts have assumed the additive noise to be independent next we develop decision rules for checking whether the observed signal has only one sinusoid or two close sinusoids all of the earlier studies assumed that there are two close sinusoids in the signal another model discrimination problem considered is the determination of the causal structure of the observed periodicity a rule is given to test whether the observation comes from a sinusoid plus additive noise possible colored or the observation comes from a stationary autoregressive model we also present a numerical study showing the efficacy of the robust estimation procedure for estimating the two frequencies and the decision rules for checking the number of dominant frequencies and the causal mechanism finally we compare the proposed method to well known methods used for two close sinusoids in white noise

multiple bounded solutions of differential inclusions the nielsen theory approach abstract the generalized nielsen number is defined for self maps which are composed by operators with r values on compact connected anrs then it is applied to caratheodory differential inclusions with constraints for obtaining the multiplicity criteria more precisely such problems are transformed to those for the lower estimate of fixed points of the related operators with the given properties on bounded compact connected neighbourhood retracts of frechet spaces in this way multiple solutions can be proved e g for the multivalued initial value problem on the half line or in the single valued case for boundary value problems to ordinary differential equations

application of the mesh independence principle to mesh refinement the mesh independence principle mip for a newton iteration applied to an operator equation and its discretization for different stepsizes h states for a fixed starting point the number of iterations of newton s method to obtain a fixed tolerance is eventually independent of h this mip is used to formulate an efficient strategy for the solution of nonlinear equations amounting to the work equivalent of two or three newton iterates for the smallest h examples are given

a conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields an iterative method based on the conjugate gradient cg algorithm is developed for the efficient treatment of equations involving multiple excitations examples show that significant time savings can be obtained as compared to treating each excitation individually with the conjugate gradient algorithm however these savings are not obtained without the drawback of increased memory requirements to store the additional excitations residuals and solutions the efficiency of this algorithm tends to increase as additional excitations are added

enhancement of sinusoids in colored noise and the whitening performance of exact least squares predictors the extraction of sinusoids in white noise using least squares predictors has attracted a lot of attention in the past mainly in the context of the adaptive line enhancer ale however very few results exist for the practical colored noise case or for the whitening performance of the predictors we use a matrix formulation to derive the optimal least squares coefficients and frequency response of the d step predictor or ale for sinusoids real or complex in additive colored noise several cases are considered and in particular new formulas for the amplitude gain are obtained in low pass background noise the amplitude gain of the sinusoids becomes essentially a monotonically increasing function of their frequency and a decreasing function for high pass noise for the whitening application signal to noise ratio snr bounds of the output are derived when the input is a white signal plus a sinusoidal interference we also give a state space model and stochastic interpretations of our analysis of the d step predictor providing connections to other related areas to enable filtering of nonstationary complex inputs as well as multichannel and multiexperiment data a complex vector version of the ladder algorithm is presented that can be used to implement the ale noise cancelling and noise inversion for narrow band interference rejection

continuity compactness and degree theory for operators in systems involving p laplacians and inclusions the study of weak solutions for systems of nonlinear partial differential equations of elliptic type with inclusions leads to a multivalued operator of superposition type in sobolev spaces we show that under natural assumptions this operator has the properties which allow to apply degree theory fixed point index for multivalued maps more precisely this operator is upper semicontinuous and compact with nonempty convex compact values for the particular case of systems involving p laplacians we show that there is a homeomorphism transforming the whole system to a situation for which a fixed point index is available

on controllability of duffing equation in this paper by using the topological degree theory for multivalued maps and the method of a priori bounds we study the controllability problem for a system governed by the duffing equation an example is given

