mismatch in high rate entropy constrained vector quantization bucklew s 1984 high rate vector quantizer mismatch result is extended from fixed rate coding to variable rate coding using a lagrangian formulation it is shown that if an asymptotically high rate optimal sequence of variable rate codes is designed for a k dimensional probability density function pdf g and then applied to another pdf f for which f g is bounded then the resulting mismatch or loss of performance from the optimal possible is given by the relative entropy or kullback leibler 1968 divergence i f spl par g it is also shown that under the same assumptions an asymptotically optimal code sequence for g can be converted to an asymptotically optimal code sequence for a mismatched source f by modifying only the lossless component of the code applications to quantizer design using uniform and gaussian densities are described including a high rate analog to the shannon rate distortion result of sakrison 1975 and lapidoth 1997 showing that the gaussian is the worst case for lossy compression of a source with known covariance by coupling the mismatch result with composite quantizers the worst case properties of uniform and gaussian densities are extended to conditionally uniform and gaussian densities which provides a lloyd clustering algorithm for fitting mixtures to general densities

solution of 3d inverse scattering problems by combined inverse equivalent current and finite element methods an approach combining boundary integral and finite element methods is introduced for the solution of three dimensional inverse electromagnetic medium scattering problems based on the equivalence principle unknown equivalent electric and magnetic surface current densities on a closed surface are utilized to decompose the inverse medium problem into two parts a linear radiation problem and a nonlinear cavity problem the first problem is formulated by a boundary integral equation the computational burden of which is reduced by employing the multilevel fast multipole method mlfmm reconstructed cauchy data on the surface allows the utilization of the lorentz reciprocity and the poynting s theorems exploiting these theorems the noise level and an initial guess are estimated for the cavity problem moreover it is possible to determine whether the material is lossy or not in the second problem the estimated surface currents form inhomogeneous boundary conditions of the cavity problem the cavity problem is formulated by the finite element technique and solved iteratively by the gauss newton method to reconstruct the properties of the object regularization for both the first and the second problems is achieved by a krylov subspace method the proposed method is tested against both synthetic and experimental data and promising reconstruction results are obtained

inverse medium scattering problems for electromagnetic waves consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in mathbb r 3 in this paper existence and uniqueness of the variational problem for forward scattering are established an energy estimate for the scattered field with a uniform bound with respect to the wavenumber is obtained in the case of low frequency on which the born approximation is based a continuation method for the inverse medium scattering problem which reconstructs the scatterer of an inhomogeneous medium from boundary measurements of the scattered wave is developed the algorithm requires multifrequency scattering data using an initial guess from the born approximation each update is obtained via recursive linearization on the wavenumber k by solving one forward problem and one adjoint problem of maxwell s equations

a framework for the adaptive finite element solution of large scale inverse problems since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging efficient methods are indispensable in this paper we will introduce a framework for the efficient solution of such problems this comprises the use of adaptive finite element schemes solvers for the large linear systems arising from discretization and methods to treat additional information in the form of inequality constraints on the parameter to be recovered the methods to be developed will be based on an all at once approach in which the inverse problem is solved through a lagrangian formulation the main feature of the paper is the use of a continuous function space setting to formulate algorithms in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed this entails that steps such as the description of a newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions on the other hand this approach avoids the dependence of finite dimensional norms on the mesh size making individual steps of the algorithm comparable even if they used differently refined meshes numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10 000 parameters

variance mismatched fixed rate scalar quantization of laplacian sources the paper investigates the mean squared error mse distortion of a variance mismatched fixed rate scalar quantizer that is optimal or asymptotically optimal in the minimum mse sense for a laplacian density with standard deviation q but is applied to another with standard deviation p a nitadori like formula is discovered for an optimal quantizer when p q 1 2 also it is shown that the distortion expressions derived rigorously for asymptotically optimal quantile quantizers essentially confirm the heuristically obtained previous result that the distortion decreases as 1 n3 for the heavy mismatch of 3 2 as lnn n2 for the critical mismatch of 3 2 and as 1 n2 for the mild mismatch of 1 and the resulting distortion decreases as in n n2 1 p

on experimental results and explicit exact solutions for the generalized boussinesq type equation we implemented a decomposition method for approximating the solution of the generalized boussinesq type equation by using this scheme the explicit exact solution is calculated in the form of a convergent power series with easily computable components to illustrate the application of this method numerical results are derived by using the calculated components of the decomposition series the obtained results are found to be in good agreement with the exact solution

