optimization for training neural nets various techniques of optimizing criterion functions to train neural net classifiers are investigated these techniques include three standard deterministic techniques variable metric conjugate gradient and steepest descent and a new stochastic technique it is found that the stochastic technique is preferable on problems with large training sets and that the convergence rates of the variable metric and conjugate gradient techniques are similar

deterministic neural classification this letter presents a minimum classification error learning formulation for a single layer feedforward network slfn by approximating the nonlinear counting step function using a quadratic function the classification error rate is shown to be deterministically solvable essentially the derived solution is related to an existing weighted least squares method with class specific weights set according to the size of data set by considering the class specific weights as adjustable parameters the learning formulation extends the classification robustness of the slfn without sacrificing its intrinsic advantage of being a closed form algorithm while the method is applicable to other linear formulations our empirical results indicate slfn s effectiveness on classification generalization

fast exact multiplication by the hessian just storing the hessian h the matrix of second derivatives 2e wi wj of the error e with respect to each pair of weights of a large neural network is difficult since a common use of a large matrix like h is to compute its product with various vectors we derive a technique that directly calculates hv where v is an arbitrary vector to calculate hv we first define a differential operator rv f w r f w rv r 0 note that rv w hv and rv w v and then apply rv to the equations used to compute w the result is an exact and numerically stable procedure for computing hv which takes about as much computation and is about as local as a gradient evaluation we then apply the technique to a one pass gradient calculation algorithm backpropagation a relaxation gradient calculation algorithm recurrent backpropagation and two stochastic gradient calculation algorithms boltzmann machines and weight perturbation finally we show that this technique can be used at the heart of many iterative techniques for computing various properties of h obviating any need to calculate the full hessian

discriminant independent component analysis a conventional linear model based on negentropy maximization extracts statistically independent latent variables which may not be optimal to give a discriminant model with good classification performance in this paper a single stage linear semisupervised extraction of discriminative independent features is proposed discriminant independent component analysis dica presents a framework of linearly projecting multivariate data to a lower dimension where the features are maximally discriminant with minimal redundancy the optimization problem is formulated as the maximization of linear summation of negentropy and weighted functional measure of classification motivated by independence among extracted features fisher linear discriminant is used as the functional measure of classification experimental results show improved classification performance when dica features are used for recognition tasks in comparison to unsupervised principal component analysis and ica and supervised feature extraction techniques like linear discriminant analysis lda conditional ica and those based on information theoretic learning approaches dica features also give reduced data reconstruction error in comparison to lda and ica method based on negentropy maximization

the schrodinger propagator for scattering metrics let g be a scattering metric on a compact manifold x with boundary i e a smooth metric giving the interior x the structure of a complete riemannian manifold with asymptotically conic ends an example is any compactly supported perturbation of the standard metric on rn consider the operator h 1 2 v where is the positive laplacian with respect to g and v is a smooth real valued function on x vanishing to second order at iox assuming that g is nontrapping we construct a global parametrix u z w t for the kernel of the schri odinger propagator u t e ith where z w io x and t 0 the parametrix is such that the difference between u and u is smooth and rapidly decreasing both as t iae 0 and as z iae iox uniformly for w on compact subsets of x let r x 1 where x is a boundary defining function for x be an asymptotic radial variable and let w t be the kernel e ir2 2tu t using the parametrix we show that w t belongs to a class of i legendre distributionsi on x i x i r 0 previously considered by hassell vasy when the metric is trapping then the parametrix construction goes through microlocally in the nontrapping part of the phase space we apply this result to determine the singularities of u t f for any tempered distribution f and for any fixed t 0 in terms of the oscillation of f near iox if the metric is nontrapping then we precisely determine the wavefront set of u t f and hence also precisely determine its singular support more generally we are able to determine the wavefront set of u t f for t 0 resp t 0 on the non backward trapped resp non forward trapped subset of the phase space this generalizes results of craig kappeler strauss and wunsch

