a new analysis of discontinuous galerkin methods for a fourth order variational inequality abstract we study a family of discontinuous galerkin methods for the displacement obstacle problem of kirchhoff plates on two and three dimensional convex polyhedral domains which are characterized as fourth order elliptic variational inequalities of the first kind we prove that the error in an h 2 like energy norm is o h for the quadratic method where 1 2 1 is determined by the geometry of the domain under additional assumptions on the contact set such that the solution has improved regularity we derive the optimal error estimate with 1 3 2 for the cubic method numerical experiments demonstrate the performance of the methods and confirm the theoretical results

on the quadratic finite element approximation to the obstacle problem in this paper we obtain the error bound o left h frac 32 epsilon right for any epsilon 0 for the piecewise quadratic finite element approximation to the obstacle problem without the hypothesis that the free boundary has finite length see 3

efficient compression of borehole resistivity and acoustic imaging data using wavelets this paper presents a new approach to borehole resistivity and acoustic imaging data compression the method is based on the discrete wavelet transform dwt combined with the set partitioning in the hierarchical trees spiht coding method the primary advantages of the dwt spiht method are its superior compression performance simple implementation and constant compression and transmission rate control this new approach generally performs better than the traditional discrete cosine transform dct followed by huffman coding methods such as jpeg particularly for target compression ratios greater than 50 1 image block size plays a key role in using this method to achieve the best compression performance the image block size needs to be selected appropriately in general large image block sizes and more dwt decomposition levels result in higher compression ratios and or fewer reconstruction errors whenever possible image block sizes that allow the maximum levels of dyadic dwt decomposition should be used when insufficient data samples are available in either depth or azimuthal direction a small number of extra data samples can be added without sacrificing the overall compression performance the dwt spiht method also shows good fidelity in representing major features in borehole images such as fractures and sinusoids

a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis a discontinuous galerkin formulation is developed and analyzed for the cases of classical and gradient plasticity the model of gradient plasticity is based on the von mises yield function in which dependence is on the isotropic hardening parameter and its laplacian the problem takes the form of a variational inequality of the second kind the discontinuous galerkin formulation is shown to be consistent and convergent error estimates are obtained for the cases of semi and fully discrete formulations these mimic the error estimates obtained for classical plasticity with the conventional galerkin formulation

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

gtm the generative topographic mapping latent variable models represent the probability density of data in a space of several dimensions in terms of a smaller number of latent or hidden variables a familiar example is factor analysis which is based on a linear transformations between the latent space and the data space in this paper we introduce a form of non linear latent variable model called the generative topographic mapping for which the parameters of the model can be determined using the em algorithm gtm provides a principled alternative to the widely used self organizing map som of kohonen 1982 and overcomes most of the significant limitations of the som we demonstrate the performance of the gtm algorithm on a toy problem and on simulated data from flow diagnostics for a multi phase oil pipeline

a two level algorithm for an obstacle problem abstract due to the inequality feature of the obstacle problem the standard quadratic finite element method for solving the problem can only achieve an error bound of the form o n 3 4 n being the total number of degrees of freedom and 0 arbitrary to achieve a better error bound the key lies in how to capture the free boundary accurately in this paper we propose a two level algorithm for solving the obstacle problem the first part of the algorithm is through the use of the linear elements on a quasi uniform mesh then information on the approximate free boundary from the linear element solution is used in the construction of a quadratic finite element method under some assumptions it is shown that the numerical solution from the two level algorithm is expected to have a nearly optimal error bound of o n 1 0 arbitrary such an expected convergence order is observed numerically in numerical examples

emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces olshausen and field 1996 applied the principle of independence maximization by sparse coding to extract features from natural images this leads to the emergence of oriented linear filters that have simultaneous localization in space and in frequency thus resembling gabor functions and simple cell receptive fields in this article we show that the same principle of independence maximization can explain the emergence of phase and shift invariant features similar to those found in complex cells this new kind of emergence is obtained by maximizing the independence between norms of projections on linear subspaces instead of the independence of simple linear filter outputs the norms of the projections on such independent feature subspaces then indicate the values of invariant features

