error analysis of trigonometric integrators for semilinear wave equations an error analysis of trigonometric integrators or exponential integrators applied to spatial semidiscretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given in particular optimal second order convergence is shown requiring only that the exact solution is of finite energy the analysis is uniform in the spatial discretization parameter it covers the impulse method which coincides with the method of deuflhard and the mollified impulse method of garcia archilla sanz serna and skeel as well as the trigonometric methods proposed by hairer and lubich and by grimm and hochbruck the analysis can also be used to explain the convergence behavior of the stormer verlet leapfrog time discretization

sample adaptive product quantization asymptotic analysis and examples vector quantization vq is an efficient data compression technique for low bit rate applications however the major disadvantage of vq is that its encoding complexity increases dramatically with bit rate and vector dimension even though one can use a modified vq such as the tree structured vq to reduce the encoding complexity it is practically infeasible to implement such a vq at a high bit rate or for large vector dimensions because of the huge memory requirement for its codebook and for the very large training sequence requirement to overcome this difficulty a structurally constrained vq called the sample adaptive product quantizer sapq has recently been proposed we extensively study the sapq that is based on scalar quantizers in order to exploit the simplicity of scalar quantization through an asymptotic distortion result we discuss the achievable performance and the relationship between distortion and encoding complexity we illustrate that even when sapq is based on scalar quantizers it can provide vq level performance we also provide numerical results that show a 2 3 db improvement over the lloyd max 1982 1960 quantizers for data rates above 4 b point

numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

deformed heisenberg algebra origin of q calculus the intimate connection between q deformed heisenberg uncertainty relation and the jackson derivative jd based on q basic numbers has been noted in the literature the purpose of this work is to establish this connection in a clear and self consistent formulation and to show explicitly how the jd arises naturally we utilize a holomorphic representation to arrive at the correct algebra to describe q deformed bosons we investigate the algebra of q fermions and point out how different it is from the theory of q bosons we show that the holomorphic representation for q fermions is indeed feasible in the framework of the theory of generalized fermions we also examine several different q algebras in the context of the modified heisenberg equation of motion

general embedded quantization for wavelet based lossy image coding embedded quantization is a mechanism employed by many lossy image codecs to progressively refine the distortion of a transformed image currently the most common approach to do so in the context of wavelet based image coding is to couple uniform scalar deadzone quantization usdq with bitplane coding bpc usdq bpc is convenient for its practicality and has proved to achieve competitive coding performance but the quantizer established by this scheme does not allow major variations this paper introduces a multistage quantization scheme named general embedded quantization geq that provides more flexibility to the quantizer geq schemes can be devised for specific decoding rates achieving optimal coding performance practical approaches of geq schemes achieve coding performance similar to that of usdq bpc while requiring fewer quantization stages the performance achieved by geq is evaluated in this paper through experimental results carried out in the framework of modern image coding systems

overcoming order reduction in diffusion reaction splitting part 2 oblique boundary conditions splitting methods constitute a well established class of numerical schemes for the time integration of partial differential equations their main advantages over more traditional schemes are computational efficiency and superior geometric properties in the presence of nontrivial boundary conditions however splitting methods usually suffer from order reduction and some additional loss of accuracy for diffusion reaction equations with inhomogeneous oblique boundary conditions a modification of the classic second order strang splitting is proposed that successfully solves the problem of order reduction the same correction also improves the accuracy of the classic first order lie splitting the proposed modification only depends on the available boundary data and in the case of non dirichlet boundary conditions on the computed numerical solution consequently this modification can be implemented in an efficient way which makes the modified splitting schemes superior to their classic versions the fr

quantification of errors for operator split advection diffusion calculations abstract multiphysics simulations frequently are composed from highly optimized solvers for physical subprocesses through the use of temporal operator splitting the subphysics are evolved sequentially passing information between solver components as needed it is often useful yet difficult to determine how the simulation error depends on the temporal splitting method the time step size and the discretization parameters in the solver components this paper proposes a framework to decompose the total error in a quantity of interest for an advection diffusion simulation into two primary contributions that due to the operator splitting as implied by the structure of the code and that due to the discretization errors from the component solvers the method is applied to the advection diffusion equation with boundaries the advection and diffusion operators require separate boundary conditions upon splitting the specification of these impacts the splitting error contribution computational examples demonstrate that the proposed method successfully identifies the splitting error contribution including that induced by a suboptimal imposition of boundary conditions and the decrease in the splitting error contribution in moving to a higher order splitting method the discretization error contribution is decomposed further into contributions attributed to the separate advection and diffusion solvers

avoiding order reduction when integrating reaction diffusion boundary value problems with exponential splitting methods abstract in this paper we suggest a technique to avoid order reduction in time when integrating reaction diffusion boundary value problems under non homogeneous boundary conditions with exponential splitting methods more precisely we consider lie trotter and strang splitting methods and dirichlet neumann and robin boundary conditions beginning from an abstract framework in banach spaces a thorough error analysis after full discretization is performed and some numerical results are shown which corroborate the theoretical results