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

new results on employing cumulants for retrieving sinusoids in colored non gaussian noise a novel analysis is presented which shows that signals consisting of sinusoids plus non gaussian noise have a new signal to noise ratio nsnr associated with their fourth order cumulants this nsnr is a multiple of the original snr when the noise is colored and the original snr is equal to or greater than one therefore employing cumulants for harmonic retrieval problems is now an approach capable of handling not only gaussian noise but also colored non gaussian noise the presented analysis is verified via computer simulations

minres and minerr are better than symmlq in eigenpair computations the symmlq and minres methods for solving sparse indefinite symmetric systems of equations siam j numer anal 12 1975 pp 617 629 can be used inside the inverse or rayleigh quotient iterations for computing the eigenpairs of large sparse generalized eigenproblems a y lambda b y it is shown in this paper that the common opinion concerning the superiority of symmlq over minres is not true in this context several results taken from quantum chemistry acoustics and structural mechanics are presented that show the superiority of minres and the new minerr method over symmlq especially for ill conditioned eigenproblems where symmlq may even diverge theoretical explanations of the observed phenomena are given the use of minres or minerr instead of symmlq may thus lead to a significant increase in efficiency in large scale eigenvalue computations

a two level stochastic collocation method for semilinear elliptic equations with random coefficients in this work we propose a novel two level discretization for solving semilinear elliptic equations with random coefficients motivated by the two grid method for deterministic partial differential equations pdes introduced by xu 1994 our two level stochastic collocation method utilizes a two grid finite element discretization in the physical space and a two level collocation method in the random domain in particular we solve semilinear equations on a coarse mesh t h with a low level stochastic collocation corresponding to the polynomial space p p and solve linearized equations on a fine mesh t h using high level stochastic collocation corresponding to the polynomial space p p we prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with t h and p p the two level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients numerical experiments are provided to verify the theoretical results

on the zeros of the linear prediction error filter for deterministic signals the zeros of a linear prediction error filter polynomial for a class of deterministic signals that are a sum of samples of m exponentially damped undamped sinusoids is studied it is assumed that n samples are available for processing and that they are uncorrupted by noise it is shown that the exponent parameters of the m signals can be determined from m zeros called signal zeros of an lth degree prediction error filter polynomial l m if l lies in between m and n m n m 2 in a special case the rest of the l m zeros of the filter polynomial switch are called extraneous zeros are shown to be approximately uniformly distributed with in the unit circle regardless of the type of exponentials in the signal if the prediction filter coefficients are chosen to have minimum euclidean length the results obtained provide insight into the estimation problem with noisy data

estimating the parameters of exponentially damped sinusoids and pole zero modeling in noise we have presented techniques 1 6 based on linear prediction lp and singular value decomposition svd for accurate estimation of closely spaced frequencies of sinusoidal signals in noise in this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise the estimation procedure presented here makes use of backward prediction in addition to svd first the method is applied to data consisting of one and two exponentially damped sinusoids the choice of one and two signal components facilitates the comparison of estimation error in pole damping factors and pole frequencies to the appropriate cramer rao cr bounds and to traditional methods of linear prediction second our method is applied to an example due to steiglitz 8 in which the data consists of noisy values of the impulse response samples composed of many exponentially damped sinusoids of a linear system having both poles and zeros the poles of the system are accurately determined by our method and the zeros are obtained subsequently using shanks method

a novel block truncation coding of color images using a quaternion moment preserving principle block truncation coding btc is an efficient tool for image compression to compress color pixel blocks a novel color btc algorithm called quaternion moment block truncation coding qmbtc is presented analytical formulas for qmbtc whose computation time is on the order of pixel block size are derived by using quaternion arithmetic and the moment preserving principle the proposed color btc algorithm can adaptively truncate a pixel block into one or two output classes according to the distribution of the color values inside the blocks the experimental results show that the compression ratio is increased as compared with existing color btc algorithms and the picture quality of the reconstructed images is satisfactory in addition a post btc data compression scheme is proposed to further compress the subimage constructed by reproduction colors of truncated pixel blocks using a lookup table to display decoded data this postprocessing scheme can output images acceptable to human eyes

global bifurcation of the p laplacian and related operators abstract we prove some global bifurcation theorems of rabinowitz type for a large class of operators the results apply in particular to equations involving the p laplacian

an algorithm for pole zero modeling and spectral analysis an explicit connection between fitting exponential models and pole zero models to observed data is made the fitting problem is formulated as a constrained nonlinear minimization problem this problem is then solved using a simplified iterative algorithm the algorithm is applied to simulated data and the performance of the algorithm is compared to previous results