high rate vector quantization for detection we investigate high rate quantization for various detection and reconstruction loss criteria a new distortion measure is introduced which accounts for global loss in best attainable binary hypothesis testing performance the distortion criterion is related to the area under the receiver operating characteristic roc curve specifically motivated by sanov s theorem we define a performance curve as the trajectory of the pair of optimal asymptotic type i and type ii error rates of the most powerful neyman pearson test of the hypotheses the distortion measure is then defined as the difference between the area under the curve auc of the optimal pre encoded hypothesis test and the auc of the optimal post encoded hypothesis test as compared to many previously introduced distortion measures for decision making this distortion measure has the advantage of being independent of any detection thresholds or priors on the hypotheses which are generally difficult to specify in the code design process a high resolution zador gersho type of analysis is applied to characterize the point density and the inertial profile associated with the optimal high rate vector quantizer the analysis applies to a restricted class of high rate quantizers that have bounded cells with vanishing volumes the optimal point density is used to specify a lloyd type algorithm which allocates its finest resolution to regions where the gradient of the pre encoded likelihood ratio has greatest magnitude

high resolution quantization theory and the vector quantizer advantage the authors consider how much performance advantage a fixed dimensional vector quantizer can gain over a scalar quantizer they collect several results from high resolution or asymptotic in rate quantization theory and use them to identify source and system characteristics that contribute to the vector quantizer advantage one well known advantage is due to improvement in the space filling properties of polytopes as the dimension increases others depend on the source s memory and marginal density shape the advantages are used to gain insight into product transform lattice predictive pyramid and universal quantizers although numerical prediction consistently overestimated gains in low rate 1 bit sample experiments the theoretical insights may be useful even at these rates

quantization on the grassmann manifold the problem of quantization in an euclidean space with unitary constraints can be formulated as an unconstrained problem on a grassmann manifold such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver due to the constraints the distortion rate analysis developed for euclidean spaces cannot be applied directly this paper extends gersho s asymptotic large rate small distortion distortion bounds to the case when the source is distributed on the complex grassmann manifold the special structure of the grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in euclidean spaces

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

asymptotically optimal model estimation for quantization using high rate theory approximations we introduce flexible practical quantizers based on possibly non gaussian models in both the constrained resolution cr and the constrained entropy cases we derive model estimation criteria optimizing asymptotic with increasing rate quantizer performance we show that in the cr case the optimal criterion is different from the maximum likelihood criterion commonly used for that purpose and introduce a new criterion that we call constrained resolution minimum description length cr mdl we apply these principles to the generalized gaussian scaled mixture model which is accurate for many real world signals we provide an explanation of the reason why the cr mdl improves quantization performance in the cr case and show that cr mdl can compensate for a possible mismatch between model and data distribution thus this criterion is of a great interest for practical applications our experiments apply the new quantization method to controllable artificial data and to the commonly used modulated lapped transform representation of audio signals we show that both the cr mdl criterion and a non gaussian modeling have significant advantages

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

lagrangian vector quantization with combined entropy and codebook size constraints in this paper the lagrangian formulation of variable rate vector quantization is extended to quantization with simultaneous constraints on entropy and codebook size including variable and fixed rate quantization as special cases the formulation leads to a lloyd quantizer design algorithm and generalizations of gersho s approximations characterizing optimal performance for asymptotically large rate a variation of gersho s approach is shown to yield rigorous results partially characterizing the asymptotically optimal performance

entropy density and mismatch in high rate scalar quantization with renyi entropy constraint properties of scalar quantization with th power distortion and constrained renyi entropy of order are investigated for an asymptotically high rate optimal sequence of quantizers the contribution to the renyi entropy due to source values in a fixed interval is identified in terms of the entropy density of the quantizer sequence this extends results related to the well known point density concept in optimal fixed rate quantization a dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion the distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined this extends bucklew s fixed rate and gray et al s variable rate mismatch results to general values of the entropy order parameter

solitary wave solutions for a generalized hirota satsuma coupled kdv equation the solitary wave solutions of a generalized hirota satsuma coupled kdv equation are obtained exactly by using the decomposition method the solutions were calculated in the form of a convergent power series with easily computable components a variety of initial value system is considered the convergence of the method as applied to the coupled kdv equation is illustrated numerically the present algorithm performs extremely well in terms of accuracy and simplicity

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

lattice and trellis quantization with lattice and trellis bounded codebooks high rate theory for memoryless sources high rate lattice and trellis quantizers for nonuniform sources are introduced and analyzed the performance of these quantizers is determined by two separable quantities the granular gain and the boundary gain which are determined by the shapes of the granular cells and of the support region respectively the granular gain and boundary gain are the duals of shaping and coding gain in data transmission applications using this duality it is shown for gaussian sources that the ultimate achievable boundary gain with high rate lattice bounded lattice codebooks is the same as the ultimate gain that can be obtained from variable rate entropy coding it is observed that if lattice codebooks can achieve the ultimate granular gain of 0 255 b per dimension then lattice bounded lattice codebooks can approach the rate distortion limit the performance of lattice quantizers is compared to that of optimum vector quantizers

a two stage method for inverse medium scattering we present a novel numerical method to the time harmonic inverse medium scattering problem of recovering the refractive index from noisy near field scattered data the approach consists of two stages one pruning step of detecting the scatterer support and one resolution enhancing step with nonsmooth mixed regularization the first step is strictly direct and of sampling type and it faithfully detects the scatterer support the second step is an innovative application of nonsmooth mixed regularization and it accurately resolves the scatterer size as well as intensities the nonsmooth model can be efficiently solved by a semi smooth newton type method numerical results for two and three dimensional examples indicate that the new approach is accurate computationally efficient and robust with respect to data noise

asymptotic distribution of the errors in scalar and vector quantizers high rate or asymptotic quantization theory has found formulas for the average squared length more generally the qth moment of the length of the error produced by various scalar and vector quantizers with many quantization points in contrast this paper finds an asymptotic formula for the probability density of the length of the error and in certain special cases for the probability density of the multidimensional error vector itself the latter can be used to analyze the distortion of two stage vector quantization the former permits one to learn about the point density and cell shapes of a quantizer from a histogram of quantization error lengths histograms of the error lengths in simulations agree well with the derived formulas also presented are a number of properties of the error density including the relationship between the error density the point density and the cell shapes the fact that its qth moment equals bennett s integral a formula for the average distortion of a scalar or vector quantizer and the fact that for stationary sources the marginals of the multidimensional error density of an optimal vector quantizer with large dimension are approximately i i d gaussian

an infinite element for maxwell s equations a new infinite element ie approximation for steady state maxwell s equations is proposed the ie is compatible with a new class of hp finite element fe discretizations presented in 13 32 following 12 a convergence analysis for the proposed coupled fe ie approximations is presented

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

hp finite element discretisation of the electrical impedance tomography problem abstract this paper is concerned with the accurate recovery of the permittivity and conductivity from the solution of the electrical impedance tomography eit inverse problem given the knowledge of either the neumann to dirichlet or the dirichlet to neumann map the linearised eit inverse problem for the fourier regime is discretised by hp finite elements which allows the accurate calculation of the sensitivity matrix entries the variational algorithm of knowles 1998 31 is extended to allow the recovery of both material parameters by solution of the eit problem in the laplace regime the associated forward problem is again discretised by hp finite elements a series of numerical results are included to demonstrate the effectiveness of the proposed techniques

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation we find the explicit and numerical traveling wave solutions for a coupled kdv equation and a coupled mkdv equation by using the decomposition method with help of symbolic computation by using this method the solutions were calculated in the form of a convergent power series with easily computable components the convergence of the method as applied to the coupled kdv and mkdv equations is illustrated numerically

a goal oriented hp adaptive finite element approach to radar scattering problems abstract the paper presents application of an hp adaptive finite element method for scattering of electromagnetic waves the main objective of the numerical analysis is to determine the characteristics of the scattered waves indicating the power being scattered at a given direction i e the radar cross section rcs this is achieved considering the scattered far field which defines rcs and which is expressed as a linear functional of the solution techniques of error estimation for the far field are considered and an h adaptive strategy leading to the fast reduction of the error of the far field is presented the simulations are performed with a three dimensional version of an hp adaptive finite element method for electromagnetics based on the hexahedral edge elements combined with infinite elements for modeling the unbounded space surrounding the scattering object

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

two results on the asymptotic performance of quantizers a necessary and sufficient condition is presented for the normalized asymptotic r th power distortion of a mismatched multidimensional quantizer to converge to a certain optimum constant known as bennett s integral the dimensionality of our results is allowed to approach infinity in order to make some universal source coding comparisons to quantization theory

a posteriori quantization of progressive matching pursuit streams this paper proposes a rate distortion optimal a posteriori quantization scheme for matching pursuit mp coefficients the a posteriori quantization applies to an mp expansion that has been generated offline and cannot benefit of any feedback loop to the encoder in order to compensate for the quantization noise the redundancy of the mp dictionary provides an indicator of the relative importance of coefficients and atom indices and subsequently on the quantization error it is used to define a universal upper bound on the decay of the coefficients sorted in decreasing order of magnitude a new quantization scheme is then derived where this bound is used as an oracle for the design of an optimal a posteriori quantizer the latter turns the exponentially distributed coefficient entropy constrained quantization problem into a simple uniform quantization problem using simulations with random dictionaries we show that the proposed exponentially upper bounded quantization euq clearly outperforms classical schemes stepping on the ideal oracle based approach a suboptimal adaptive scheme is then designed that approximates the euq but still outperforms competing quantization methods in terms of rate distortion characteristics finally the proposed quantization method is studied in the context of image coding it performs similarly to state of the art coding methods and even better at low rates while interestingly providing a progressive stream that is very easy to transcode and adapt to changing rate constraints

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

on entropy constrained vector quantization using gaussian mixture models a flexible and low complexity entropy constrained vector quantizer ecvq scheme based on gaussian mixture models gmms lattice quantization and arithmetic coding is presented the source is assumed to have a probability density function of a gmm an input vector is first classified to one of the mixture components and the karhunen loeve transform of the selected mixture component is applied to the vector followed by quantization using a lattice structured codebook finally the scalar elements of the quantized vector are entropy coded sequentially using a specially designed arithmetic coder the computational complexity of the proposed scheme is low and independent of the coding rate in both the encoder and the decoder therefore the proposed scheme serves as a lower complexity alternative to the gmm based ecvq proposed by gardner subramaniam and rao the performance of the proposed scheme is analyzed under a high rate assumption and quantified for a given gmm the practical performance of the scheme was evaluated through simulations on both synthetic and speech line spectral frequency lsf vectors for lsf quantization the proposed scheme has a comparable performance to at rates relevant for speech coding 20 28 bits per vector with lower computational complexity

fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem we propose a method for computing the scattering of a time harmonic electromagnetic wave in a medium with a locally perturbed refractive index we prove that it converges of arbitrarily high order depending on the smoothness of the refractive index and that the time complexity is o n 3log n as an application we solve the corresponding inverse problem using a preconditioned newton method and discuss how to compute the frechet derivative of the solution operator and its adjoint by the proposed method the performance of both the forward and the inverse solver is illustrated in a number of numerical experiments

asymptotic formulas for variance mismatched fixed rate scalar quantization of a gaussian source asymptotic formulas are derived for the mean squared error mse distortion and the snr of a variance mismatched fixed rate scalar quantizer designed mse optimally for a gaussian source it is discovered that in the case that the standard deviation ratio of the applied to density to the designed for density is greater than 3 2 the snr in db increases at large rate r as 9 03r 2 15 1 1 2 log10r plus a constant that depends only on

numerical solution of an inverse medium scattering problem for maxwell s equations at fixed frequency consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in r 3 in this paper well posedness of the variational problem for the direct scattering is examined an energy estimate for the scattered field is obtained on which the born approximation is based a regularized recursive linearization method for the inverse medium scattering which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field is developed the algorithm requires only single frequency data using an initial guess from the born approximation each update is obtained via continuation on the spatial frequency of a two parameter family of plane waves by solving one direct problem and one adjoint problem of the maxwell equation

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

application of the fem with adaptivity for electromagnetic inverse medium scattering problems abstract the paper presents a study of solution of inverse medium scattering problems for 3d time harmonic electromagnetics the solution is based on minimization of the misfit function between the observed scattered fields and trial solutions of direct problems for the sought approximate distribution of electric permittivity the sources of illuminating waves and the observation points of the scattered fields are located in the nearfield the scatterer the simulations of the direct problems are performed with the finite element method application of adaptive mesh refinement to improve accuracy of solutions is illustrated tests with real and complex valued distributions of electric permittivity and with complete and incomplete measurement data are presented