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

avoiding order reduction when integrating nonlinear schrodinger equation with strang method in this paper a technique is suggested to avoid order reduction when using strang method to integrate nonlinear schrodinger equation subject to time dependent dirichlet boundary conditions the computational cost of this technique is negligible compared to that of the method itself at least when the timestepsize is fixed moreover a thorough error analysis is given as well as a modification of the technique which allows to conserve the symmetry of the method while retaining its second order

basic sturm liouville problems this paper is devoted to studying a q analogue of sturm liouville eigenvalue problems we formulate a self adjoint q difference operator in a hilbert space some of the properties of the eigenvalues and the eigenfunctions are discussed green s function is constructed and the problem in question is inverted into a q type fredholm integral operator with a symmetric kernel the set of eigenfunctions is shown to be a complete orthogonal set in the hilbert space examples involving basic trigonometric functions are involved

on sampling theory and basic sturm liouville systems we investigate the sampling theory associated with basic sturm liouville eigenvalue problems we derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of jackson s type the kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic green s function of the basic sturm liouville systems examples involving basic sine and cosine transforms are given

on uniform quantization with various distortion measures corresp upper bounds are presented for the difference in entropy between that of a uniform scalar quantizer and that of any n dimensional quantizer the bounds are universal in the sense that they suit every input density and every value of distortion bounds were found for some common distortion criteria

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

design of sample adaptive product quantizers for noisy channels channel optimized vector quantization covq has proven to be an effective joint source channel coding technique that makes the underlying quantizer robust to channel noise unfortunately covq retains the high encoding complexity of the standard vector quantizer vq for medium to high quantization dimensions and moderate to good channel conditions a technique called sample adaptive product quantization sapq was recently introduced by kim and shroff to reduce the complexity of the vq while achieving comparable distortions in this letter we generalize the design of sapq for the case of memoryless noisy channels by optimizing the quantizer with respect to both source and channel statistics numerical results demonstrate that the channel optimized sapq cosapq achieves comparable performance to the covq within 0 2 db while maintaining considerably lower encoding complexity up to half of that of covq and storage requirements robustness of the cosapq system against channel mismatch is also examined

time stampless adaptive nonuniform sampling for stochastic signals in this paper we introduce a time stampless adaptive nonuniform sampling tans framework in which time increments between samples are determined by a function of the m most recent increments and sample values since only past samples are used in computing time increments it is not necessary to save sampling times time stamps for use in the reconstruction process we focus on two tans schemes for discrete time stochastic signals a greedy method and a method based on dynamic programming we analyze the performances of these schemes by computing or bounding their trade offs between sampling rate and expected reconstruction distortion for autoregressive and markovian signals simulation results support the analysis of the sampling schemes we show that by opportunistically adapting to local signal characteristics tans may lead to improved power efficiency in some applications

sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure e f x eta the expected value of an error weighting function f x eta where x is a random variable to be quantized where the probability density function p x describing x is continuous and positive on some finite or infinite interval and zero outside it and where eta is the quantization error the function f x eta is assumed convex and symmetric in eta and zero only for eta 0 it is shown that in the cases of f x eta eta 2 and f x eta eta the simple condition of concavity of in p x is sufficient for uniqueness of a locally optimal quantizer

transform coding with backward adaptive updates the karhunen loeve transform klt is optimal for transform coding of a gaussian source this is established for all scale invariant quantizers generalizing previous results a backward adaptive technique for combating the data dependence of the klt is proposed and analyzed when the adapted transform converges to a klt the scheme is universal among transform coders a variety of convergence results are proven

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

the design of two dimensional quantizers using prequantization the theoretical advantages of two dimensional quantization over univariate quantization have been studied in the literature however in many cases there is no known implementation for the two dimensional quantizer that can operate in real time a new approach to the design of two dimensional quantizers is presented this technique called prequantization is used to design two dimensional quantizers that operate in real time the importance of prequantization is demonstrated by the design of the optimum uniform two dimensional hexagonal quantizer additional examples are given to illustrate the flexibility of this design approach

optimum quantizer performance for a class of non gaussian memoryless sources the performance of optimum quantizers subject to an entropy constraint is studied for a wide class of memoryless sources for a general distortion criterion necessary conditions are developed for optimality and a recursive algorithm is described for obtaining the optimum quantizer under a mean square error criterion the performance of entropy encoded uniform quantization of memoryless gaussian sources is well known to be within 0 255 bits sample of the rate distortion bound at relatively high rates despite claims to the contrary it is demonstrated that similar performance can be expected for a wide range of memoryless sources indeed for the cases considered the worst case performance is observed to be less than 0 3 bits sample from the rate distortion bound and in most cases this disparity is less at iow rates