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made

wave front set for solutions to schrodinger equations with long range perturbed harmonic oscillators abstract in this paper we consider a schrodinger operator with variable coefficients and harmonic potential the perturbation is assumed to be long range in a sense similar to the work of nakamura 2009 13 we construct a modified propagator and then by using this propagator and also the propagator of the unperturbed free harmonic oscillator we characterize the propagation of singularities for solutions to the equations

multigrid method and fourth order compact difference discretization scheme with unequal meshsizes for 3d poisson equation a fourth order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three dimensional 3d poisson equation on a cubic domain two multgrid methods are developed to solve the resulting sparse linear systems one is to use the full coarsening multigrid method with plane gauss seidel relaxation which uses line gauss seidel relaxation to compute each planewise solution the other is to construct a partial semi coarsening multigrid method with the traditional point or plane gauss seidel relaxations numerical experiments are conducted to test the computed accuracy of the fourth order compact difference scheme and the computational efficiency of the multigrid methods with the fourth order compact difference scheme

a new paradigm for solving navier stokes equations streamfunction velocity formulation in this paper we propose a new paradigm for solving navier stokes equations the proposed methodology is based on a streamfunction velocity formulation of the two dimensional steady state navier stokes equations representing incompressible fluid flows in two dimensional domains similar formulations are also possible for three dimensional fluid flows the main advantage of our formulation is that it avoids the difficulties associated with the computation of vorticity values especially on solid boundaries encountered when solving the streamfunction vorticity formulations our formulation also avoids the difficulties associated with solving pressure equations of the conventional velocity pressure formulations of the navier stokes equations we describe the new formulation of the navier stokes equations and use this formulation to solve a couple of fluid flow problems we use a biconjugate gradient method to obtain the numerical solutions of the fluid flow problems and provide detailed comparison data for the lid driven cavity flow problem it is discovered that our new formulation successfully provides high accuracy solutions for the benchmark problem in addition we also solve a problem of flow in a rectangular cavity with aspect ratio 2 and compare our results qualitatively and quantitatively with numerical and experimental results available in the literature in all cases we obtain high accuracy solutions with little additional cost

analytic smoothing effect for the schrodinger equation with long range perturbation we study microlocal analytic singularity of solutions to schr odinger equation with analytic coefficients using microlocal weight estimate developped for estimating the phase space tunneling we prove microlocal smoothing estimates that generalize results by l robbiano and c zuily we suppose the schr odinger operator is a long range type perturbation of the laplacian and we employ positive commutator type estimates to prove the smoothing property

on the discriminatory power of adaptive feed forward layered networks this correspondence expands the available theoretical framework that establishes a link between discriminant analysis and adaptive feed forward layered linear output networks used as mean square classifiers this has the advantages of providing more theoretical justification for the use of these nets in pattern classification and gaining a better insight into their behavior and about their use the authors prove that under reasonable assumptions minimizing the mean square error at the network output is equivalent to minimizing the following 1 the difference between the optimum value of a familiar discriminant criterion and the value of this criterion evaluated in the space spanned 2 the outputs of the final hidden layer and 3 the difference between the values of the same discriminant criterion evaluated in desired output and actual output subspaces the authors also illustrate under specific constraints how to solve the following problem given a feature extraction criterion how the target coding scheme can be selected such that this criterion is maximized at the output of the network final hidden layer other properties for these networks are explored

feature extraction with deep neural networks by a generalized discriminant analysis we present an approach to feature extraction that is a generalization of the classical linear discriminant analysis lda on the basis of deep neural networks dnns as for lda discriminative features generated from independent gaussian class conditionals are assumed this modeling has the advantages that the intrinsic dimensionality of the feature space is bounded by the number of classes and that the optimal discriminant function is linear unfortunately linear transformations are insufficient to extract optimal discriminative features from arbitrarily distributed raw measurements the generalized discriminant analysis gerda proposed in this paper uses nonlinear transformations that are learnt by dnns in a semisupervised fashion we show that the feature extraction based on our approach displays excellent performance on real world recognition and detection tasks such as handwritten digit recognition and face detection in a series of experiments we evaluate gerda features with respect to dimensionality reduction visualization classification and detection moreover we show that gerda dnns can preprocess truly high dimensional input data to low dimensional representations that facilitate accurate predictions even if simple linear predictors or measures of similarity are used