reversible integer to integer wavelet transforms for image compression performance evaluation and analysis in the context of image coding a number of reversible integer to integer wavelet transforms are compared on the basis of their lossy compression performance lossless compression performance and computational complexity of the transforms considered several were found to perform particularly well with the best choice for a given application depending on the relative importance of the preceding criteria reversible integer to integer versions of numerous transforms are also compared to their conventional i e nonreversible real to real counterparts for lossy compression at low bit rates reversible integer to integer and conventional versions of transforms were found to often yield results of comparable quality factors affecting the compression performance of reversible integer to integer wavelet transforms are also presented supported by both experimental data and theoretical arguments

discontinuous galerkin methods for solving a quasistatic contact problem we consider the numerical solution of a nonlinear evolutionary variational inequality arising in the study of quasistatic contact problems we study spatially semi discrete and fully discrete schemes for the problem with several discontinuous galerkin discretizations in space and finite difference discretization in time under appropriate regularity assumptions on the solution a unified error analysis is established for the schemes reaching the optimal convergence order for linear elements numerical results are presented on a two dimensional test problem to illustrate numerical convergence orders

noise contrastive estimation of unnormalized statistical models with applications to natural image statistics we consider the task of estimating from observed data a probabilistic model that is parameterized by a finite number of parameters in particular we are considering the situation where the model probability density function is unnormalized that is the model is only specified up to the partition function the partition function normalizes a model so that it integrates to one for any choice of the parameters however it is often impossible to obtain it in closed form gibbs distributions markov and multi layer networks are examples of models where analytical normalization is often impossible maximum likelihood estimation can then not be used without resorting to numerical approximations which are often computationally expensive we propose here a new objective function for the estimation of both normalized and unnormalized models the basic idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise with this approach the normalizing partition function can be estimated like any other parameter we prove that the new estimation method leads to a consistent convergent estimator of the parameters for large noise sample sizes the new estimator is furthermore shown to behave like the maximum likelihood estimator in the estimation of unnormalized models there is a trade off between statistical and computational performance we show that the new method strikes a competitive trade off in comparison to other estimation methods for unnormalized models as an application to real data we estimate novel two layer models of natural image statistics with spline nonlinearities

inhomogeneous dirichlet boundary condition in the a posteriori error control of the obstacle problem abstract we study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous dirichlet boundary condition the results in the article are two fold firstly we address the influence of the inhomogeneous dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem secondly by rewriting the obstacle problem in an equivalent form we derive a posteriori error bounds which are in simpler form and efficient to accomplish this we construct and use a post processed solution u h of the discrete solution u h which satisfies the exact boundary conditions sharply although the discrete solution u h may not satisfy we propose two post processing methods and analyze them namely the harmonic extension and a linear extension the theoretical results are illustrated by the numerical results

overlearning in marginal distribution based ica analysis and solutions the present paper is written as a word of caution with users of independent component analysis ica in mind to overlearning phenomena that are often observed we consider two types of overlearning typical to high order statistics based ica these algorithms can be seen to maximise the negentropy of the source estimates the first kind of overlearning results in the generation of spike like signals if there are not enough samples in the data or there is a considerable amount of noise present it is argued that if the data has power spectrum characterised by 1 f curve we face a more severe problem which cannot be solved inside the strict ica model this overlearning is better characterised by bumps instead of spikes both overlearning types are demonstrated in the case of artificial signals as well as magnetoencephalograms meg several methods are suggested to circumvent both types either by making the estimation of the ica model more robust or by including further modelling of the data

learning nonlinear statistical regularities in natural images by modeling the outer product of image intensities it is well known that there exist nonlinear statistical regularities in natural images existing approaches for capturing such regularities always model the image intensities by assuming a parameterized distribution for the intensities and learn the parameters in the letter we propose to model the outer product of image intensities by assuming a gaussian distribution for it a two layer structure is presented where the first layer is nonlinear and the second layer is linear trained on natural images the first layer bases resemble the receptive fields of simple cells in the primary visual cortex v1 while the second layer units exhibit some properties of the complex cells in v1 including phase invariance and masking effect the model can be seen as an approximation of the covariance model proposed in karklin and lewicki 2009 but has more robust and efficient learning algorithms

separation theorem for independent subspace analysis and its consequences independent component analysis ica the theory of mixed independent non gaussian sources has a central role in signal processing computer vision and pattern recognition one of the most fundamental conjectures of this research field is that independent subspace analysis isa the extension of the ica problem where groups of sources are independent can be solved by traditional ica followed by grouping the ica components the conjecture called isa separation principle i has been rigorously proven for some distribution types recently ii forms the basis of the state of the art isa solvers iii enables one to estimate the unknown number and the dimensions of the sources efficiently and iv can be extended to generalizations of the isa task such as different linear controlled post nonlinear complex valued partially observed problems as well as to problems dealing with nonparametric source dynamics here we shall review the advances on this field