acceleration of a two grid method for eigenvalue problems this paper provides a new two grid discretization method for solving partial differential equation or integral equation eigenvalue problems in 2001 xu and zhou introduced a scheme that reduces the solution of an eigenvalue problem on a finite element grid to that of one single linear problem on the same grid together with a similar eigenvalue problem on a much coarser grid by solving a slightly different linear problem on the fine grid the new algorithm in this paper significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate numerical examples are also provided to demonstrate the efficiency of the new method

optimizing schrodinger functionals using sobolev gradients applications to quantum mechanics and nonlinear optics in this paper we study the application of the sobolev gradients technique to the problem of minimizing several schrodinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics we show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete fourier basis and a fast fourier transform we show that the sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators

periodic solutions to impulsive differential inclusions with constraints abstract the existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non convex set of state constraints is established by the use a lefschetz type fixed point theorem for set valued maps

multigrid bifurcation iteration we analyze a multigrid technique for the numerical computation of bifurcating solutions of large sparse systems of nonlinear parameter dependent equations which arise from discretizations of nonlinear elliptic eigenvalue problems by finite differences or finite elements the algorithm consists of a nested iteration procedure which employs a multigrid method for singular problems the underlying iteration scheme related to the lyapunov schmidt method has been widely used for proving the existence of bifurcating branches and also sometimes for numerical calculations our multi grid technique is shown to be a very efficient and theoretically sound tool for computing bifurcating branches of solutions of nonlinear elliptic eigenvalue problems on arbitrarily bounded domains in mathbb r 2 several numerical examples confirm this

spectral analysis based on the canonical autoregressive decomposition statistical inference for mixed spectral problems based on a parametric time series model is studied the model used is based on the canonical autoregressive decomposition card and represents the underlying random process as the sum of an autoregressive process and sinusoids maximum likelihood estimation of the unknown parameters in the model is considered the entire estimation problem can be shown to require a numerical maximization with respect to only the sinusoidal frequencies an iterative algorithm to efficiently implement this maximization is presented this enables us to examine a host of issues associated with a practical implementation of inferential procedures for mixed spectral problems some of the topics are accuracy of parameter estimates selection of model orders and sensitivity and robustness of the spectral estimates to modeling inaccuracies the modeling approach together with the inferential procedures overcome many of the difficulties encountered in current spectral estimation techniques

adaptive multigrid techniques for large scale eigenvalue problems solutions of the schrodinger problem in two and three dimensions multigrid mg algorithms for large scale eigenvalue problems ep obtained from discretizations of partial differential ep have often been shown to be more efficient than single level eigenvalue algorithms this paper describes a robust and efficient adaptive mg eigenvalue algorithm the robustness of the present approach is a result of a combination of mg techniques introduced here i e the completion of clusters the adaptive treatment of clusters the simultaneous treatment of solutions in each cluster the miltigrid projection mgp coupled with backrotations and robustness tests due to the mgp the algorithm achieves a better computational complexity and better convergence rates than previous mg eigenvalue algorithms that use only fine level projections these techniques overcome major computational difficulties related to equal and closely clustered eigenvalues some of these difficulties were not treated in previous mg algorithms computational examples for the schr odinger eigenvalue problem in two and three dimensions are demonstrated for cases of special computational difficulties which are due to equal and closely clustered eigenvalues for these cases the algorithm requires o qn operations for the calculation of q eigenvectors of size n using a second order approximation the total computational cost is equivalent to only a few gause seidel relaxations per eigenvector

investigations on two kinds of two grid mixed finite element methods for the elliptic eigenvalue problem two kinds of two grid new mixed finite element schemes for the elliptic eigenvalue problem based on less regularity of flux are considered in the new mixed variational formulation the flux belongs to the square integrable space instead of the classical h div and the choices of finite element pairs become simple and easy the two grid methods consist of solving a small elliptic eigenvalue problem on the coarse mesh and then solving a linear algebraic system on the fine mesh which can still maintain asymptotically optimal accuracy finally the performance of two kinds of two grid schemes are compared in efficiency and precision aspects by a series of numerical experiments numerical results show that the accelerated two grid method is a viable choice for the lowest order approximations of the elliptic eigenvalue problem