high order compact exponential finite difference methods for convection diffusion type problems a class of high order compact hoc exponential finite difference fd methods is proposed for solving one and two dimensional steady state convection diffusion problems the newly proposed hoc exponential fd schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection dominated problems the o h 4 compact exponential fd schemes developed for the one dimensional 1d problems produce diagonally dominant tri diagonal system of equations which can be solved by applying the tridiagonal thomas algorithm for the two dimensional 2d problems o h 4 k 4 compact exponential fd schemes are formulated on the nine point 2d stencil and the line iterative approach with alternating direction implicit adi procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time to validate the present hoc exponential fd methods three linear and nonlinear problems mostly with boundary or internal layers where sharp gradients may appear due to high peclet or reynolds numbers are numerically solved comparisons are made between analytical solutions and numerical results for the currently proposed hoc exponential fd methods and some previously published hoc methods the present hoc exponential fd methods produce excellent results for all test problems it is shown that besides including the excellent performances in computational accuracy efficiency and stability the present method has the advantage of better scale resolution the method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2d incompressible navier stokes equations using the stream function vorticity formulation

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

semisupervised generalized discriminant analysis generalized discriminant analysis gda is a commonly used method for dimensionality reduction in its general form it seeks a nonlinear projection that simultaneously maximizes the between class dissimilarity and minimizes the within class dissimilarity to increase class separability in real world applications where labeled data are scarce gda may not work very well however unlabeled data are often available in large quantities at very low cost in this paper we propose a novel gda algorithm which is abbreviated as semisupervised generalized discriminant analysis ssgda we utilize unlabeled data to maximize an optimality criterion of gda and formulate the problem as an optimization problem that is solved using the constrained concave convex procedure the optimization procedure leads to estimation of the class labels for the unlabeled data we propose a novel confidence measure and a method for selecting those unlabeled data points whose labels are estimated with high confidence the selected unlabeled data can then be used to augment the original labeled dataset for performing gda we also propose a variant of ssgda called m ssgda which adopts the manifold assumption to utilize the unlabeled data extensive experiments on many benchmark datasets demonstrate the effectiveness of our proposed methods

singularities of solutions to the schrodinger equation on scattering manifold in this paper we study microlocal singularities of solutions to schr odinger equations on scattering manifolds i e noncompact riemannian manifolds with asymptotically conic ends we characterize the wave front set of the solutions in terms of the initial condition and the classical scattering maps under the nontrapping condition our result is closely related to a recent work by hassell and wunsch though our model is more general and the method which relies heavily on scattering theoretical ideas is simple and quite different in particular we use an egorov type argument in the standard pseudodifferential symbol classes and avoid using legendre distributions in the proof we employ a microlocal smoothing property in terms of the it radially homogenous wave front set which is more precise than the preceding results

analytic wave front set for solutions to schrodinger equations this paper is a continuation of a martinez s nakamura v sordoni analytic smoothing effect for the schrodinger equation with long range perturbation comm pure appl math lix 2006 1330 1351 where an analytic smoothing effect was proved for long range type perturbations of the laplacian h0 on rn in this paper we consider short range type perturbations h of the laplacian on rn and we characterize the analytic wave front set of the solution to the schrodinger equation e ithf in terms of that of the free solution e ith0f for t 0 in the backward non trapping region this result is an analytic analogue of results by hassel and wunsch a hassel j wunsch the schrodinger propagator for scattering metrics ann of math 162 2005 487 523 and nakamura s nakamura wave front set for solutions to schrodinger equations j funct anal 256 2009 1299 1309

a perturbational h 4 exponential finite difference scheme for the convective diffusion equation a perturbational h4 compact exponential finite difference scheme with diagonally dominant coefficient matrix and upwind effect is developed for the convective diffusion equation perturbations of second order are exerted on the convective coefficients and source term of an h2 exponential finite difference scheme proposed in this paper based on a transformation to eliminate the upwind effect of the convective diffusion equation four numerical examples including one to three dimensional model equations of fluid flow and a problem of natural convective heat transfer are given to illustrate the excellent behavior of the present exponential schemes the h4 accuracy of the perturbational scheme is verified using double precision arithmetic

