local discontinuous galerkin methods for nonlinear schrodinger equations in this paper we develop a local discontinuous galerkin method to solve the generalized nonlinear schrodinger equation and the coupled nonlinear schrodinger equation l 2 stability of the schemes are obtained for both of these nonlinear equations numerical examples are shown to demonstrate the accuracy and capability of these methods

conservative modified cranknicolson and time splitting wavelet methods for modeling boseeinstein condensates in delta potentials this paper explores two wavelet based energy conserving algorithms for the grosspitaevskii equation with delta potentials in boseeinstein condensates named modified cranknicolson wavelet method and time splitting wavelet method respectively both proposed methods can preserve the intrinsic properties of original problems as much as possible meanwhile the rigorous error estimates and some conservative properties are investigated they are proved to preserve the charge conservation exactly the global energy conservation laws can be preserved under several conditions in practical computations to avoid a large drift in energy values caused by discontinuous potential well an improved discrete delta function is implemented numerical experiments for attractive and repulsive cases are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis

a unified constructive network model for problem solving abstract we develop a neural network model that relieves time consuming trial and error computer experiments usually performed in problem solving with networks where problems including the traveling salesman problem pattern matching and pattern classification learning are formulated as optimization problems with constraint first we specify and uniquely distinguish the model as a set of constituent functions that should comply with restrictive conditions next we demonstrate that it is unified i e it yields most current networks finally we verify that it is constructive i e we show a standard method that systematically constructs from a given optimization problem a particular network in that model to solve it

symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

the solution of nonlinear schrodinger equations using orthogonal spline collocation abstract an orthogonal spline collocation semidiscretization previously applied in the numerical solution of the cubic schrodinger equation is extended to the solution of schrodinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives the accuracy of the method is examined as well as its ability to conserve approximations to certain invariants associated with the theoretical solution the results of numerical experiments are presented in which the integration in time is performed using a routine from a software library particular attention is paid to the value and limitations of conservation as an indicator of accuracy

a fourth order explicit schemes for the coupled nonlinear schrodinger equation abstract in this paper we derive numerical methods for solving the coupled nonlinear schrodinger equation we discretize the space derivative by central difference formulas of fourth order the resulting ordinary differential system is solved by the fourth order explicit runge kutta method neumann and periodic boundary conditions are used the method is tested for accuracy and the conserved quantities these methods conserve the three conserved quantities conserve exactly for at least five decimal places a comparison has been made with some existing methods

scattering matrices and weyl functions for a scattering system a a 0 consisting of self adjoint extensions a and a 0 of a symmetric operator a with finite deficiency indices the scattering matrix s and a spectral shift function are calculated in terms of the weyl function associated with a boundary triplet for a and a simple proof of the krein birman formula is given the results are applied to singular sturm liouville operators with scalar and matrix potentials to dirac operators and to schrodinger operators with point interactions

constrained clustering as an optimization method a deterministic annealing approach to clustering is derived on the basis of the principle of maximum entropy this approach is independent of the initial state and produces natural hierarchical clustering solutions by going through a sequence of phase transitions it is modified for a larger class of optimization problems by adding constraints to the free energy the concept of constrained clustering is explained and three examples are are given in which it is used to introduce deterministic annealing the previous clustering method is improved by adding cluster mass variables and a total mass constraint the traveling salesman problem is reformulated as constrained clustering yielding the elastic net en approach to the problem more insight is gained by identifying a second lagrange multiplier that is related to the tour length and can also be used to control the annealing process the open path constraint formulation is shown to relate to dimensionality reduction by self organization in unsupervised learning a similar annealing procedure is applicable in this case as well

numerical solutions of coupled kleingordonzakharov equations by quintic b spline differential quadrature method numerical solutions of the coupled kleingordonzakharov equations are obtained by using quintic b spline based differential quadrature method a rungekutta fourth method is used for time integration stability of the scheme is studied using matrix stability analysis the accuracy and efficiency of the presented method is shown by conducting some numerical experiments on test problems which includes the motion of single soliton and interactions of two solitons the numerical results are found in good agreement with the exact solutions