a stochastic self organizing map for proximity data we derive an efficient algorithm for topographic mapping of proximity data tmp which can be seen as an extension of kohonen s self organizing map to arbitrary distance measures the tmp cost function is derived in a baysian framework of folded markov chains for the description of autoencoders it incorporates the data by a dissimilarity matrix and the topographic neighborhood by a matrix of transition probabilities from the principle of maximum entropy a nonfactorizing gibbs distribution is obtained which is approximated in a mean field fashion this allows for maximum likelihood estimation using an expectation maximization algorithm in analogy to the transition from topographic vector quantization to the self organizing map we suggest an approximation to tmp that is computationally more efficient in order to prevent convergence to local minima an annealing scheme in the temperature parameter is introduced for which the critical temperature of the first phase transition is calculated in terms o

error estimates of the weakly over penalized symmetric interior penalty method for two variational inequalities in this paper we apply the weakly over penalized symmetric interior penalty method to solve some variational inequalities which include the signorini problem and the obstacle problem optimal a priori error estimates in energy norm are derived some numerical tests are provided to confirm our theoretical analysis

discrete ordinate discontinuous galerkin methods for solving the radiative transfer equation the radiative transfer equation rte occurs in a wide variety of applications in this paper we study discrete ordinate discontinuous galerkin methods for solving the rte the numerical methods are formed in two steps in the first step the discrete ordinate technique is applied to discretize the integral operator for the angular variable resulting in a semidiscrete hyperbolic system in the second step the spatial discontinuous galerkin method is applied to discretize the semidiscrete system a stability and error analysis is performed on the numerical methods some numerical examples are included to demonstrate the convergence behavior of the methods

stabilized hp finite element methods for first order hyperbolic problems we analyze the hp version of the streamline diffusion finite element method sdfem and of the discontinuous galerkin finite element method dgfem for first order linear hyperbolic problems for both methods we derive new error estimates on general finite element meshes which are sharp in the mesh width h and in the spectral order p of the method assuming that the stabilization parameter is o h p for piecewise analytic solutions exponential convergence is established on quadrilateral meshes for the dgfem we admit very general irregular meshes and for the sdfem we allow meshes which contain hanging nodes numerical experiments confirm the theoretical results

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle abstract superconvergence analysis of nonconforming finite element methods fems are discussed for solving the second order variational inequality problem with displacement obstacle the elements employed have a common typical character i e the consistency error can reach order o h 3 2 nearly 1 2 order higher than their interpolation error when the exact solution of the considered problem belongs to h 5 2 for any e 0 by making full use of special properties of the element s interpolations and bramble hilbert lemma the superconvergence error estimates of order o h 3 2 in the broken h1 norm are derived finally some numerical results are provided to confirm the theoretical results

analysis of a damped nonlinear multilevel method in this paper we present a new algorithm that is obtained by introducing a damping parameter in the classical nonlinear multilevel method we analyse this damped nonlinear multilevel method in particular we prove global convergence and local efficiency for a suitable class of problems

a fast fixed point algorithm for independent component analysis we introduce a novel fast algorithm for independent component analysis which can be used for blind source separation and feature extraction we show how a neural network learning rule can be transformed into a fixedpoint iteration which provides an algorithm that is very simple does not depend on any user defined parameters and is fast to converge to the most accurate solution allowed by the data the algorithm finds one at a time all nongaussian independent components regardless of their probability distributions the computations can be performed in either batch mode or a semiadaptive manner the convergence of the algorithm is rigorously proved and the convergence speed is shown to be cubic some comparisons to gradient based algorithms are made showing that the new algorithm is usually 10 to 100 times faster sometimes giving the solution in just a few iterations

sparse code shrinkage denoising of nongaussian data by maximum likelihood estimation sparse coding is a method for finding a representation of data in which each of the components of the representation is only rarely significantly active such a representation is closely related to redundancy reduction and independent component analysis and has some neurophysiological plausibility in this article we show how sparse coding can be used for denoising using maximum likelihood estimation of nongaussian variables corrupted by gaussian noise we show how to apply a soft thresholding shrinkage operator on the components of sparse coding so as to reduce noise our method is closely related to the method of wavelet shrinkage but it has the important benefit over wavelet methods that the representation is determined solely by the statistical properties of the data the wavelet representation on the other hand relies heavily on certain mathematical properties like self similarity that may be only weakly related to the properties of natural data