deep linear discriminant analysis on fisher networks person re identification is to seek a correct match for a person of interest across different camera views among a large number of impostors it typically involves two procedures of non linear feature extractions against dramatic appearance changes and subsequent discriminative analysis in order to reduce intra personal variations while enlarging inter personal differences in this paper we introduce a hybrid deep architecture which combines fisher vectors and deep neural networks to learn non linear transformations of pedestrian images to a deep space where data can be linearly separable the proposed method starts from fisher vector encoding which computes a sequence of local feature extraction aggregation and encoding the resulting fisher vector output are fed into stacked supervised layer to seek non linear transformation into a deep space on top of the deep neural network linear discriminant analysis lda is reinforced such that linearly separable latent representations can be learned in an end to end fashion by optimizing an objective function modified from lda the network is enforced to produce feature distributions which have a low variance within the same class and high variance between classes the objective is essentially derived from the general lda eigenvalue problem and allows to train the network with stochastic gradient descent and back propagate lda gradients to compute gaussian mixture model gmm gradients in fisher vector encoding for empirical evaluations we test our approach on four benchmark data sets in person re identification viper 1 cuhk03 2 cuhk01 3 and market 1501 4 extensive experiments on these benchmarks show that our method can achieve state of the art results highlightsa hybrid architecture that combines fisher vectors and deep neural networks an end to end training with linear discriminant analysis as objective deep features are linearly separable and class separability is maximally preserved

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

flexible manifold embedding a framework for semi supervised and unsupervised dimension reduction we propose a unified manifold learning framework for semi supervised and unsupervised dimension reduction by employing a simple but effective linear regression function to map the new data points for semi supervised dimension reduction we aim to find the optimal prediction labels f for all the training samples x the linear regression function h x and the regression residue f0 f h x simultaneously our new objective function integrates two terms related to label fitness and manifold smoothness as well as a flexible penalty term defined on the residue f0 our semi supervised learning framework referred to as flexible manifold embedding fme can effectively utilize label information from labeled data as well as a manifold structure from both labeled and unlabeled data by modeling the mismatch between h x and f we show that fme relaxes the hard linear constraint f h x in manifold regularization mr making it better cope with the data sampled from a nonlinear manifold in addition we propose a simplified version referred to as fme u for unsupervised dimension reduction we also show that our proposed framework provides a unified view to explain and understand many semi supervised supervised and unsupervised dimension reduction techniques comprehensive experiments on several benchmark databases demonstrate the significant improvement over existing dimension reduction algorithms

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a fast learning algorithm for deep belief nets we show how to use complementary priors to eliminate the explaining away effects that make inference difficult in densely connected belief nets that have many hidden layers using complementary priors we derive a fast greedy algorithm that can learn deep directed belief networks one layer at a time provided the top two layers form an undirected associative memory the fast greedy algorithm is used to initialize a slower learning procedure that fine tunes the weights using a contrastive version of the wake sleep algorithm after fine tuning a network with three hidden layers forms a very good generative model of the joint distribution of handwritten digit images and their labels this generative model gives better digit classification than the best discriminative learning algorithms the low dimensional manifolds on which the digits lie are modeled by long ravines in the free energy landscape of the top level associative memory and it is easy to explore these ravines by using the directed connections to display what the associative memory has in mind