parameter sensitivity of the elastic net approach to the traveling salesman problem durbin and willshaw s elastic net algorithm can find good solutions to the tsp the purpose of this paper is to point out that for certain ranges of parameter values the algorithm converges into local minima that do not correspond to valid tours the key parameter is the ratio governing the relative strengths of the two competing terms in the elastic net energy function based on recent work by durbin szeliski and yuille the parameter regime in which the net may visit some cities twice is examined further analysis predicts the regime in which the net may fail to visit some cities at all understanding these limitations allows one to select the parameter value most likely to avoid either type of problem simulation data support the theoretical work

a mathematical aspect of a tunnel junction for spintronic qubit abstract we consider the dirac particle that lives in the 1 dimensional configuration space consisting of two quantum wires and a junction between the two we regard the spin of a dirac particle as spintronic qubit we give concrete formulae explicitly expressing the one to one correspondence between every self adjoint extension of the minimal dirac operator and its corresponding boundary condition of the wave functions of the dirac particle we then show that all the boundary conditions can be classified into just two types the two types are characterized by whether the electron passes through the junction or not we also show how the tunneling produces its own phase factor and what is the relation between the phase factor and the spintronic qubit in the tunneling boundary condition

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

an hr adaptive method for the cubic nonlinear schrodinger equation abstract the nonlinear schrodinger equation nlse is one of the most important equations in quantum mechanics and appears in a wide range of applications including optical fibre communications plasma physics and biomolecule dynamics it is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients adaptive moving mesh methods r adaptive attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features this approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest adaptive mesh refinement h adaptive where the mesh is locally coarsened or refined is an alternative adaptive strategy which is popular for time independent problems in this paper we consider the effectiveness of a combined method h r adaptive to solve the nlse in one space dimension simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches the method is also shown to control the spatial error based on the user s input error tolerance evidence is also presented indicating second order spatial convergence using a novel monitor function to generate the adaptive moving mesh

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

morphable surface models we describe a novel automatic technique for finding a dense correspondence between a pair of n dimensional surfaces with arbitrary topologies this method employs a different formulation than previous correspondence algorithms such as optical flow and includes images as a special case we use this correspondence algorithm to build morphable surface models an extension of morphable models from examples we present a method for matching the model to new surfaces and demonstrate their use for analysis synthesis and clustering

are visual cortex maps optimized for coverage the elegant regularity of maps of variables such as ocular dominance orientation and spatial frequency in primary visual cortex has prompted many people to suggest that their structure could be explained by an optimization principle up to now the standard way to test this hypothesis has been to generate artificial maps by optimizing a hypothesized objective function and then to compare these artificial maps with real maps using a variety of quantitative criteria if the artificial maps are similar to the real maps this provides some evidence that the real cortex may be optimizing a similar function to the one hypothesized recently a more direct method has been proposed for testing whether real maps represent local optima of an objective function swindale shoham grinvald bonhoeffer hubener 2000 in this approach the value of the hypothesized function is calculated for a real map and then the real map is perturbed in certain ways and the function recalculated if each of these perturbations leads to a worsening of the function it is tempting to conclude that the real map is quite likely to represent a local optimum of that function in this article we argue that such perturbation results provide only weak evidence in favor of the optimization hypothesis

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

numerical methods based on additive splittings for hyperbolic partial differential equations we derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds these techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales some of which are sometimes treated exactly the efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator this is analyzed and a technique is discussed for reducing this error through a simple change of variables a procedure for generating the appropriate boundary data for the intermediate solutions is also presented

the bootstrap widrow hoff rule as a cluster formation algorithm an algorithm that is widely used for adaptive equalization in current modems is the bootstrap or decision directed version of the widrow hoff rule we show that this algorithm can be viewed as an unsupervised clustering algorithm in which the data points are transformed so that they form two clusters that are as tight as possible the standard algorithm performs gradient ascent in a crude model of the log likelihood of generating the transformed data points from two gaussian distributions with fixed centers better convergence is achieved by using the exact gradient of the log likelihood

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

simulations of solitons using quadratic spline finite elements abstract a numerical solution to the korteweg de vries equation is obtained using the bubnov galerkin method with quadratic spline finite elements the element matrices are set up algebraically using reduce a linear stability analysis of the scheme shows the method to be unconditionally stable classical problems concerning the development motion and interaction of solitons are used to validate the method the numerical scheme is compared with published algorithms and shown to be both accurate and efficient

a quadratic b spline finite element method for solving nonlinear schrodinger equation a numerical solution of the nonlinear schrodinger nls equation is obtained by using the quadratic b spline finite element method test problems concerning motion birth and bound states of solitons of the nlg equation are studied to evaluate the method results are compared with various numerical methods

a hybrid neural network model for solving optimization problems a hybrid neural network model for solving optimization problems is proposed an energy function which contains the constraints and cost criteria of an optimization problem is derived and then the neural network is used to find the global minimum or maximum of the energy function which corresponds to a solution of the optimization problem the network contains two subnets a constraint network and a goal network the constraint network models the constraints of an optimization problem and computes the gradient updating value of each neuron such that the energy function monotonically converges to satisfy all constraints of the problem the goal network points out the direction of convergence for finding an optimal value for the cost criteria these two subnets ensure that the neural network finds feasible as well as optimal or near optimal solutions the traveling salesman problem and the hamiltonian cycle problem are used to demonstrate the method

resolution based complexity control for gaussian mixture models in the domain of unsupervised learning mixtures of gaussians have become a popular tool for statistical modeling for this class of generative models we present a complexity control scheme which provides an effective means for avoiding the problem of overfitting usually encountered with unconstrained mixtures of gaussians in high dimensions according to some prespecified level of resolution as implied by a fixed variance noise model the scheme provides an automatic selection of the dimensionalities of some local signal subspaces by maximum likelihood estimation together with a resolution based control scheme for adjusting the number of mixture components we arrive at an incremental model refinement procedure within a common deterministic annealing framework which enables an efficient exploration of the model space the advantages of the resolution based framework are illustrated by experimental results on synthetic and high dimensional real world data

finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions in this paper we use two symplectic schemes to simulate the ablowitz ladik model associated to the cubic nonlinear schrodinger equation and we compare them with nonsymplectic methods

a tau method based on non uniform space time elements for the numerical simulation of solitons abstract in this paper we discuss the numerical simulation of soliton solutions of schrodnger s system of nonlinear partial differential equations the technique used is based on a novel approach to the tau method based on space time elements the original operator is not discretized in either space or time the perturbation term which in the tau method is a by product of the computation is used to measure the error in the equation or defect and therefore to control the accuracy of the computation in an adaptive way this leads to an accurate and economical numerical procedure which even for approximations of a very low degree can simulate surfaces with very sharp gradients in the direction of both space and time the tau method formulation presented in this paper preserves very accurately the principle of conservation of energy however our numerical examples support the idea that the latter is a necessary but not a sufficient condition to guarantee an efficient approximation of soliton solutions of schrodinger s equation by means of a method of numerical simulation

vector quantization by deterministic annealing a deterministic annealing approach is suggested to search for the optimal vector quantizer given a set of training data the problem is reformulated within a probabilistic framework no prior knowledge is assumed on the source density and the principle of maximum entropy is used to obtain the association probabilities at a given average distortion the corresponding lagrange multiplier is inversely related to the temperature and is used to control the annealing process in this process as the temperature is lowered the system undergoes a sequence of phase transitions when existing clusters split naturally without use of heuristics the resulting codebook is independent of the codebook used to initialize the iterations

stability satisfied numerical approximates to the non analytical solutions of the cubic schrodinger equation abstract the time dependent complex schrodinger equation with cubic non linearity is solved by constructing differential quadrature algorithm based on sinc functions reduction the equation to a coupled system of real equations enables to approximate the space derivative terms by the proposed method the resultant ordinary differential equation system is integrated with respect to the time variable by using various explicit methods of lower and higher orders some initial boundary value problems containing some analytical and non analytical initial data are solved for experimental illustrations the computational errors between the analytical and numerical solutions are measured by the discrete maximum error norm in case the analytical solutions exist the two conserved quantities are calculated by using the numerical results in all cases the matrix stability analysis is implemented to control the time step size

statistical physics mixtures of distributions and the em algorithm we show that there are strong relationships between approaches to optmization and learning based on statistical physics or mixtures of experts in particular the em algorithm can be interpreted as converging either to a local maximum of the mixtures model or to a saddle point solution to the statistical physics system an advantage of the statistical physics approach is that it naturally gives rise to a heuristic continuation method deterministic annealing for finding good solutions

a class of constrained clustering algorithms for object boundary extraction boundary extraction is a key task in many image analysis operations this paper describes a class of constrained clustering algorithms for object boundary extraction that includes several well known algorithms proposed in different fields deformable models constrained clustering data ordering and traveling salesman problems the algorithms belonging to this class are obtained by the minimization of a cost function with two terms a quadratic regularization term and an image dependent term defined by a set of weighting functions the minimization of the cost function is achieved by lowpass filtering the previous model shape and by attracting the model units toward the centroids of their attraction regions to define a new algorithm belonging to this class the user has to specify a regularization matrix and a set of weighting functions that control the attraction of the model units toward the data the usefulness of this approach is twofold it provides a unified framework for many existing algorithms in pattern recognition and deformable models and allows the design of new recursive schemes

operator formulation of wigner s r matrix theories for the schrodinger and dirac equations the r matrix theories for the schrodinger and dirac equations are formulated in the language of integral operators in the nonrelativistic theory the central role is played by an integral operator rb e relating function values to normal derivatives on a surface s of a closed volume v inside which the function satisfies the schrodinger equation at energy e in the relativistic theory the same role is played by two integral operators rb e and rb e linking on the surface s values of upper and lower components of spinor wave functions satisfying in the volume v the dirac equation at energy e systematic procedures for constructing the operators rb e and rb e generalizing the methods due to kapur and peierls and to wigner are presented

arbitrary elastic topologies and ocular dominance the elastic net which has been used to produce accounts of the formation of topology preserving maps and ocular dominance stripes od embodies a nearest neighbor topology a hebbian account of od is not so restricted and indeed makes the prediction that the width of the stripes depends on the nature of the more general neighborhood relations elastic and hebbian accounts have recently been unified raising a question mark about their different determiners of stripe widths this paper considers this issue and demonstrates theoretically that it is possible to use more general topologies in the elastic net including those effectively adopted in the hebbian model

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

hyperparameter selection for self organizing maps the self organizing map som algorithm for finite data is derived as an approximate maximum a posteriori estimation algorithm for a gaussian mixture model with a gaussian smoothing prior which is equivalent to a generalized deformable model gdm for this model objective criteria for selecting hyperparameters are obtained on the basis of empirical bayesian estimation and cross validation which are representative model selection methods the properties of these criteria are compared by simulation experiments these experiments show that the cross validation methods favor more complex structures than the expected log likelihood supports which is a measure of compatibility between a model and data distribution on the other hand the empirical bayesian methods have the opposite bias

a note on a paper by a g bratsos m ehrhardt and i th famelis in this short note an addition to the paper a g bratsos m ehrhardt i th famelis a discrete adomian decomposition method for discrete nonlinear schrodinger equations appl math comput 197 1 2008 190 205 using the modulus of the terms evaluated from the adomian decomposition method on p 194 and their relation to the convergence of the resulting series is presented conclusions for the accuracy of the approximated solution are derived

normalized adaptive channel equalizer based on minimal symbol error rate existing minimum symbol error rate equalizers were derived based on the symbol error rate objective function due to the complexity of the objective function the derivation is not straightforward in this paper we present a new approach to derive the minimum symbol error rate adaptive equalizers the problem is formulated as minimizing the norm between two subsequent parameter vectors under the constraint of symbol error rate minimization the constrained optimization problem then is solved with the lagrange multiplier method which results in an adaptive algorithm with normalization simulation results show that the proposed algorithm outperforms the existing adaptive minimum symbol error rate equalizer in convergence speed and steady state performance

eigensolutions of the wigner eisenbud problem for a cylindrical nanowire within finite volume method we present a node centered finite volume method for computing a representative range of eigenvalues and eigenvectors of the schrodinger operator on a three dimensional cylindrically symmetric bounded domain with mixed boundary conditions the three dimensional schrodinger operator is reduced to a family of two dimensional schrodinger operators distinguished by a centrifugal potential we consider a uniform boundary conforming delaunay mesh which additionally conforms to the material interfaces we study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first k eigenvalues and eigenvectors and their sequential arrangement there exists an optimal uniform delaunay discretization with matching anisotropy with respect to the effective masses of the host material for a centrifugal potential one retrieves the theoretically established first order convergence while second order convergence is recovered only on uniform grids with an anisotropy correction

solving optimization problems with variable constraint by an extended cohen grossberg model abstract a variety of real world problems can be formulated as continuous optimization problems with variable constraint it is however well known that it is quite hard to develop a unified method for obtaining their feasible solutions we have recognized that the recent work of solving the traveling salesman problem by the hopfield model explores an innovative approach to them as well as combinatorial optimization problems the hopfield model is generalized into the cohen grossberg model to which a specific liapunov function has been found this paper thus extends the hopfield method onto the cohen grossberg model in order to develop a unified solving method of continuous optimization problems with variable constraint specifically we consider a certain class of continuous optimization problems with a constraint equation including the hopfield version of the traveling salesman problem as a particular member then we theoretically develop a method that from any given problem of that class derives a network of an extended cohen grossberg model to provide feasible solutions to it the main idea for constructing that extended cohen grossberg model lies in adding to the cohen grossberg model a synapse dynamical system concurrently operating with its current unit dynamical system so that the constraint equation can be enforced to satisfaction at final states this construction is also motivated by previous neuron models in biophysics and learning algorithms in neural networks

on eisenbud s and wigner s r matrix a general approach abstract the main objective of this paper is to give a rigorous treatment of wigner s and eisenbud s r matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices in the framework of boundary triplets and associated weyl functions an abstract generalization of the r matrix method is developed and the results are applied to schrodinger operators on the real axis

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

linearly implicit methods for the nonlinear schrodinger equation in nonhomogeneous media the nonlinear schrodinger equation in one dimensional cartesian coordinates is studied numerically by means of three linearly implicit finite difference methods that result in block tridiagonal matrices at each time level as a function of the damping pumping and changes in the refraction index the numerical experiments performed to determine the effects of the time step spatial step size and implicitness parameters of the dispersion and nonlinear terms show that the most accurate results are obtained when both the dispersion and nonlinear terms are treated by means of a second order accurate trapezoidal discretization and full linearization of the nonlinear terms it is also shown that if either the dispersion or the nonlinear terms are treated with a first order accurate approximation substantial errors are found and these errors are largest when the nonlinear terms are discretized by means of a first order difference and do not decrease substantially as the number of grid points is increased it has been observed that when solitons move from a medium to another one characterized by a higher refraction index the amplitude and speed of the soliton increase whereas its width decreases upon approaching the interface the amplitude of the soliton was found to increase downstream while some radiation was observed upstream both the increase in amplitude and radiation were found to increase with the change in the refraction index the effect of the distance where the index of refraction changes was found to be small if this distance is smaller than the width of the soliton for solitons propagating from a medium to another one of lower refraction index the amplitude and speed of the soliton decrease whereas its width increases the amplitude of the soliton increases downstream and spreads towards the boundary where it is reflected when the index of refraction decreases below a threshold it was found that the soliton may become trapped at the interface the results presented in this paper indicate that theories based on treating solitons as particles perturbation methods inverse scattering and variational formulations can provide accurate results only when the change in the refraction index is sufficiently small

b spline finite element studies of the non linear schrodinger equation abstract the non linear schrodinger equation is solved numerically by a b spline finite element method the approach used is based on collocation of cubic b splines over spatial finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range time integration of the resulting system of ordinary differential equations is effected using a crank nicolson approximation standard problems are used to validate the algorithm comparisons are made with published numerical and analytical solutions the proposed method exhibits good conservation properties and performs well the development of temporally recurrent states from spatially periodic initial conditions is examined through computer experiments the birth of mobile as well as static solitons out of maxwellian initial data is observed