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

a general procedure for learning mixtures of independent component analyzers this paper presents a new procedure for learning mixtures of independent component analyzers the procedure includes non parametric estimation of the source densities supervised unsupervised learning of the model parameters incorporation of any independent component analysis ica algorithm into the learning of the ica mixtures and estimation of residual dependencies after training for correction of the posterior probability of every class to the testing observation vector we demonstrate the performance of the procedure in the classification of ica mixtures of two three and four classes of synthetic data and in the classification of defective materials consisting of 3d finite element models and lab specimens in non destructive testing using the impact echo technique the application of the proposed posterior probability correction demonstrates an improvement in the classification accuracy semi supervised learning shows that unlabeled data can degrade the performance of the classifier when they do not fit the generative model comparative results of the proposed method and standard ica algorithms for blind source separation in one and multiple ica data mixtures show the suitability of the non parametric ica mixture based method for data modeling

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

evolution strategies based adaptive lp ls svm not only different databases but two classes of data within a database can also have different data structures svm and ls svm typically minimize the empirical f risk regularized versions subject to fixed penalty l 2 or l 1 penalty are non adaptive since their penalty forms are pre determined they often perform well only for certain types of situations for example ls svm with l 2 penalty is not preferred if the underlying model is sparse this paper proposes an adaptive penalty learning procedure called evolution strategies es based adaptive l p least squares support vector machine es based l p ls svm to address the above issue by introducing multiple kernels a l p penalty based nonlinear objective function is derived the iterative re weighted minimal solver irms algorithm is used to solve the nonlinear function then evolution strategies es is used to solve the multi parameters optimization problem penalty parameterp kernel and regularized parameters are adaptively selected by the proposed es based algorithm in the process of training the data which makes it easier to achieve the optimal solution numerical experiments are conducted on two artificial data sets and six real world data sets the experiment results show that the proposed procedure offer better generalization performance than the standard svm the ls svm and other improved algorithms

multilevel projection algorithm for solving obstacle problems abstract obstacle problems are nonlinear free boundary problems and the computation of approximate solutions can be difficult and expensive little work has been done on effective numerical methods of such problems this paper addresses some aspects of this issue discretizing the problem in a continuous piecewise linear finite element space gives a quadratic programming problem with inequality constraints a new method called the multilevel projection mp method is established in this paper the mp algorithm extends the multigrid method for linear equations to nonlinear obstacle problems the convergence theorems of this method are also proved a numerical example presented shows our error estimate is sharp and the mp algorithm is robust

convergence analysis of symmetric dual wind discontinuous galerkin approximation methods for the obstacle problem abstract this paper formulates and analyzes symmetric dual wind discontinuous galerkin dg methods for second order elliptic obstacle problem these new methods follow from the dg differential calculus framework that defines discrete differential operators to replace the continuous differential operators when discretizing a partial differential equation pde we establish optimal a priori error estimates for both linear and quadratic elements provided the exact solution is sufficiently regular these results are also shown to hold for some non positive penalty parameters with the emphasis on zero penalization across all interior and boundary edges numerical experiments are provided to validate the theoretical results and gauge the performance of the proposed methods

a new image representation algorithm inspired by image submodality models redundancy reduction and learning in biological vision we develop a new biologically motivated algorithm for representing natural images using successive projections into complementary subspaces an image is first projected into an edge subspace spanned using an ica basis adapted to natural images which captures the sharp features of an image like edges and curves the residual image obtained after extraction of the sharp image features is approximated using a mixture of probabilistic principal component analyzers mppca model the model is consistent with cellular functional information theoretic and learning paradigms in visual pathway modeling we demonstrate the efficiency of our model for representing different attributes of natural images like color and luminance we compare the performance of our model in terms of quality of representation against commonly used basis like the discrete cosine transform dct independent component analysts ica and principal components analysis pca based on their entropies chrominance and luminance components of images are represented using codes having lower entropy than dct ica or pca for similar visual quality the model attains considerable simplification for learning from images by using a sparse independent code for representing edges and explicitly evaluating probabilities in the residual subspace