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

some iterative finite element methods for steady navier stokes equations with different viscosities two level iterative finite element methods are designed to solve numerically the steady 2d 3d navier stokes equations for a large viscosity n such that a strong uniqueness condition holds moreover the one level oseen iterative finite element method based on a fine mesh with small mesh size h is designed to solve numerically the steady 2d 3d navier stokes equations for small viscosity n such that a weak uniqueness condition holds meanwhile the numerical investigations are provided to show that the proposed methods are efficient for solving the 2d 3d steady navier stokes equations for different viscosities

numerical simulation of 2d square driven cavity using fourth order compact finite difference schemes abstract fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large reynolds numbers the governing navier stokes equation is linearized in streamfunction and vorticity formulation the fourth order compact approximation schemes are coupled with fourth order approximations for velocities and vorticity boundaries numerical solutions are obtained for square driven cavity flow at high reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods

wave front set for solutions to schrodinger equations abstract we consider solutions to schrodinger equation on r d with variable coefficients let h be the schrodinger operator and let u t e i t h u 0 be the solution to the schrodinger equation with the initial condition u 0 l 2 r d we show that the wave front set of u t in the nontrapping region can be characterized by the wave front set of e i t h 0 u 0 where h 0 is the free schrodinger operator the characterization of the wave front set is given by the wave operator for the corresponding classical mechanical scattering or equivalently by the asymptotics of the geodesic flow

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

high order finite difference methods for the helmholtz equation high order finite difference methods for solving the helmholtz equation are developed and analyzed in one and two dimensions on uniform grids the standard pointwise representation has a second order accurate local truncation error we also study two schemes which have a fourth order accurate local truncation error one of the high order schemes is based on generalizations of the pade approximation the second scheme is based on high order approximation to the derivative calculated from the helmholtz equation itself a symmetric high order representation is developed for a neumann boundary condition numerical results are presented on model problems approximated with the developed schemes

higher order solutions of square driven cavity flow using a variable order multi grid method a new higher order method is devised for the numerical simulation of square driven cavity flows the spatial derivatives of the navier stokes equations are discretized by means of the modified differential quadrature mdq method the resulting system of ordinary differential equations odes in time is then integrated by the classical fourth order runge kutta gill rkg scheme the elliptic poisson equation is solved by means of a new variable order multi grid method the numerical simulations of the square driven cavity flows are carried out with spatial order of accuracy up to 10th order the results suggest that the higher order solutions are more reliable than the well known results obtained by ghia et al

a novel radial basis function neural network for discriminant analysis a novel radial basis function neural network for discriminant analysis is presented in this paper in contrast to many other researches this work focuses on the exploitation of the weight structure of radial basis function neural networks using the bayesian method it is expected that the performance of a radial basis function neural network with a well explored weight structure can be improved as the weight structure of a radial basis function neural network is commonly unknown the bayesian method is therefore used in this paper to study this a priori structure two weight structures are investigated in this study i e a single gaussian structure and a two gaussian structure an expectation maximization learning algorithm is used to estimate the weights the simulation results showed that the proposed radial basis function neural network with a weight structure of two gaussians outperformed the other algorithms

a compact streamfunction velocity scheme on nonuniform grids for the 2d steady incompressible navier stokes equations in this paper an effective compact finite difference approximation which carries streamfunction and its first derivatives velocities as the unknown variables for the streamfunction velocity formulation of the steady two dimensional incompressible navier stokes equation is developed on non uniform orthogonal cartesian grids to solve the resulting system of equations a multigrid iterative strategy on nonuniform grids is introduced by using the interpolation techniques numerical experiments involving two test problems with analytical solutions and the lid driven square cavity flow problem are carried out to display the superiority of the currently developed method on nonuniform grid numerical results show that the present method on nonuniform grids gets as similarly efficient convergence rate as on uniform grids viz second order accuracy and the resolution of the computed solutions for the problems with the sharp changes can be significantly improved when the nonuniform grid strategy is utilized the backward facing step flow is also calculated by the present method to exhibit the capability to simulate the distant field using fewer grid points the solution for the natural convection problem reveals further the wide applications of the present method not only in the flow problems but also in the heat transfer problems all of these numerical results demonstrate the accuracy and efficiency of the currently proposed schemes

reconstruction of singularities for solutions of schrodinger s equation we determine the behavior in time of singularities of solutions to some schrodinger equations onr n we assume the hamiltonians are of the formh 0 v where h 0 1 2 delta 1 2 sum limits k 1 n omega k 2 x k 2 and wherev is bounded and smooth with decaying derivatives when all k 0 the kernelk t x y of exp ith is smooth inx for every fixed t y when all 1 are equal but non zero the initial singularity reconstructs at times t frac m pi omega 1 and positionsx 1 m y just as ifv 0 k is otherwise regular in the general case the singular support is shown to be contained in the union of the hyperplanes x x js 1 l js y js when j t l j forj j 1 j r