computing ground states of spin 1 bose einstein condensates by the normalized gradient flow in this paper we propose an efficient and accurate numerical method for computing the ground state of spin 1 bose einstein condensates becs by using the normalized gradient flow or imaginary time method the key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization this allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of a single component bec to compute the ground state of spin 1 becs an efficient and accurate discretization scheme the backward forward euler sine pseudospectral method is proposed to discretize the normalized gradient flow extensive numerical results on ground states of spin 1 becs with ferromagnetic antiferromagnetic interaction and harmonic optical lattice potential in one three dimensions are reported to demonstrate the efficiency of our new numerical method

statistical analysis of effective singular values in matrix rank determination a major problem in using svd singular value decomposition as a tool in determining the effective rank of a perturbed matrix is that of distinguishing between significantly small and significantly large singular values to the end conference regions are derived for the perturbed singular values of matrices with noisy observation data the analysis is based on the theories of perturbations of singular values and statistical significance test threshold bounds for perturbation due to finite precision and i i d random models are evaluated in random models the threshold bounds depend on the dimension of the matrix the noisy variance and predefined statistical level of significance results applied to the problem of determining the effective order of a linear autoregressive system from the approximate rank of a sample autocorrelation matrix are considered various numerical examples illustrating the usefulness of these bounds and comparisons to other previously known approaches are given

geometric methods for determining system poles from transient response the problem treated is that of identifying the poles of a finite order system by observing its transient decay after cessation of input for a limited time using possibly multiple observation points and experimental repetition various approaches are studied having the common characteristic that a homogeneous matrix equation must be solved several techniques that have been given scant attention in the literature are consolidated by a geometric treatment together with new results including an analytical treatment of the consequences of assuming an excessively high system order derivation of a statistically unbiased estimate for an intermediate parameter in the solution new theorems on error effects a recipe for effective use of the singular value decomposition and a new method for suppression of extraneous poles

parameter estimation of exponentially damped sinusoids using a higher order correlation based approach a very common problem in signal processing is parameter estimation of exponentially damped sinusoids from a finite subset of noisy observations when the signal is contaminated with colored noise of unknown power spectral density a cumulant based approach provides an appropriate solution to this problem we propose a new class of estimator namely a covariance type estimator which reduces the deterministic errors associated with imperfect estimation of higher order correlations from finite data length this estimator allows a higher order correlation sequence to be modeled as a damped exponential model in certain slices of the moments plane this result shows a useful link with well known linear prediction based methods such as the minimum norm principal eigenvector method of kumaresan and tufts 1982 which can be subsequently applied to extracting frequencies and damping coefficients from the 1 d correlation sequence this paper discusses the slices allowed in the moments plane the uses and limitations of this estimator using multiple realizations and a single record in a noisy environment monte carlo simulations applied to standard examples are also performed and the results are compared with the kt method and the standard biased estimator based approach the comparison shows the effectiveness of the proposed estimator in terms of bias and mean square error when the signals are contaminated with additive gaussian noise and a single data record with short data length is available

a two grid method of the non conforming crouzeix raviart element for the steklov eigenvalue problem abstract this paper discusses a high efficient scheme for the steklov eigenvalue problem a two grid discretization scheme of nonconforming crouzeix raviart element is established with this scheme the solution of a steklov eigenvalue problem on a fine grid h is reduced to the solution of the eigenvalue problem on a much coarser grid h and the solution of a linear algebraic system on the fine grid h by using spectral approximation theory and nitsche lascaux lesaint technique in space h 1 2 we prove that the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking h h and the numerical experiments indicate that when the eigenvalues k h of nonconforming crouzeix raviart element approximate the exact eigenvalues from below the approximate eigenvalues k h obtained by the two grid discretization scheme also approximate the exact ones from below and the accuracy of k h is higher than that of k h

statistical characterization of sample fourth order cumulants of a noisy complex sinusoidal process the paper deals with the statistical characterization of sample estimates of the fourth order cumulants of a random process consisting of multiple complex sinusoids and additive colored gaussian noise in particular it presents necessary and sufficient conditions for strong consistency of the sample cumulants of arbitrary orders and derives expressions for the asymptotic covariance of the sample estimates of the fourth order cumulants it is shown that the fourth order cumulant c sub 4y spl tau sub 1 spl tau sub 4 can be written as a function of a single argument spl tau spl tau sub 3 spl tau sub 4 spl tau sub 1 spl tau sub 2 which implies large flexibility in estimating the cumulant it is recommended that the estimate be based upon lags such that spl tau sub 1 is distant from spl tau sub 2 and spl tau sub 3 is distant from spl tau sub 4 and or as a linear combination of such terms the asymptotic variance of a cumulant based frequency estimator is shown to have the form c sub 2 spl middot snr sup 2 c sub 3 spl middot snr sup 3 c sub 4 spl middot snr sup 4 where the coefficient c sub 2 may possibly vanish the theory is illustrated via numerical examples the results of this paper will be useful in analyzing the performance of various cumulant based frequency estimation algorithms

color image processing by using binary quaternion moment preserving thresholding technique this paper presents a new moment preserving thresholding technique called the binary quaternion moment preserving bqmp thresholding for color image data based on representing color data by the quaternions the statistical parameters of color data can be expressed through the definition of quaternion moments analytical formulas of the bqmp thresholding can thus be determined by using the algebra of the quaternions the computation time for the bqmp thresholding is of order of the data size by using the bqmp thresholding quaternion moment based operators are designed for the application of color image processing such as color image compression multiclass clustering of color data and subpixel color edge detection the experimental results show that the proposed operator for color image compression can have output picture quality acceptable to human eyes in addition the proposed edge operator can detect the color edge at the subpixel level therefore the proposed bqmp thresholding can be used as a tool for color image processing

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

a two grid mixed finite element method for a nonlinear fourth order reaction diffusion problem with time fractional derivative in this article we develop a two grid algorithm based on the mixed finite element mfe method for a nonlinear fourth order reaction diffusion equation with the time fractional derivative of caputo type we formulate the problem as a nonlinear fully discrete mfe system where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the mfe method to solve the nonlinear mfe system more efficiently we propose a two grid algorithm which is composed of two steps we first solve a nonlinear mfe system on a coarse grid by nonlinear iterations then solve the linearized mfe system on the fine grid by newton iteration numerical stability and optimal error estimate o k 2 h r 1 h 2 r 2 in l 2 norm are proved for our two grid scheme where k h and h are the time step size coarse grid mesh size and fine grid mesh size respectively we implement the two grid algorithm and present the numerical results justifying our theoretical error estimate the numerical tests also show that the two grid method is much more efficient than solving the nonlinear mfe system directly

nonlinear filtering methods for harmonic retrieval and model order selection in gaussian and non gaussian noise this paper addresses the problem of high resolution parameter estimation harmonic retrieval and model order selection for superimposed sinusoids the harmonic retrieval problem is analyzed using a nonlinear parameter estimation approach estimation is performed using several nonlinear estimators with signals embedded in white and colored gaussian noise simulation results demonstrate that the nonlinear filters perform close to the cramer rao bound model order selection is performed in gaussian and non gaussian noise the problem is formulated using a multiple hypothesis testing approach with assumed known a priori probabilities for each hypothesis parameter estimation is performed using the extended kalman filter when the noise is gaussian the extended high order filter ehof is implemented in non gaussian noise simulation results demonstrate excellent performance in selecting the correct model order and estimating the signal parameters

a two grid discretization scheme for eigenvalue problems a two grid discretization scheme is proposed for solving eigenvalue problems including both partial differential equations and integral equations with this new scheme the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy

parameter estimation of exponentially damped sinusoids using higher order statistics and matrix pencil it is demonstrated that the higher order statistics based algorithm proposed by c k papadopoulos and c l nikias see ieee trans acoust speech signal process vol 38 no 5 p 814 24 1990 can be improved by using the matrix pencil approach of y hua and t k sarkar see ieee trans acoust speech signal process vol 38 no 8 p 1424 36 1990 the matrix pencil algorithm is computationally more efficient than the papadopoulos nikias algorithm since the computation of the k roots of the k degree polynomial is not needed in the matrix pencil algorithm furthermore it has been shown that the matrix pencil algorithm is less sensitive to noise than the kumaresan tufts method

wigner higher order moment spectra definition properties computation and application to transient signal analysis the wigner higher order moment spectra whos are defined as extensions of the wigner ville distribution wd to higher order moment spectra domains a general class of time frequency higher order moment spectra is also defined in terms of arbitrary higher order moments of the signal as gen eralizations of the cohen s general class of time frequency rep resentations the properties of the general class of time fre quency higher order moment spectra can be related to the properties of whos which are in fact extensions of the prop erties of the wd discrete time and frequency wigner higher order moment spectra dtf whos distributions are intro duced for signal processing applications and are shown to be implemented with two fft based algorithms one application is presented where the wigner bispectrum wb which is a whos in the third order moment domain is utilized for the detection of transient signals embedded in noise the wb is compared with the wd in terms of simulation examples and analysis of real sonar data it is shown that better detection schemes can be derived in low signal to noise ratio when the wb is applied

arma bispectrum approach to nonminimum phase system identification an identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase fir finite impulse response system the method is based on parametric modeling of the third moments of the output sequence and uses causal and anticausal autoregressive moving average arma models the magnitude and phase response of the system are expressed in terms of the ar parameters of the arma models in particular the ar part of the causal arma model captures the minimum phase component of the system and the ar part of the anticausal arma captures the maximum phase component both sets of parameters are obtained by solving overdetermined linear systems of equations a model order selection criterion based on third order moments is proposed the arma bispectrum approach is compared to more conventional approaches for magnitude and phase reconstruction it is demonstrated that the proposed identification procedure exhibits improved modeling performance the method does not require knowledge of the non gaussian noise distribution

efficient numerical methods for computing ground states of spin 1 bose einstein condensates based on their characterizations in this paper we propose efficient numerical methods for computing ground states of spin 1 bose einstein condensates becs with without the ioffe pritchard magnetic field b x when b x 0 a numerical method is introduced to compute the ground states and it is also applied to study properties of ground states numerical results suggest that the densities of m f 1 components in ground states are identical for any nonzero b x in particular if b x b 0 is a constant the ground states satisfy the single mode approximation when b x 0 efficient and simpler numerical methods are presented to solve the ground states of spin 1 becs based on their ferromagnetic antiferromagnetic characterizations numerical simulations show that our methods are more efficient than those in the literature in addition some conjectures are made from our numerical observations

a two grid discretization scheme for semilinear elliptic eigenvalue problems we present a two grid finite element discretization scheme with a two loop continuation algorithm for tracing solution branches of semilinear elliptic eigenvalue problems first we use the predictor corrector continuation method to compute an approximating point for the solution curve on the coarse grid then we use this approximating point as a predicted point for the solution curve on the fine grid in the corrector step we solve the first and the second order approximations of the nonlinear pde to obtain corrections for the state variable on the fine grid and the coarse grid respectively the continuation parameter is updated by computing the rayleigh quotient on the fine space to guarantee the approximating point we just obtained lies on the solution curve we perform newton s method we repeat the process described above until the solution curve on the fine space is obtained we show how the singular points such as folds and bifurcation points can be well approximated comprehensive numerical experiments show that the two grid finite element discretization scheme with a two loop continuation algorithm is efficient and robust for solving second order semilinear elliptic eigenvalue problems

the cross bicepstrum definition properties and application for simultaneous reconstruction of three nonminimum phase signals the recovery of three signals from their cross bispectrum or the identification of the impulse responses of three parallel linear time invariant lti systems from the cross bispectrum of their system outputs by computing the complex cepstrum of the cross bispectrum as long as the signals or impulse responses have no zeros on the unit circle is discussed it is shown that the three signals can be separated completely and approximately recovered in the cross bicepstrum domain except for their magnitude and linear phase factors the computation of the cross bicepstrum can be seen as a method for the simultaneous computation of the ordinary complex cepstra of three nonminimum phase signals without the need for phase unwrapping both least squares and fast fourier transform fft based methods for computing the bicepstral coefficients are presented simulation examples of signal reconstruction in gaussian white and nongaussian colored noise and of system identification are included the results are extended to nth order cross spectra and the factorization problem for these spectra is discussed

two grid discretizations and a local finite element scheme for a non selfadjoint stekloff eigenvalue problem abstract in this paper for a new stekloff eigenvalue problem which is non selfadjoint and not h 1 elliptic we establish and analyze two kinds of two grid discretization scheme and a local finite element scheme we present the error estimates of approximations of two grid discretizations we also prove a local error estimate which is suitable for the case that the local refined region contains singular points lying on the boundary of domain numerical experiments are reported finally to show the efficiency of our schemes

superconvergence of high order fems for eigenvalue problems with periodic boundary conditions we study adini s elements for nonlinear schrodinger equations nls defined in a square box with periodic boundary conditions first we transform the time dependent nls to a time independent stationary state equation which is a nonlinear eigenvalue problem nep a predictor corrector continuation method is exploited to trace solution curves of the nep we are concerned with energy levels and superfluid densities of the nls we analyze superconvergence of the adini elements for the linear schrodinger equation defined in the unit square the optimal convergence rate o h 6 is obtained for quasiuniform ele

application of a coincidence index to some classes of impulsive control systems we suggest the oriented coincidence index theory for pairs consisting of nonlinear zero index fredholm operators and multivalued maps which may be represented as compositions of multimaps with aspheric values we consider sequentially finite dimensional compact and condensing cases the theory developed is applied to the study of a feedback impulsive control system

a spectral galerkin continuation method for numerical solutions of the gross pitaevskii equation abstract we study spectral galerkin methods sgm for nonlinear eigenvalue problems where the legendre polynomials are used as the basis functions for the trial function space the sgm is applied to find the ground state solution of the gross pitaevskii equation gpe and the gpe in a periodic potential when the sgm is incorporated in the context of continuation methods for curve tracking a manifest advantage is that the target points are very close to the bifurcation points thus it takes less continuation steps to reach the target point when compared with the centered difference methods cdm or finite element methods fem we also implement the spectral collocation method scm so that the computational cost for the numerical integration in the sgm can be saved comprehensive numerical experiments on the gpe using various numerical methods described in this paper are reported

superconvergence two grid scheme based on shifted inverse power method for eigenvalue problems by function value recovery abstract in the paper an improved two grid scheme based on shifted inverse power method is proposed to solve the elliptic eigenvalue problems with this new scheme the solution of the elliptic eigenvalue problem on a fine grid t h is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid t h and the solution of an elliptic boundary value problem and the recovered solution on the fine grid t h theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate finally some numerical experiments are carried out to confirm the theoretical analysis

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

on parameter estimates of the lossy wave equation the cramer rao bound crb and maximum likelihood estimates in gaussian noise for the parameters of a sampled signal that satisfies the lossy wave equation such as the voltage sampled along a transmission line are derived results for both equally and unequally spaced samples are given it is shown that for certain values of the wave number frequency the estimates are sensitive to the amount of the ldquoreflected waverdquo in the signal while for other values of the wave number the estimates are insensitive it is also shown that for certain special cases it is possible to improve on standard frequency estimation up to 3 6 db by reflecting some of the noise free signal energy into a backward propagating wave and applying the maximum likelihood estimator mle to the new measured signal

spaces and maps approximation and fixed points abstract a general approximation property for topological spaces is studied in relation with fixed point theory for set valued maps a particular instance of this property is the admissibility in the sense of klee examples of convex sets of topological spaces equipped with a local topological convexity structure as well as general classes of approximative neighborhood retracts are shown to have this approximation property a general topological principle on the preservation of the fixed point property under this space approximation is proved it allows the passage from basic classes of spaces to more elaborate ones for general classes of nonconvex set valued maps