bayesian estimation of causal direction in acyclic structural equation models with individual specific confounder variables and non gaussian distributions several existing methods have been shown to consistently estimate causal direction assuming linear or some form of nonlinear relationship and no latent confounders however the estimation results could be distorted if either assumption is violated we develop an approach to determining the possible causal direction between two observed variables when latent confounding variables are present we first propose a new linear non gaussian acyclic structural equation model with individual specific effects that are sometimes the source of confounding thus modeling individual specific effects as latent variables allows latent confounding to be considered we then propose an empirical bayesian approach for estimating possible causal direction using the new model we demonstrate the effectiveness of our method using artificial and real world data

a model for learning topographically organized parts based representations of objects in visual cortex topographic nonnegative matrix factorization object representation in the inferior temporal cortex it an area of visual cortex critical for object recognition in the primate exhibits two prominent properties 1 objects are represented by the combined activity of columnar clusters of neurons with each cluster representing component features or parts of objects and 2 closely related features are continuously represented along the tangential direction of individual columnar clusters here we propose a learning model that reflects these properties of parts based representation and topographic organization in a unified framework this model is based on a nonnegative matrix factorization nmf basis decomposition method nmf alone provides a parts based representation where nonnegative inputs are approximated by additive combinations of nonnegative basis functions our proposed model of topographic nmf tnmf incorporates neighborhood connections between nmf basis functions arranged on a topographic map and attains the topographic property without losing the parts based property of the nmf the tnmf represents an input by multiple activity peaks to describe diverse information whereas conventional topographic models such as the self organizing map som represent an input by a single activity peak in a topographic map we demonstrate the parts based and topographic properties of the tnmf by constructing a hierarchical model for object recognition where the tnmf is at the top tier for learning high level object features the tnmf showed better generalization performance over nmf for a data set of continuous view change of an image and more robustly preserving the continuity of the view change in its object representation comparison of the outputs of our model with actual neural responses recorded in the it indicates that the tnmf reconstructs the neuronal responses better than the som giving plausibility to the parts based learning of the model

a hierarchical bayesian model for learning nonlinear statistical regularities in nonstationary natural signals capturing statistical regularities in complex high dimensional data is an important problem in machine learning and signal processing models such as principal component analysis pca and independent component analysis ica make few assumptions about the structure in the data and have good scaling properties but they are limited to representing linear statistical regularities and assume that the distribution of the data is stationary for many natural complex signals the latent variables often exhibit residual dependencies as well as nonstationary statistics here we present a hierarchical bayesian model that is able to capture higher order nonlinear structure and represent nonstationary data distributions the model is a generalization of ica in which the basis function coefficients are no longer assumed to be independent instead the dependencies in their magnitudes are captured by a set of density components each density component describes a common pattern of deviation from the marginal density of the pattern ensemble in different combinations they can describe nonstationary distributions adapting the model to image or audio data yields a nonlinear distributed code for higher order statistical regularities that reflect more abstract invariant properties of the signal

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

the two dimensional gabor function adapted to natural image statistics a model of simple cell receptive fields and sparse structure in images the two dimensional gabor function is adapted to natural image statistics leading to a tractable probabilistic generative model that can be used to model simple cell receptive field profiles or generate basis functions for sparse coding applications learning is found to be most pronounced in three gabor function parameters representing the size and spatial frequency of the two dimensional gabor function and characterized by a nonuniform probability distribution with heavy tails all three parameters are found to be strongly correlated resulting in a basis of multiscale gabor functions with similar aspect ratios and size dependent spatial frequencies a key finding is that the distribution of receptive field sizes is scale invariant over a wide range of values so there is no characteristic receptive field size selected by natural image statistics the gabor function aspect ratio is found to be approximately conserved by the learning rules and is therefore not well determined by natural image statistics this allows for three distinct solutions a basis of gabor functions with sharp orientation resolution at the expense of spatial frequency resolution a basis of gabor functions with sharp spatial frequency resolution at the expense of orientation resolution or a basis with unit aspect ratio arbitrary mixtures of all three cases are also possible two parameters controlling the shape of the marginal distributions in a probabilistic generative model fully account for all three solutions the best performing probabilistic generative model for sparse coding applications is found to be a gaussian copula with pareto marginal probability density functions