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

a rational high order compact difference method for the steady state stream functionvorticity formulation of the navierstokes equations a rational high order compact rhoc finite difference fd method on the nine point stencil is proposed for solving the steady state two dimensional navierstokes equations in the stream functionvorticity form the resulting system of algebra equations can be solved by using the point successive over or under relaxation sor iteration numerical experiments involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward facing step flows are carried out to validate the performance of the newly proposed method numerical solutions of the driven cavity problem with different grid mesh sizes maximum being 513513 for reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature

universal approximation using incremental constructive feedforward networks with random hidden nodes according to conventional neural network theories single hidden layer feedforward networks slfns with additive or radial basis function rbf hidden nodes are universal approximators when all the parameters of the networks are allowed adjustable however as observed in most neural network implementations tuning all the parameters of the networks may cause learning complicated and inefficient and it may be difficult to train networks with nondifferential activation functions such as threshold networks unlike conventional neural network theories this paper proves in an incremental constructive method that in order to let slfns work as universal approximators one may simply randomly choose hidden nodes and then only need to adjust the output weights linking the hidden layer and the output layer in such slfns implementations the activation functions for additive nodes can be any bounded nonconstant piecewise continuous functions g r r and the activation functions for rbf nodes can be any integrable piecewise continuous functions g r r and rg x dx 0 the proposed incremental method is efficient not only for sflns with continuous including nondifferentiable activation functions but also for slfns with piecewise continuous such as threshold activation functions compared to other popular methods such a new network is fully automatic and users need not intervene the learning process by manually tuning control parameters

dimensionality reduction of multimodal labeled data by local fisher discriminant analysis reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high dimensional data analysis fisher discriminant analysis fda is a traditional technique for supervised dimensionality reduction but it tends to give undesired results if samples in a class are multimodal an unsupervised dimensionality reduction method called locality preserving projection lpp can work well with multimodal data due to its locality preserving property however since lpp does not take the label information into account it is not necessarily useful in supervised learning scenarios in this paper we propose a new linear supervised dimensionality reduction method called local fisher discriminant analysis lfda which effectively combines the ideas of fda and lpp lfda has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem we demonstrate the practical usefulness and high scalability of the lfda method in data visualization and classification tasks through extensive simulation studies we also show that lfda can be extended to non linear dimensionality reduction scenarios by applying the kernel trick

training products of experts by minimizing contrastive divergence it is possible to combine multiple latent variable models of the same data by multiplying their probability distributions together and then renormalizing this way of combining individual expert models makes it hard to generate samples from the combined model but easy to infer the values of the latent variables of each expert because the combination rule ensures that the latent variables of different experts are conditionally independent when given the data a product of experts poe is therefore an interesting candidate for a perceptual system in which rapid inference is vital and generation is unnecessary training a poe by maximizing the likelihood of the data is difficult because it is hard even to approximate the derivatives of the renormalization term in the combination rule fortunately a poe can be trained using a different objective function called contrastive divergence whose derivatives with regard to the parameters can be approximated accurately and efficiently examples are presented of contrastive divergence learning using several types of expert on several types of data

