a sparse collocation method for solving time dependent hjb equations using multivariate b splines this paper presents a sparse collocation method for solving the time dependent hamilton jacobi bellman hjb equation associated with the continuous time optimal control problem on a fixed finite time horizon with integral cost functional through casting the problem in a recursive framework using the value iteration procedure the value functions of every iteration step is approximated with a time varying multivariate simplex b spline on a certain state domain of interest in the collocation scheme the time dependent coefficients of the spline function are further approximated with ordinary univariate b splines to yield a discretization for the value function fully in terms of piece wise polynomials the b spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems the proposed algorithm is demonstrated on a pair of benchmark example problems simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation

a radial basis collocation method for hamilton jacobi bellman equations in this paper we propose a semi meshless discretization method for the approximation of viscosity solutions to a first order hamilton jacobi bellman hjb equation governing a class of nonlinear optimal feedback control problems in this method the spatial discretization is based on a collocation scheme using the global radial basis functions rbfs and the time variable is discretized by a standard two level time stepping scheme with a splitting parameter q a stability analysis is performed showing that even for the explicit scheme that q 0 the method is stable in time since the time discretization is consistent the method is also convergent in time numerical results performed to verify the usefulness of the method demonstrate that the method gives accurate approximations to both of the control and state variables

the hermite transform applications it is demonstrated how the hermite transform can be used for image coding and analysis hierarchical coding structures based on increasingly specified basic patterns i e general 2 d patterns general 1 d patterns and specific 1 d patterns such as edges and corners are presented in the image coding application the relation with existing pyramid coders is described a new coding scheme based on local one dimensional image approximations is introduced in the image analysis application the relation between the hermite transform and existing line edge detection schemes is described it is shown that by concentrating on more specific patterns the coding efficiency can be increased since fewer coefficients have to be coded meanwhile sufficient descriptive power can be maintained for approximating the most interesting features in natural images

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

classified vector quantization of images vector quantization vq provides many attractive features for image coding with high compression ratios however initial studies of image coding with vq have revealed several difficulties most notably edge degradation and high computational complexity we address these two problems and propose a new coding method classified vector quantization cvq which is based on a composite source model blocks with distinct perceptual features such as edges are generated from different subsources i e belong to different classes in cvq a classifier determines the class for each block and the block is then coded with a vector quantizer designed specifically for that class we obtain better perceptual quality with significantly lower complexity with cvq when compared to ordinary vq we demonstrate with cvq visual quality which is comparable to that produced by existing coders of similar complexity for rates in the range 0 6 1 0 bits pixel

bivariate splines of various degrees for numerical solution of partial differential equations bivariate splines with various degrees are considered in this paper a matrix form of the extended smoothness conditions for these splines is presented upon this form the multivariate spline method for numerical solution of partial differential equations pdes proposed by awanou lai and wenston in the multivariate spline method for scattered data fitting and numerical solutions of partial differential equations in wavelets and splines g chen and m j lai eds nashboro press brentwood tn 2006 pp 24 76 is generalized to obtain a new spline method it is observed that combined with prelocal refinement of triangulation and automatic degree raising over triangles of interest the new spline method of bivariate splines of various degrees is able to solve linear pdes very effectively and efficiently

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

a relevant two qubit bell inequality inequivalent to the chsh inequality we computationally investigate the complete polytope of bell inequalities for two particles with small numbers of possible measurements and outcomes our approach is limited by pitowsky s connection of this problem to the computationally hard np problem despite this we find that there are very few relevant inequivalent inequalities for small numbers for example in the case with three possible 2 outcome measurements on each particle there is just one new inequality we describe mixed 2 qubit states which violate this inequality but not the chsh the new inequality also illustrates a sharing of bi partite non locality between three qubits something not seen using the chsh inequality it also inspires us to discover a class of bell inequalities with m possible n outcome measurements on each particle

a p version two level spline method for 2d navier stokes equations we propose a p version two level method for the spline solutions of the navier stokes equations in stream function formulation the new two level method can significantly accelerate the corresponding one level method especially when the degrees of the spline spaces are greater than 5 in two dimensional case the convergence analysis as well as the error estimation is given which is asymptotic optimal with respect to the degrees of the spline spaces several numerical examples are presented to validate the theoretical analysis and show the efficiency of the proposed two level spline method

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

feature specific vector quantization of images a new interband vector quantization of a human vision based image representation is presented the feature specific vector quantizer fvq is suited for data compression beyond second order decorrelation the scheme is derived from statistical investigations of natural images and the processing principles of biological vision systems the initial stage of the coding algorithm is a hierarchical and orientation selective analytic bandpass decomposition realized by even and odd symmetric filter pairs that are modeled after the simple cells of the visual cortex the outputs of each even and odd symmetric filter pair are interpreted as real and imaginary parts of an analytic bandpass signal which is transformed into a local amplitude and a local phase component according to the operation of cortical complex cells feature specific multidimensional vector quantization is realized by combining the amplitude phase samples of all orientation filters of one resolution layer the resulting vectors are suited for a classification of the local image features with respect to their intrinsic dimensionality and enable the exploitation of higher order statistical dependencies between the subbands this final step is closely related to the operation of cortical hypercomplex or end stopped cells the codebook design is based on statistical as well as psychophysical and neurophysiological considerations and avoids the common shortcomings of perceptually implausible mathematical error criteria the resulting perceptual quality of compressed images is superior to that obtained with standard vector quantizers of comparable complexity

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

a vector quantizer for the laplace source the low complexity nearly optimal vector quantizer vq is a generalization of t r fischer s 1986 pyramid vq and is similar in structure to the unrestricted polar quantizers previously presented for the independent gaussian source an analysis of performance is presented with results for both the product code pyramid vq and the unrestricted version this analysis although asymptotic in nature helps to demonstrate the performance advantages of the vq implementation issues of the vq are discussed nonasymptotic results are considered in particular the author presents an approximate design algorithm for finite bit rate and demonstrates the usefulness of this vq through several example designs with monte carlo simulations of performance for the restricted form the pyramid vq the author provides further implementational information and low dimension analytical results

stronger violation of local theories with equalities bell type inequalities are used to test local realism against quantum theory in this paper we consider a two party system with two settings and two possible outcomes on each side and derive equalities in local theories which are violated by quantum theory by a factor of 1 522 tolerating 0 586 fraction of white noise admixture which is twice that of the previous results

trivariate spline approximations of 3d navier stokes equations we present numerical approximations of the 3d steady state navier stokes equations in velocity pressure formulation using trivariate splines of arbitrary degree d and arbitrary smoothness r with r d using functional arguments we derive the discrete navier stokes equations in terms of b coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain smoothness conditions boundary conditions and the divergence free condition are enforced through lagrange multipliers the pressure is computed by solving a poisson equation with neumann boundary conditions we have implemented this approach in matlab and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem

model reduction of parabolic pdes using multivariate splines abstracta new methodology is presented for model reduction of linear parabolic partial differential equations pdes on general geometries using multivariate splines on triangulations state space descriptions are derived that can be used for control design this method uses galerkin projection with b splines to derive a finite set of ordinary differential equations odes any desired smoothness conditions between elements as well as the boundary conditions are flexibly imposed as a system of side constraints on the set of odes projection of the set of odes on the null space of the system of side constraints naturally produces a reduced order model that satisfies these constraints this method can be applied for both in domain control and boundary control of parabolic pdes with spatially varying coefficients on general geometries the reduction method is applied to design and implement feedback controllers for stabilisation of a 1 d unstable heat equation and a more challenging 2 d reaction convection d

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

brief paper a neural network solution for fixed final time optimal control of nonlinear systems in this paper fixed final time optimal control laws using neural networks and hjb equations for general affine in the input nonlinear systems are proposed the method utilizes kronecker matrix methods along with neural network approximation over a compact set to solve a time varying hjb equation the result is a neural network feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated on an example

online actor critic algorithm to solve the continuous time infinite horizon optimal control problem in this paper we discuss an online algorithm based on policy iteration for learning the continuous time ct optimal control solution with infinite horizon cost for nonlinear systems with known dynamics that is the algorithm learns online in real time the solution to the optimal control design hj equation this method finds in real time suitable approximations of both the optimal cost and the optimal control policy while also guaranteeing closed loop stability we present an online adaptive algorithm implemented as an actor critic structure which involves simultaneous continuous time adaptation of both actor and critic neural networks we call this synchronous policy iteration a persistence of excitation condition is shown to guarantee convergence of the critic to the actual optimal value function novel tuning algorithms are given for both critic and actor networks with extra nonstandard terms in the actor tuning law being required to guarantee closed loop dynamical stability the convergence to the optimal controller is proven and the stability of the system is also guaranteed simulation examples show the effectiveness of the new algorithm

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

fast simplicial quadrature based finite element operators using bernstein polynomials we derive low complexity matrix free finite element algorithms for simplicial bernstein polynomials on simplices our techniques based on a sparse representation of differentiation and special block structure in the matrices evaluating b form polynomials at warped gauss points apply to variable coefficient problems as well as constant coefficient ones thus extending our results in kirby numer math 2011 in press

differential constraints for bounded recursive identification with multivariate splines the ability to perform online model identification for nonlinear systems with unknown dynamics is essential to any adaptive model based control system in this paper a new differential equality constrained recursive least squares estimator for multivariate simplex splines is presented that is able to perform online model identification and bounded model extrapolation in the framework of a model based control system a new type of linear constraints the differential constraints are used as differential boundary conditions within the recursive estimator which limit polynomial divergence when extrapolating data the differential constraints are derived with a new one step matrix form of the de casteljau algorithm which reduces their formulation into a single matrix multiplication the recursive estimator is demonstrated on a bivariate dataset where it is shown to provide a speedup of two orders of magnitude over an ordinary least squares batch method additionally it is demonstrated that inclusion of differential constraints in the least squares optimization scheme can prevent polynomial divergence close to edges of the model domain where local data coverage may be insufficient a situation often encountered with global recursive data approximation

spline element methods allowing multiple level hanging nodes abstract in this paper a systematic algorithm of spline element method allowing multiple level hanging nodes on triangular mesh is proposed the classical smoothness condition of the piecewise polynomials on triangular mesh is generalized to the case of meshes with hanging nodes and is treated as the linear constraints of the system equations we then derive a concise formulation for such linear constraints according to the special hierarchical configurations of the hanging nodes moreover an efficient linear equation solver is proposed when applied to solve the elliptic partial differential equations numerical examples are illustrated to show the accuracy of the proposed methods and the comparisons are made between different levels of hanging nodes

subband coding of images using vector quantization a novel two dimensional subband coding technique is presented that can be applied to images as well as speech a frequency band decomposition of the image is carried out by means of 2d separable quadrature mirror filters which split the image spectrum into 16 equal rate subbands these 16 parallel subband signals are regarded as a 16 dimensional vector source and coded as such using vector quantization in the asymptotic case of high bit rates a theoretical analysis yields that a lower bound to the gain is attainable by choosing this approach over scalar quantization of each subband with an optimal bit allocation it is shown that vector quantization in this scheme has several advantages over coding the subbands separately experimental results are given and it is shown the scheme has a performance that is comparable to that of more complex coding techniques

brief paper a new approach to linear regression with multivariate splines a new methodology for creating highly accurate static nonlinear maps from scattered multivariate data is presented this new methodology uses the b form polynomials of multivariate simplex splines in a new linear regression scheme this allows the use of standard parameter estimation techniques for estimating the b coefficients of the multivariate simplex splines we present a generalized least squares estimator for the b coefficients and show how the estimated b coefficient variances lead to a new model quality assessment measure in the form of the b coefficient variance surface the new modeling methodology is demonstrated on a nonlinear scattered bivariate dataset

the hermite transform theory the author introduces a scheme for the local processing of visual information called the hermite transform the problem is addressed from the point of view of image coding and therefore the scheme is presented as an analysis resynthesis system the objectives of the present work however are not restricted to coding the analysis part is designed so that it can also serve applications in the area of computer vision indeed derivatives of gaussians which have found widespread application in feature detection over the past few years play a central role in the hermite analysis it is also argued that the proposed processing scheme is in close agreement with current insight into the image processing that is carried out by the human visual system in particular it is demonstrated that the hermite transform is in better agreement with human visual modeling than gabor expansions

bell inequalities for arbitrarily high dimensional systems we develop a novel approach to bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy this is used to construct a family of bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise in particular our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

continuity and convergence in rational triangular bezier spline based isogeometric analysis abstract this paper presents a method for isogeometric analysis using rational triangular bezier splines rtbs where optimal convergence rates are achieved in this method both the geometry and the physical field are represented by bivariate splines in bernstein bezier form over the triangulation of a domain from a given physical domain bounded by nurbs curves a parametric domain and its triangulation are constructed by imposing continuity constraints on bezier ordinates we obtain a set of global c r smooth basis functions convergence analysis shows that isogeometric analysis with such c r rtbs basis can deliver the optimal rate of convergence provided that the c r geometric map remains unchanged during the refinement process this condition can be satisfied by constructing a pre refinement geometric map that is sufficiently smooth numerical experiments verify that optimal rates of convergence are achieved for poisson and linear elasticity problems

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author