a mixed discontinuous galerkin method with symmetric stress for brinkman problem based on the velocity pseudostress formulation abstract the brinkman equations can be regarded as a combination of the stokes and darcy equations which model transitions between the fast flow in channels governed by stokes equations and the slow flow in porous media governed by darcy s law the numerical challenge for this model is the designing of a numerical scheme which is stable for both the stokes dominated high permeability and the darcy dominated low permeability equations in this paper we solve the brinkman model in n dimensions n 2 3 by using the mixed discontinuous galerkin mdg method which meets this challenge this mdg method is based on the pseudostress velocity formulation and uses a discontinuous piecewise polynomial pair p k 1 s p k k 0 where the stress field is symmetric the main unknowns are the pseudostress and the velocity whereas the pressure is easily recovered through a simple postprocessing a key step in the analysis is to establish the parameter robust inf sup stability through specific parameter dependent norms at both continuous and discrete levels therefore the stability results presented here are uniform with respect to the permeability thanks to the parameter robust stability analysis we obtain optimal error estimates for the stress in broken h div norm and velocity in l 2 norm furthermore the l 2 error estimate for pseudostress is derived under certain conditions finally numerical experiments are provided to support the theoretical results and to show the robustness accuracy and flexibility of the mdg method

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

discontinuous galerkin methods for solving elliptic variational inequalities we study discontinuous galerkin methods for solving elliptic variational inequalities of both the first and second kinds analysis of numerous discontinuous galerkin schemes for elliptic boundary value problems is extended to the variational inequalities we establish a priori error estimates for the discontinuous galerkin methods which reach optimal order for linear elements results from some numerical examples are reported

a posteriori error control of discontinuous galerkin methods for elliptic obstacle problems in this article we derive an a posteriori error estimator for various discontinuous galerkin dg methods that are proposed in wang han and cheng siam j numer anal 48 708 733 2010 for an elliptic obstacle problem using a key property of dg methods we perform the analysis in a general framework the error estimator we have obtained for dg methods is comparable with the estimator for the conforming galerkin cg finite element method in the analysis we construct a non linear smoothing function mapping dg finite element space to cg finite element space and use it as a key tool the error estimator consists of a discrete lagrange multiplier associated with the obstacle constraint it is shown for non over penalized dg methods that the discrete lagrange multiplier is uniformly stable on non uniform meshes finally numerical results demonstrating the performance of the error estimator are presented

learning intermediate level representations of form and motion from natural movies we present a model of intermediate level visual representation that is based on learning invariances from movies of the natural environment the model is composed of two stages of processing an early feature representation layer and a second layer in which invariances are explicitly represented invariances are learned as the result of factoring apart the temporally stable and dynamic components embedded in the early feature representation the structure contained in these components is made explicit in the activities of second layer units that capture invariances in both form and motion when trained on natural movies the first layer produces a factorization or separation of image content into a temporally persistent part representing local edge structure and a dynamic part representing local motion structure consistent with known response properties in early visual cortex area v1 this factorization linearizes statistical dependencies among the first layer units making them learnable by the second layer the second layer units are split into two populations according to the factorization in the first layer the form selective units receive their input from the temporally persistent part local edge structure and after training result in a diverse set of higher order shape features consisting of extended contours multiscale edges textures and texture boundaries the motion selective units receive their input from the dynamic part local motion structure and after training result in a representation of image translation over different spatial scales and directions in addition to more complex deformations these representations provide a rich description of dynamic natural images and testable hypotheses regarding intermediate level representation in visual cortex

gradient schemes for the signorini and the obstacle problems and application to hybrid mimetic mixed methods gradient schemes is a framework which enables the unified convergence analysis of many different methods such as finite elements conforming non conforming and mixed and finite volumes methods for 2nd order diffusion equations we show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed dirichlet neumann and signorini boundary conditions this extension allows us to provide error estimates for numerical approximations of such models recovering known convergence rates for some methods and establishing new convergence rates for schemes not previously studied for variational inequalities the general framework we develop also enables us to design a new numerical method for the obstacle and signorini problems based on hybrid mimetic mixed schemes we provide numerical results that demonstrate the accuracy of these schemes and confirm our theoretical rates of convergence