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

2 step scalar deadzone quantization for bitplane image coding modern lossy image coding systems generate a quality progressive codestream that truncated at increasing rates produces an image with decreasing distortion quality progressivity is commonly provided by an embedded quantizer that employs uniform scalar deadzone quantization usdq together with a bitplane coding strategy this paper introduces a 2 step scalar deadzone quantization 2sdq scheme that achieves same coding performance as that of usdq while reducing the coding passes and the emitted symbols of the bitplane coding engine this serves to reduce the computational costs of the codec and or to code high dynamic range images the main insights behind 2sdq are the use of two quantization step sizes that approximate wavelet coefficients with more or less precision depending on their density and a rate distortion optimization technique that adjusts the distortion decreases produced when coding 2sdq indexes the integration of 2sdq in current codecs is straightforward the applicability and efficiency of 2sdq are demonstrated within the framework of jpeg2000

an algorithm for uniform vector quantizer design a vector quantizer maps a k dimensional vector into one of a finite set of output vectors or points although certain lattices have been shown to have desirable properties for vector quantization applications there are as yet no algorithms available in the quantization literature for building quantizers based on these lattices an algorithm for designing vector quantizers based on the root lattices a n d n and e n and their duals is presented also a coding scheme that has general applicability to all vector quantizers is presented a four dimensional uniform vector quantizer is used to encode laplacian and gamma distributed sources at entropy rates of one and two bits sample and is demonstrated to achieve performance that compares favorably with the rate distortion bound and other scalar and vector quantizers finally an application using uniform four and eight dimensional vector quantizers for encoding the discrete cosine transform coefficients of an image at 0 5 bit pel is presented which visibly illustrates the performance advantage of vector quantization over scalar quantization

efficient boundary corrected strang splitting abstract strang splitting is a well established tool for the numerical integration of evolution equations it allows the application of tailored integrators for different parts of the vector field however it is also prone to order reduction in the case of non trivial boundary conditions this order reduction can be remedied by correcting the boundary values of the intermediate splitting step in this paper three different approaches for constructing such a correction in the case of inhomogeneous dirichlet neumann and mixed boundary conditions are presented numerical examples that illustrate the effectiveness and benefits of these corrections are included

on error bounds for the gautschi type exponential integrator applied to oscillatory second order differential equations this paper studies a numerical method for second order oscillatory differential equations in which high frequency oscillations are generated by a linear time and or solution dependent part for constant linear part it is known that the method allows second order error bounds independent of the product of the step size with the frequencies and is therefore a long time step method most real world problems are not of that kind and it is important to study more general equations the analysis in this paper shows that one obtains second order error bounds even in the case of a time and or solution dependent linear part if the matrix is evaluated at averaged positions

radii of starlikeness and convexity of some q bessel functions abstract geometric properties of the jackson and hahn exton q bessel functions are studied for each of them three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane for each of the six functions we determine the radii of starlikeness and convexity precisely by using their hadamard factorization these are q generalizations of some known results for bessel functions of the first kind the characterization of entire functions from the laguerre polya class via hyperbolic polynomials plays an important role in this paper moreover the interlacing property of the zeros of jackson and hahn exton q bessel functions and their derivatives is also useful in the proof of the main results we also deduce a necessary and sufficient condition for the close to convexity of a normalized jackson q bessel function and its derivatives some open problems are proposed at the end of the paper

a modified gautschi s method without order reduction when integrating boundary value nonlinear wave problems abstract in this paper we analyse the order reduction which turns up when integrating nonlinear wave problems with non homogeneous and time dependent boundary conditions with the well known gautschi s method moreover a technique is suggested to avoid that order reduction so that the classical local order 4 and global order 2 are recovered on the other hand the usual approximation for the derivative which is used together with this method is also analysed and a substantial improvement is suggested some numerical results are shown which corroborate the performed analysis

sampling theory associated with q difference equations of the sturm liouville type we will show that if the kernel k x t in the representation f t a0k x t u t dqt with u l2q 0 a is a solution of a second order q sturm liouville boundary problem then f admits a representation as a sampling formula of the form f t n 0f n w t w n t n where n is the nth eigenvalue of the associated q sturm liouville boundary problem and w t is the q wronskian of two solutions selected in a specified way