a rank one update algorithm for fast solving kernel foley sammon optimal discriminant vectors discriminant analysis plays an important role in statistical pattern recognition a popular method is the foley sammon optimal discriminant vectors fsodvs method which aims to find an optimal set of discriminant vectors that maximize the fisher discriminant criterion under the orthogonal constraint the fsodvs method outperforms the classic fisher linear discriminant analysis flda method in the sense that it can solve more discriminant vectors for recognition kernel foley sammon optimal discriminant vectors kfsodvs is a nonlinear extension of fsodvs via the kernel trick however the current kfsodvs algorithm may suffer from the heavy computation problem since it involves computing the inverse of matrices when solving each discriminant vector resulting in a cubic complexity for each discriminant vector this is costly when the number of discriminant vectors to be computed is large in this paper we propose a fast algorithm for solving the kfsodvs which is based on rank one update rou of the eigensytems it only requires a square complexity for each discriminant vector moreover we also generalize our method to efficiently solve a family of optimally constrained generalized rayleigh quotient ocgrq problems which include many existing dimensionality reduction techniques we conduct extensive experiments on several real data sets to demonstrate the effectiveness of the proposed algorithms

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

microlocal dispersive smoothing for the schrodinger equation this paper establishes a connection between the microlocal smoothness of solutions of the initial value problem for schrodinger s equation and the global behavior of bicharacteristics of the principal symbol in particular localized initial data gives rise to solutions which are microlocally smooth at all points which are not trapped backwards by the bicharacteristic flow the origin of the phenomenon is in the dispersive nature of the equation the results imply microlocal regularity properties of the fundamental solution 1995 john wiley sons inc

multigrid method based on the transformation free hoc scheme on nonuniform grids for 2d convection diffusion problems in this paper a multigrid method based on the high order compact hoc difference scheme on nonuniform grids which has been proposed by kalita et al j c kalita a k dass d c dalal a transformation free hoc scheme for steady convection diffusion on non uniform grids int j numer methods fluids 44 2004 33 53 is proposed to solve the two dimensional 2d convection diffusion equation the hoc scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids consequently the multigrid method is brand new for solving the discrete system arising from the difference equation on nonuniform grids the corresponding multigrid projection and interpolation operators are constructed by the area ratio some boundary layer and local singularity problems are used to demonstrate the superiority of the present method numerical results show that the multigrid method with the hoc scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems the present method is also applied to solve the 2d incompressible navier stokes equations using the stream function vorticity formulation and the numerical solutions of the lid driven cavity flow problem are obtained and compared with solutions available in the literature

preconditioned iterative methods and finite difference schemes for convection diffusion we conduct experimental study on numerical solution of the two dimensional convection diffusion equation discretized by three finite difference schemes the traditional central difference scheme the standard upwind scheme and the fourth order compact scheme we study the computed accuracy achievable by each scheme the algebraic properties of the coefficient matrices arising from different schemes and the performance of the gauss seidel iterative method the preconditioned gmres iterative method and the multigrid method for solving linear systems arising from these schemes

fully compact higher order computation of steady state natural convection in a square cavity the flow in a thermally driven square cavity with adiabatic top and bottom walls and differentially heated vertical walls for a wide range of rayleigh numbers 10 3 or ra or 10 7 has been computed with a fourth order accurate higher order compact scheme which was used earlier only for the stream function vorticity psi omega form of the two dimensional steady state navier stokes equations the boundary conditions used are also compact and of identical accuracy in particular a compact fourth order accurate neumann boundary condition has been developed for temperature at the adiabatic walls the treatment of the derivative source term is also compact and has been done in such a way as to give fourth order accuracy and easy assimilation with the solution procedure as the discretization for the psi omega formulation boundary conditions and source term treatment are all fourth order accurate highly accurate solutions are obtained on relatively coarser grids unlike other compact solution procedure in literature for this physical configuration the present method is fully compact and fully higher order accurate also use of conjugate gradient and hybrid biconjugate gradient stabilized algorithms to solve the symmetric and nonsymmetric algebraic systems at every outer iteration ensures good convergence behavior of the method even at higher rayleigh numbers

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions

analyticity in time and smoothing effect of solutions to nonlinear schrodinger equations in this paper we consider analyticity in time and smoothing effect of solutions to nonlinear schrodinger equations where we prove that if satisfies then there exists a unique solution of 1 and positive constants t c 0 c 1 such that is analytic in time and space variables for and and has an analytic continuation on and in the case the condition 2 can be relaxed as follows where m 0 if n 1 p 1 m 1 if n 2 and m 1 if n 3 p 1