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

an analysis of the elastic net approach to the traveling salesman problem this paper analyzes the elastic net approach durbin and willshaw 1987 to the traveling salesman problem of finding the shortest path through a set of cities the elastic net approach jointly minimizes the length of an arbitrary path in the plane and the distance between the path points and the cities the tradeoff between these two requirements is controlled by a scale parameter k a global minimum is found for large k and is then tracked to a small value in this paper we show that 1 in the small k limit the elastic path passes arbitrarily close to all the cities but that only one path point is attracted to each city 2 in the large k limit the net lies at the center of the set of cities and 3 at a critical value of k the energy function bifurcates we also show that this method can be interpreted in terms of extremizing a probability distribution controlled by k the minimum at a given k corresponds to the maximum a posteriori map bayesian estimate of the tour under a natural statistical interpretation the analysis presented in this paper gives us a better understanding of the behavior of the elastic net allows us to better choose the parameters for the optimization and suggests how to extend the underlying ideas to other domains

a soft decision directed lms algorithm for blind equalization an adaptation algorithm for equalizers operating on very distorted channels is presented the algorithm is based on the idea of adjusting the equalizer tap gains to maximize the likelihood that the equalizer outputs would be generated by a mixture of two gaussians with known means the decision directed least mean square algorithm is shown to be an approximation to maximizing the likelihood that the equalizer outputs come from such an independently and identically distributed source the algorithm is developed in the context of a binary pulse amplitude modulation channel and simulations demonstrate that the algorithm converges in channels for which the decision directed lms algorithms does not converge

self organizing mixture networks for probability density estimation a self organizing mixture network somn is derived for learning arbitrary density functions the network minimizes the kullback leibler information metric by means of stochastic approximation methods the density functions are modeled as mixtures of parametric distributions a mixture needs not to be homogenous i e it can have different density profiles the first layer of the network is similar to kohonen s self organizing map som but with the parameters of the component densities as the learning weights the winning mechanism is based on maximum posterior probability and updating of the weights is limited to a small neighborhood around the winner the second layer accumulates the responses of these local nodes weighted by the learned mixing parameters the network possesses a simple structure and computational form yet yields fast and robust convergence the network has a generalization ability due to the relative entropy criterion used applications to density profile estimation and pattern classification are presented the somn can also provide an insight to the role of neighborhood function used in the som

annealing and the normalized n cut we describe an annealing procedure that computes the normalized n cut of a weighted graph g the first phase transition computes the solution of the approximate normalized 2 cut problem while the low temperature solution computes the normalized n cut the intermediate solutions provide a sequence of refinements of the 2 cut that can be used to split the data to k clusters with 2

solving the generalized nonlinear schrodinger equation via quartic spline approximation this paper is concerned with a new conservative finite difference method for solving the generalized nonlinear schrodinger gnls equation iut uxx f u 2 u 0 the numerical scheme is constructed through the semidiscretization and an application of the quartic spline approximation central difference and extrapolation formulae are used for approximating the neumann boundary conditions introduced both continuous and discrete energy conservation and the stability property are investigated the numerical method provides an efficient and reliable way for computing long time solitary solutions given by the gnls equation numerical examples are given to demonstrate our conclusions

quartic and quintic b spline methods for advection diffusion equation differential quadrature methods based on b spline functions of degree four and five have been introduced to solve advection diffusion equation numerically two initial boundary value problems modeling the transportation of a concentration and distribution of an initial pulse are simulated using both methods the errors of the numerical results obtained by both methods have been computed stability analysis for both methods is also studied by the use of matrix stability

topology and geometry of single hidden layer network least squares weight solutions in this paper the topological and geometric properties of the weight solutions for multilayer perceptron mlp networks under the mse error criterion are characterized the characterization is obtained by analyzing a homotopy from linear to nonlinear networks in which the hidden node function is slowly transformed from a linear to the final sigmoidal nonlinearity two different geometric perspectives for this optimization process are developed the generic topology of the nonlinear mlp weight solutions is described and related to the geometric interpretations error surfaces and homotopy paths both analytically and using carefully constructed examples these results illustrate that although the natural homotopy provides a practically valuable heuristic for training it suffers from a number of theoretical and practical difficulties the linear system is a bifurcation point of the homotopy equations and solution paths are therefore generically discontinuous bifurcations and infinite solutions further occur for data sets that are not of measure zero these results weaken the guarantees on global convergence and exhaustive behavior normally associated with homotopy methods however the analyses presented provide a clear understanding of the relationship between linear and nonlinear perceptron networks and thus a firm foundation for development of more powerful training methods the geometric perspectives and generic topological results describing the nature of the solutions are further generally applicable to network analysis and algorithm evaluation

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

an investigation into the effect of product approximation in the numerical solution of the cubic nonlinear schro dinger equation abstract we study the effect of product approximation on the galerkin solutions of the one dimensional cubic schrodinger equation a crank nicolson scheme is used to discretize in time the paper describes two numerical experiments in the first we examine the approximation obtained by the standard galerkin method and discuss the possibility of enforcing discrete analogs of the conservation laws satisfied by the exact solution in the second experiment numerical results obtained by the product approximation version of the galerkin method are compared and the effectiveness of the method for different combinations of test and trial functions is also investigated

high order split step exponential methods for solving coupled nonlinear schrodinger equations numerical integration schemes for coupled time dependent nonlinear schrodinger equations are examined using exponential splitting step methods exponentiation of the nonlinear potential term is reduced to the exponential of a kinetic energy term which can be calculated by fast fourier transforms high order iteration schemes involving a minimum number of product operators are shown to yield highly accurate amplitude and phase these new splitting methods are shown to be highly efficient both with respect to accuracy and integration time

numerical simulations of time dependent partial differential equations when a time dependent partial differential equation pde is discretized in space with a spectral approximation the result is a coupled system of ordinary differential equations odes in time this is the notion of the method of lines mol and the resulting set of odes is stiff the stiffness may be even exacerbated sometimes the linear terms are the primarily responsible for the stiffness with a rapid exponential decay of some modes as in a dissipative pde or a rapid oscillation of some modes as in a dispersive pde therefore for a time dependent pde which combines low order nonlinear terms with higher order linear terms it is desirable to use a higher order approximation both in space and in time along our research we have focused on a particular case of spectral methods the so called pseudo spectral methods to solve numerically time dependent pdes using different techniques an integrating factor in de la hoz and vadillo 2010 an exponential time differencing method in de la hoz and vadillo 2008 and differentiation matrices in the theoretical frame of matrix differential equations in de la hoz and vadillo 2012 2013a b this paper which is a unified review of those contributions aims at providing a better understanding of those methods by illustrating their variety and more importantly their power furthermore we also give emphasis to choosing adequate schemes to advance in time

symplectic methods for the ablowitz ladik discrete nonlinear schrodinger equation using several kinds of coordinate transformations we standardize the noncanonical symplectic structure of the ablowitz ladik model a l model of nonlinear schrodinger equation nlse then we employ some symplectic scheme to simulate the solitons motion and test the evolution of the discrete invariants of the a l model and also the conserved quantities of the original nlse in comparison with a higher order non symplectic scheme applied directly to the a l model we show the overwhelming superiorities of the symplectic method we also compare the implementation of the same symplectic scheme to different standardized hamiltonian systems resulting from different coordinate transformations and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one in preserving the invariants of the a l model and the original nlse

a robust global and local mixture distance based non rigid point set registration we present a robust global and local mixture distance glmd based non rigid point set registration method which consists of an alternating two step process correspondence estimation and transformation updating we first define two distance features for measuring global and local structural differences between two point sets respectively the two distances are then combined to form a glmd based cost matrix which provides a flexible way to estimate correspondences by minimizing global or local structural differences using a linear assignment solution to improve the correspondence estimation and enhance the interaction between the two steps an annealing scheme is designed to gradually change the cost minimization from local to global and the thin plate spline transformation from rigid to non rigid during registration we test the performance of our method in contour registration sequence images and real images and compare with six state of the art methods where our method shows the best alignments in most scenarios highlightswe develop a multi feature based non rigid point set registration algorithm we define two distance based features a multi feature based cost matrix is designed for correspondence estimation we solve issues in the current iterative non rigid point set registration methods the performance of our method outperforms state of the art methods in this area

a differential quadrature algorithm for simulations of nonlinear schrodinger equation numerical simulations of nonlinear schrodinger equation are studied using differential quadrature method based on cosine expansion propogation of a soliton interaction of two solitons birth of standing and mobile solitons and bound state solutions are simulated the accuracy of the method dq is measured using maximum error norm the results are compared with some earlier works the lowest two conserved quantities are computed numerically for all cases

a bayesian method for fitting parametric and nonparametric models to noisy data we present a simple paradigm for fitting models parametric and nonparametric to noisy data which resolves some of the problems associated with classical mse algorithms this is done by considering each point on the model as a possible source for each data point the paradigm can be used to solve problems which are ill posed in the classical mse approach such as fitting a segment as opposed to a line it is shown to be nonbiased and to achieve excellent results for general curves even in the presence of strong discontinuities results are shown for a number of fitting problems including lines circles elliptic arcs segments rectangles and general curves contaminated by gaussian and uniform noise

hierarchical unsupervised learning with growing via phase transitions we address unsupervised learning subject to structural constraints with particular emphasis placed on clustering with an imposed decision tree structure most known methods are greedy optimizing one node of the tree at a time to minimize a local cost by constrast we develop a joint optimization method derived based on information theoretic principles and closely related to known methods in statistical physics the approach is inspired by the deterministic annealing algorithm for unstructured data clustering which was based on maximum entropy inference the new approach is founded on the principle of minimum cross entropy using informative priors to approximate the unstructured clustering solution while imposing the structural constraint the resulting method incorporates supervised learning principles applied in an unsupervised problem setting in our approach the tree grows by a sequence of bifurcations that occur while optimizing an effective free energy cost at decreasing temperature scales thus estimates of the tree size and structure are naturally obtained at each temperature in the process examples demonstrate considerable improvement over known methods

an engineer s guide to soliton phenomena application of the finite element method abstract the paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton type solutions namely the korteweg de vries equation the nonlinear schrodinger equation and the sine gordon equation throughout this study a semidiscrete galerkin method is applied using a finite element discretization in space and a step by step time integration of the resulting system of nonlinear ordinary differential equations depending upon the special type of the evolutionary equation the application of a petrov galerkin procedure may increase significantly the numerical stability accuracy and effectivity of the different approaches are demonstrated on a series of computer plots

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

the numerical treatment of the nonlinear schrodinger equation abstract the implicit midpoint rule for the integration in time of a fourth order accurate semidiscretization in space of the nonlinear schrodinger equation is studied the stability analysis for the present method is carried out by the conservation of the energy in time three increasingly difficult problems are tested by the method mentioned comparison with other methods is presented the results obtained here seem to be in better agreement with the exact solutions

a unified construction of variational r matrix methods i the schrodinger equation properties of two linear integral operators and relating function values to normal derivatives on a surface of a closed volume inside which the function satisfies the schrodinger equation at energy e are discussed variational principles for matrix elements and eigenvalues of these operators are constructed in a systematic way by using an approach of gerjuoy et al some of the variational principles derived were already known in the non relativistic r matrix theory of scattering but others seem to be new applications of the rayleigh ritz linear trial functions are presented

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained

haar wavelet approximation for the solution of cubic nonlinear schrodinger equations abstract in this study haar wavelet collocation method is used for the numerical solution of 1d and 2d cubic nonlinear schrodinger equations with initial and dirichlet boundary conditions the space derivatives are estimated through haar wavelet collocation method whereas for time derivative we have used crank nicolson scheme the proposed method is implemented upon several test problems and the numerical results of these test problems establish that the proposed method is accurate

kapur peierls and wigner r matrix theories for the dirac equation an r matrix theory for the dirac equation is shown to exist in spite of incompleteness of a relativistic r matrix basis on the reaction surface a phenomenon which does not occur in the non relativistic case the theory is constructed for the most general boundary conditions imposed on expansion basis functions it is shown that the incompleteness of the expansion basis on the reaction surface results in a matrix correction appearing in the eigenfunction expansion of the r matrix the correction vanishes in the non relativistic limit the approach is applied to the relativistic generalizations of the kapur peierls and wigner resonance reaction theories

on the problem of correspondence in range data and some inelastic uses for elastic nets in this work the authors propose a novel method to obtain correspondence between range data across image frames using neural like mechanisms the method is computationally efficient and tolerant of noise and missing points elastic nets which evolved out of research into mechanisms to establish ordered neural projections between structures of similar geometry are used to cast correspondence as an optimization problem this formulation is then used to obtain approximations to the motion parameters under the assumption of rigidity inelasticity these parameter scan be used to recover correspondence experimental results are presented to establish the veracity of the scheme and the method is compared to earlier attempts in this direction

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library