simultaneous multigrid techniques for nonlinear eigenvalue problems solutions of the nonlinear schrodinger poisson eigenvalue problem in two and three dimensions algorithms for nonlinear eigenvalue problems ep s often require solving self consistently a large number of ep s convergence difficulties may occur if the solution is not sought in an appropriate region if global constraints have to be satisfied or if close or equal eigenvalues are present multigrid mg algorithms for nonlinear problems and for ep s obtained from discretizations of partial differential ep have often been shown to be more efficient than single level algorithms this paper presents mg techniques and a mg algorithm for nonlinear schr odinger poisson ep s the algorithm overcomes the above mentioned difficulties combining the following techniques a mg simultaneous treatment of the eigenvectors and nonlinearity and with the global constrains mg stable subspace continuation techniques for the treatment of nonlinearity and a mg projection coupled with backrotations for separation of solutions these techniques keep the solutions in an appropriate region where the algorithm converges fast and reduce the large number of self consistent iterations to only a few or one mg simultaneous iteration the mg projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues computational examples for the nonlinear schr odinger poisson ep in two and three dimensions presenting special computational difficulties that are due to the nonlinearity and to the equal and closely clustered eigenvalues are demonstrated for these cases the algorithm requires o qn operations for the calculation of q eigenvectors of size n and for the corresponding eigenvalues one mg simultaneous cycle per fine level was performed the total computational cost is equivalent to only a few gauss seidel relaxations per eigenvector an asymptotic convergence rate of 0 15 per mg cycle is attained

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

two grid discretization schemes for nonlinear schrodinger equations we study efficient two grid discretization schemes with two loop continuation algorithms for computing wave functions of two coupled nonlinear schrodinger equations defined on the unit square and the unit disk both linear and quadratic approximations of the operator equations are exploited to derive the schemes the centered difference approximations the six node triangular elements and the adini elements are used to discretize the pdes defined on the unit square the proposed schemes also can compute stationary solutions of parameter dependent reaction diffusion systems our numerical results show that it is unnecessary to perform quadratic approximations

an adaptive multigrid scheme for bose einstein condensates in a periodic potential we present a novel multigrid continuation method for treating parameter dependent problems the proposed algorithm which can be flexibly implemented is a generalization of the two grid discretization schemes c s chien b w jeng a two grid discretization scheme for semilinear elliptic eigenvalue problems siam j sci comput 27 2006 1287 1304 that is approximating points on a solution curve do not necessarily lie on the same fine grid we apply the algorithm to compute energy levels and superfluid densities of bose einstein condensates bec in a periodic potential both positive and negative scattering lengths are considered in our numerical experiments for positive scattering length if the chemical potential is large enough and the domain is properly chosen the results show that the number of peaks of the first few energy states of the 2d bec in a periodic potential depends on the wave number of the periodic potential moreover for bright solitons the number of peaks of the ground state solutions is 1d 1 2 and 1d 2 where the periodic potential is expressed in terms of the sine or the cosine functions respectively however these formulae do not hold if the scattering length is negative the numerical study is extended to the two component 1d and 2d bec in a periodic potential

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

rational function neural network in this paper we observe that a particular class of rational function rf approximations may be viewed as feedforward networks like the radial basis function rbf network the training of the rf network may be performed using a linear adaptive filtering algorithm we illustrate the application of the rf network by considering two nonlinear signal processing problems the first problem concerns the one step prediction of a time series consisting of a pair of complex sinusoid in the presence of colored non gaussian noise simulated data were used for this problem in the second problem we use the rf network to build a nonlinear dynamic model of sea clutter radar backscattering from a sea surface here real life data were used for the study

a generalized petviashvili iteration method for scalar and vector hamiltonian equations with arbitrary form of nonlinearity the petviashvili s iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power law nonlinearity mu up 0 where m is a positive definite self adjoint operator and p const in this paper we propose a systematic generalization of this method to both scalar and vector hamiltonian equations with arbitrary form of nonlinearity and potential functions for scalar equations our generalized method requires only slightly more computational effort than the original petviashvili method