a note on the gautschi type method for oscillatory second order differential equations the gautschi type method has been proposed by hochbruck and lubich for oscillatory second order differential equations they conjecture that this method allows for a uniform error bound independent of the size of the system the conjecture is proved in this note

an algorithm for source coding subject to a fidelity criterion based on string matching a practical suboptimal universal block source coding scheme subject to a fidelity criterion is proposed the algorithm is an extension of the lempel ziv algorithm and is based on string matching with distortion it is shown that given average distortion d 0 the algorithm achieves a rate of exceeding r d 2 for a large class of sources and distortion measures tighter bounds on the rate are derived for discrete memoryless sources and for memoryless gaussian sources

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

order reduction and how to avoid it when explicit runge kutta nystrom methods are used to solve linear partial differential equations in this paper we study the order reduction which turns up when explicit runge kutta nystrom methods are used to discretize linear second order hyperbolic equations by means of the method of lines the order observed in practice including its fractional part is obtained it is also proved that the order reduction can be completely avoided taking the boundary values of the intermediate stages of the time semidiscretization the numerical experiments confirm that the optimal order can be reached

sampling theorems associated with biorthogonal q bessel functions this paper deals with the derivation of sampling theorems associated with q biorthogonal systems we derive interpolation expansions for qhankel transforms whose kernels are the second type q bessel functions j 2 z q 0 0 q 1 we investigate the eigenvalue problem whose solutions are the q bessel functions as well as its adjoint special cases and applications involving the associated q sine function are given the results are based on the conjecture that a family of q bessel functions of the second kind is a riesz basis clues are given to support our claim

trellis based scalar vector quantizer for memoryless sources the paper describes a structured vector quantization approach for stationary memoryless sources that combines the scalar vector quantizer svq ideas laroia and farvardin 1993 with trellis coded quantization marcellin and fischer 1990 the resulting quantizer is called the trellis based scalar vector quantizer tb svq the svq structure allows the tb svq to realize a large boundary gain while the underlying trellis code enables it to achieve a significant portion of the total granular gain for large block lengths and powerful possibly complex trellis codes the tb svq can in principle achieve the rate distortion bound as indicated by the results obtained even for reasonable block lengths and relatively simple trellis codes the tb svq outperforms all other fixed rate quantizers at reasonable complexity

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

image subband coding using context based classification and adaptive quantization adaptive compression methods have been a key component of many proposed subband or wavelet image coding techniques this paper deals with a particular type of adaptive subband image coding where we focus on the image coder s ability to adjust itself on the fly to the spatially varying statistical nature of image contents this backward adaptation is distinguished from more frequently used forward adaptation in that forward adaptation selects the best operating parameters from a predesigned set and thus uses considerable amount of side information in order for the encoder and the decoder to operate with the same parameters specifically we present backward adaptive quantization using a new context based classification technique which classifies each subband coefficient based on the surrounding quantized coefficients we couple this classification with online parametric adaptation of the quantizer applied to each class a simple uniform threshold quantizer is employed as the baseline quantizer for which adaptation is achieved our subband image coder based on the proposed adaptive classification quantization idea exhibits excellent rate distortion performance in particular at very low rates for popular test images it is comparable or superior to most of the state of the art coders in the literature

spectral zeta functions for q bessel equations explicit formulae for the zeta functions of the zeros of hahn exton and jackson s q bessel functions are derived they can be regarded as spectral sum rules for some discrete quantum billiards

adaptive scalar quantization without side information in this paper we introduce a novel technique for adaptive scalar quantization adaptivity is useful in applications including image compression where the statistics of the source are either not known a priori or will change over time our algorithm uses previously quantized samples to estimate the distribution of the source and does not require that side information be sent in order to adapt to changing source statistics our quantization scheme is thus backward adaptive we propose that an adaptive quantizer can be separated into two building blocks namely model estimation and quantizer design the model estimation produces an estimate of the changing source probability density function which is then used to redesign the quantizer using standard techniques we introduce nonparametric estimation techniques that only assume smoothness of the input distribution we discuss the various sources of error in our estimation and argue that for a wide class of sources with a smooth probability density function pdf we provide a good approximation to a universal quantizer with the approximation becoming better as the rate increases we study the performance of our scheme and show how the loss due to adaptivity is minimal in typical scenarios in particular we provide examples and show how our technique can achieve signal to noise ratios within 0 05 db of the optimal lloyd max quantizer for a memoryless source while achieving over 1 5 db gain over a fixed quantizer for a bimodal source