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

adaptive discontinuous galerkin methods for solving an incompressible stokes flow problem with slip boundary condition of frictional type abstract we consider the numerical solution of a variational inequality arising in the study of an incompressible stokes flow with a nonlinear slip boundary condition of frictional type we study several discontinuous galerkin dg schemes for solving the problem and derive reliable and efficient residual type a posteriori error estimators an adaptive dg method is constructed based on the error estimators to solve the problem some numerical examples are presented to illustrate the efficiency of the adaptive dg method

equivariant adaptive source separation source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed this paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence easi the easi algorithms are based on the idea of serial updating this specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures most importantly the performance of an easi algorithm does not depend on the mixing matrix in particular convergence rates stability conditions and interference rejection levels depend only on the normalized distributions of the source signals closed form expressions of these quantities are given via an asymptotic performance analysis the theme of equivariance is stressed throughout the paper the source separation problem has an underlying multiplicative structure the parameter space forms a matrix multiplicative group we explore the favorable consequences of this fact on implementation performance and optimization of easi algorithms

beyond independent components trees and clusters we present a generalization of independent component analysis ica where instead of looking for a linear transform that makes the data components independent we look for a transform that makes the data components well fit by a tree structured graphical model this tree dependent component analysis tca provides a tractable and flexible approach to weakening the assumption of independence in ica in particular tca allows the underlying graph to have multiple connected components and thus the method is able to find clusters of components such that components are dependent within a cluster and independent between clusters finally we make use of a notion of graphical models for time series due to brillinger 1996 to extend these ideas to the temporal setting in particular we are able to fit models that incorporate tree structured dependencies among multiple time series

kernel reconstruction ica for sparse representation independent component analysis with soft reconstruction cost rica has been recently proposed to linearly learn sparse representation with an overcomplete basis and this technique exhibits promising performance even on unwhitened data however linear rica may not be effective for the majority of real world data because nonlinearly separable data structure pervasively exists in original data space meanwhile rica is essentially an unsupervised method and does not employ class information motivated by the success of the kernel trick that maps a nonlinearly separable data structure into a linearly separable case in a high dimensional feature space we propose a kernel rica krica model to nonlinearly capture sparse representation in feature space furthermore we extend the unsupervised krica to a supervised one by introducing a class driven discrimination constraint such that the data samples from the same class are well represented on the basis of the corresponding subset of basis vectors this discrimination constraint minimizes inhomogeneous representation energy and maximizes homogeneous representation energy simultaneously which is essentially equivalent to maximizing between class scatter and minimizing within class scatter at the same time in an implicit manner experimental results demonstrate that the proposed algorithm is more effective than other state of the art methods on several datasets

a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis this work is the second of a two part investigation into the use of discontinuous galerkin methods for obtaining approximate solutions to problems of classical and gradient plasticity part i j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 presented the formulation and analysis of such problems this part focusses on algorithmic and computational aspects of the problem in particular it is shown that the predictor corrector algorithms of classical plasticity are readily extended to the case of gradient plasticity and to discontinuous galerkin formulations conditions for convergence of the algorithms are presented for the elastic secant and consistent tangent predictors the form of the consistent tangent modulus is established for the case of gradient plasticity a selection of numerical examples is presented and discussed with a view to illustrating aspects of the approximation scheme and algorithms as well as features of the model of gradient plasticity adopted here

mica a multilinear ica decomposition for natural scene modeling we refine the classical independent component analysis ica decomposition using a multilinear expansion of the probability density function of the source statistics in particular we introduce a specific nonlinear system that allows us to elegantly capture the statistical dependences between the responses of the multilinear ica mica filters the resulting multilinear probability density is analytically tractable and does not require monte carlo simulations to estimate the model parameters we demonstrate the mica model on natural image textures and envision that the new model will prove useful for analyzing nonstationarity natural images using natural scene statistics models

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution

weak galerkin finite element methods for a second order elliptic variational inequality abstract a weak galerkin finite element method is used to determine approximate solutions of an elliptic variational inequality three sets of basis functions are employed the first has constant values inside each element and on each edge the second has constant values inside each element but is a linear polynomial on each edge and the third has linear polynomials inside each element and on each edge error estimates including convergence rates are derived for all three cases numerical results were provided to illustrate the theoretical results to show that the choice of how the obstacle function is approximated affects those rates and to show the super convergence for piecewise constant basis functions

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form

topographic independent component analysis in ordinary independent component analysis the components are assumed to be completely independent and they do not necessarily have any meaningful order relationships in practice however the estimated independent components are often not at all independent we propose that this residual dependence structure could be used to define a topographic order for the components in particular a distance between two components could be defined using their higher order correlations and this distance could be used to create a topographic representation thus we obtain a linear decomposition into approximately independent components where the dependence of two components is approximated by the proximity of the components in the topographic representation

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed