backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

convergence of time splitting energy conserved symplectic schemes for 3d maxwell s equations we propose two symplectic and two non symplectic schemes for 3d maxwell s equations based on the exponential operator splitting technique and fourier pseudo spectral method these schemes are efficient and unconditionally stable and also preserve four discrete energy conservation laws simultaneously the error estimates of the schemes are obtained by using some special techniques and the energy method numerical results confirm the theoretical analysis the numerical comparison with some existing methods show the good performance of the proposed schemes

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

optimal error estimate for energy preserving splitting schemes for maxwell s equations abstract two efficient splitting schemes are developed for 3d maxwell s equations the schemes are energy preserving and unconditionally stable while being implemented explicitly rigorous optimal error estimates are established for the proposed schemes and especially the constant in the error estimates is only o t numerical results confirm the theoretical analysis and numerical comparison with some existing methods shows the good performance of the present schemes

sharp results in the integral form john nirenberg inequality we consider the strong form of the john nirenberg inequality for the l 2 based bmo we construct explicit bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant as well as the precise bound on the inequality s range of validity both previously unknown the results for the two cases are substantially different the paper not only gives another instance in the short list of such explicit calculations but also presents the bellman function method as a sequence of clear steps adaptable to a wide variety of applications

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

dispersive properties of multisymplectic integrators multisymplectic ms integrators i e numerical schemes which exactly preserve a discrete space time symplectic structure are a new class of structure preserving algorithms for solving hamiltonian pdes in this paper we examine the dispersive properties of ms integrators for the linear wave and sine gordon equations in particular a leapfrog in space and time scheme a member of the lobatto runge kutta family of methods and the preissman box scheme are considered we find the numerical dispersion relations are monotonic and that the sign of the group velocity is preserved the group velocity dispersion gvd is found to provide significant information and succinctly explain the qualitative differences in the numerical solutions obtained with the different schemes further the numerical dispersion relations for the linearized sine gordon equation provides information on the ability of the ms integrators to capture the sine gordon dynamics we are able to link the numerical dispersion relations to the total energy of the various methods thus providing information on the coarse grid behavior of ms integrators in the nonlinear regime

paraunitary filter banks over finite fields in real and complex fields unitary and paraunitary pu matrices have found many applications in signal processing there has been interest in extending these ideas to the case of finite fields we study the theory of pu filter banks fbs in gf q with q prime various properties of unitary and pu matrices in finite fields are studied in particular a number of factorization theorems are given we show that i all unitary matrices in gf q are factorizable in terms of householder like matrices and permutation matrices and ii the class of first order pu matrices the lapped orthogonal transform in finite fields can always be expressed as a product of degree one or degree two building blocks if q 2 we do not need degree two building blocks while many properties of pu matrices in finite fields are similar to those of pu matrices in complex field there are a number of differences for example unlike the conventional pu systems in finite fields there are pu systems that are unfactorizable in terms of smaller building blocks in fact in the special case of 2 spl times 2 systems all pu matrices that are factorizable in terms of degree one building blocks are diagonal matrices we derive results for both the cases of gf 2 and gf q with q 2 even though they share some similarities there are many differences between these two cases

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

an energy conserving method for stochastic maxwell equations with multiplicative noise abstract in this paper it is shown that three dimensional stochastic maxwell equations with multiplicative noise are stochastic hamiltonian partial differential equations possessing a geometric structure i e stochastic multi symplectic conservation law and the energy of system is a conservative quantity almost surely we propose a stochastic multi symplectic energy conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy the mean square convergence result of the method in temporal direction is tested numerically and numerical comparisons with finite difference method are also investigated

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

wavelet descriptors for multiresolution recognition of handprinted characters we present a novel set of shape descriptors that represents a digitized pattern in a concise way and that is particularly well suited for the recognition of handprinted characters the descriptor set is derived from the wavelet transform of a pattern s contour the approach is closely related to feature extraction methods by fourier series expansion the motivation to use an orthonormal wavelet basis rather than the fourier basis is that wavelet coefficients provide localized frequency information and that wavelets allow us to decompose a function into a multiresolution hierarchy of localized frequency bands we describe a character recognition system that relies upon wavelet descriptors to simultaneously analyze character shape at multiple levels of resolution the system was trained and tested on a large database of more than 6000 samples of handprinted alphanumeric characters the results show that wavelet descriptors are an efficient representation that can provide for reliable recognition in problems with large input variability

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

theory of locally concave functions and its applications to sharp estimates of integral functionals we prove a duality theorem the computation of certain bellman functions is usually based on as a byproduct we obtain sharp results about the norms of monotonic rearrangements the main novelty of our approach is a special class of martingales and an extremal problem on this class which is dual to the minimization problem for locally concave functions

new multisymplectic self adjoint scheme and its composition scheme for the time domain maxwell s equations in this paper we investigate euler box scheme for bridges multisymplectic form of maxwell s equations a new multisymplectic scheme is derived for maxwell s equations we prove that it is also a self adjoint scheme in time direction the multisymplecticity of composition schemes based on the new scheme is also discussed two numerical examples are proposed to indicate that the derived multisymplectic schemes are effective when used to integrate the 2 1 dimensional maxwell s equations

wavelet thresholding techniques for power spectrum estimation estimation of the power spectrum s f of a stationary random process can be viewed as a nonparametric statistical estimation problem we introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown s f this approach offers the ability to capture statistically significant components of ln s f at different resolution levels and guarantees nonnegativity of the spectrum estimator the spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non gaussian noise we propose a wavelet thresholding technique to solve this problem under specified noise resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables the thresholds are computed using a saddle point approximation to the distribution of the noise coefficients

a perceptually lossless model based texture compression technique in natural scenes still images as well as sequences backgrounds and objects surfaces usually have a textural structure therefore in order to efficiently code images it is crucial to investigate the texture compression problem in this paper a perceptually lossless synthesis by analysis texture coding method is presented the proposed approach is model based the parameters of the model consist of a binary excitation signal and the parsimonious representation of the reconstruction filter the estimated parameters which allow to one synthesize at the decoder site a texture that is perceptually indistinguishable from the original one are then compressed using a lossless strategy which is based on a fast binary wavelet transformation specifically tailored to binary images the proposed method leads to very good perceptual results superior to those of existing techniques

wavelet transforms associated with finite cyclic groups multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers an important class of multiresolution decompositions are the laplacian pyramid schemes in which the resolution is successively halved by recursively low pass filtering the signal under analysis and decimating it by a factor of two in general the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete time fourier analysis of sequences of complex numbers an analogous framework is developed for the multiresolution analysis of finite length sequences of elements from arbitrary fields attention is restricted to sequences of length 2 sup n for n a positive integer so that the resolution may be recursively halved to completion as in finite length fourier analysis a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields

hartley transforms over finite fields a general framework is presented for constructing transforms in the field of the input which have a convolution like property the construction is carried out over finite fields but is shown to be valid over the real and complex fields as well it is shown that these basefield transforms can be viewed as projections of the discrete fourier transform dft and that they exist for all lengths n for which the dft is defined the convolution property of the basefield transforms is derived and a condition for such transforms to have the self inverse property is given also fast algorithms for these basefield transforms are developed showing gains when compared to computations using the fft application of the methodology to hartley transforms over r leads to a simple derivation of fast algorithms for computing real hartley transforms

multi symplectic runge kutta nystrom methods for nonlinear schrodinger equations with variable coefficients in this paper we consider runge kutta nystrom rkn methods applied to nonlinear schrodinger equations with variable coefficients nlsevc concatenating symplectic nystrom methods in spatial direction and symplectic runge kutta methods in temporal direction for nlsevc leads to multi symplectic integrators i e to numerical methods which preserve the multi symplectic conservation law mscl we present the corresponding discrete version of mscl it is shown that the multi symplectic rkn methods preserve not only the global symplectic structure in time but also local and global discrete charge conservation laws under periodic boundary conditions we present a 4 order multi symplectic rkn method and use it in numerical simulation of quasi periodically solitary waves for nlsevc and we compare the multi symplectic rkn method with a non multi symplectic rkn method on the errors of numerical solutions the numerical errors of discrete energy discrete momentum and discrete charge the precise conservation of discrete charge under the multi symplectic rkn discretizations is attested numerically some numerical superiorities of the multi symplectic rkn methods are revealed

on the preservation of phase space structure under multisymplectic discretization in this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and fourier spectral approximations multisymplectic ms schemes are developed for two benchmark nonlinear wave equations the sine gordon and nonlinear schrodinger equations we examine the implications of preserving the ms structure under discretization on the numerical scheme s ability to preserve phase space structure as measured by the nonlinear spectrum of the governing equation we find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws dynamical invariants and complicated phase space structures

a wreath product group approach to signal and image processing i multiresolution analysis we propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and in particular image processing we pay significant attention to groups constructed as wreath products of cyclic groups within this large class of groups our approach recovers the discrete fourier transform dft the haar wavelet transform various multichannel pyramid filter banks and other aspects of multiresolution analysis as special cases of a more general phenomenon in addition the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals our approach relies on a synthesis of ideas found in the early work of holmes 1987 1990 karpovsky and trachtenberg 1985 and others on noncommutative filtering as well as diaconis s 1989 spectral analysis approach to understanding data

energy dissipation preserving birkhoffian multi symplectic methods for maxwell s equations with dissipation terms in this paper we propose two new energy dissipation preserving birkhoffian multi symplectic methods birkhoffian and birkhoffian box for maxwell s equations with dissipation terms after investigating the non autonomous and autonomous birkhoffian formalism for maxwell s equations with dissipation terms we first apply a novel generating functional theory to the non autonomous birkhoffian formalism to propose our birkhoffian scheme and then implement a central box method to the autonomous birkhoffian formalism to derive the birkhoffian box scheme we have obtained four formal local conservation laws and three formal energy global conservation laws we have also proved that both of our derived schemes preserve the discrete version of the global local conservation laws furthermore the stability dissipation and dispersion relations are also investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable dissipation preserving for maxwell s equations in a perfectly matched layer pml medium and have second order accuracy in both time and space numerical experiments for problems with exact theoretical results are given to demonstrate that the birkhoffian multi symplectic schemes are much more accurate in preserving energy than both the exponential finite difference time domain fdtd method and traditional hamiltonian scheme we also solve the electromagnetic pulse emp propagation problem and the numerical results show that the birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation

bellman function for extremal problems in bmo in this paper we develop the method of nding sharp estimates by using a bellman function in such a form the method appears in the proofs of the classical john nirenberg inequality and l p estimations of bmo functions in the present paper we elaborate a method of solving the boundary value problem for the homogeneous monge amp ere equation in a parabolic strip for suciently smooth boundary conditions in such a way we have obtained an algorithm for constructing an exact bellman function for a large class of integral functionals on the bmo space

texture classification using logical operators in this paper a new algorithm for texture classification based on logical operators is presented operators constructed from logical building blocks are convolved with texture images an optimal set of six operators are selected based on their texture discrimination ability the responses are then converted to standard deviation matrices computed over a sliding window zonal sampling features are computed from these matrices a feature selection process is applied and the new set of features are used for texture classification classification of several natural and synthetic texture images are presented demonstrating the excellent performance of the logical operator method the computational superiority and classification accuracy of the algorithm is demonstrated by comparison with other popular methods experiments with different classifiers and feature normalization are also presented the euclidean distance classifier is found to perform best with this algorithm the algorithm involves only convolutions and simple arithmetic in the various stages which allows faster implementations the algorithm is applicable to different types of classification problems which is demonstrated by segmentation of remote sensing images compressed and reconstructed images and industrial images

multi symplectic methods for the ito type coupled kdv equation abstract in this paper we find that the ito type coupled kdv equation can be written as a multi symplectic hamiltonian partial differential equation pde then multi symplectic fourier pseudospectral method and multi symlpectic wavelet collocation method are constructed for this equation in the numerical experiments we show the effectiveness of the proposed methods some comparisons between the proposed methods are also made with respect to global conservation properties

wavelet transform domain filters a spatially selective noise filtration technique wavelet transforms are multiresolution decompositions that can be used to analyze signals and images they describe a signal by the power at each scale and position edges can be located very effectively in the wavelet transform domain a spatially selective noise filtration technique based on the direct spatial correlation of the wavelet transform at several adjacent scales is introduced a high correlation is used to infer that there is a significant feature at the position that should be passed through the filter the authors have tested the technique on simulated signals phantom images and real mr images it is found that the technique can reduce noise contents in signals and images by more than 80 while maintaining at least 80 of the value of the gradient at most edges the authors did not observe any gibbs ringing or significant resolution loss on the filtered images artifacts that arose from the filtration are very small and local the noise filtration technique is quite robust there are many possible extensions of the technique the authors see its applications in spatially dependent noise filtration edge detection and enhancement image restoration and motion artifact removal they have compared the performance of the technique to that of the weiner filter and found it to be superior

tilings of the time frequency plane construction of arbitrary orthogonal bases and fast tiling algorithms the authors consider expansions which give arbitrary orthonormal tilings of the time frequency plane these differ from the short time fourier transform wavelet transform and wavelet packets tilings in that they change over time they show how this can be achieved using time varying orthogonal tree structures which preserve orthogonality even across transitions the method is based on the construction of boundary and transition filters these allow us to construct essentially arbitrary tilings time varying modulated lapped transforms are a special case where both boundary and overlapping solutions are possible with filters obtained by modulation they present a double tree algorithm which for a given signal decides on the best binary segmentation in both time and frequency that is it is a joint optimization of time and frequency splitting the algorithm is optimal for additive cost functions e g rate distortion and results in time varying best bases the main application of which is for compression of nonstationary signals experiments on test signals are presented

numerical analysis of a multi symplectic scheme for the time domain maxwell s equations in this paper we analyze qualitative properties of the first multi symplectic scheme for two dimensional maxwell s equations we prove that the scheme is unconditionally stable and convergent non dissipative and divergence free the numerical dispersion relation of the scheme is shown to converge to the exact dispersion relation of the maxwell equations we also present some numerical results to confirm our theoretical results

multi symplectic runge kutta methods for nonlinear dirac equations in this paper we consider the multi symplectic runge kutta msrk methods applied to the nonlinear dirac equation in relativistic quantum physics based on a discovery of the multi symplecticity of the equation in particular the conservation of energy momentum and charge under msrk discretizations is investigated by means of numerical experiments and numerical comparisons with non msrk methods numerical experiments presented reveal that msrk methods applied to the nonlinear dirac equation preserve exactly conservation laws of charge and momentum and conserve the energy conservation in the corresponding numerical accuracy to the method utilized it is verified numerically that msrk methods are stable and convergent with respect to the conservation laws of energy momentum and charge and msrk methods preserve not only the inner geometric structure of the equation but also some crucial conservative properties in quantum physics a remarkable advantage of msrk methods applied to the nonlinear dirac equation is the precise preservation of charge conservation law

high order compact splitting multisymplectic method for the coupled nonlinear schrodinger equations in this paper we develop a new kind of multisymplectic integrator for the coupled nonlinear schrodinger cnls equations the cnls equations are cast into multisymplectic formulation then it is split into a linear multisymplectic formulation and a nonlinear hamiltonian system the space of the linear subproblem is approximated by a high order compact hoc method which is new in multisymplectic context the nonlinear subproblem is integrated exactly for splitting and approximation we utilize an hoc sms integrator its stability and conservation laws are investigated in theory numerical results are presented to demonstrate the accuracy conservation laws and to simulate various solitons as well for the hoc sms integrator they are consistent with our theoretical analysis

a discrete line integral method of order two for the lorentz force system in this paper we apply the boole discrete line integral to solve the lorentz force system which is written as a non canonical hamiltonian system the method is exactly energy conserving for polynomial hamiltonians of degree 4 in any other case the energy can also be conserved approximatively with comparison to well used boris method numerical experiments are presented to demonstrate the energy preserving property of the method

symplectic structure preserving integrators for the two dimensional gross pitaevskii equation for bec a b s t r a c t symplectic integrators have been developed for solving the two dimensional gross pitaevskii equation the equation is transformed into a hamiltonian form with symplectic structure then symplectic integrators including the midpoint rule and a splitting symplectic scheme are developed for treating this equation it is shown that the proposed codes fulfill the discrete charge conservation law furthermore the global error of the numerical solution is theoretically estimated the theoretical analysis is supported by some numerical simulations

wavelet based representations for a class of self similar signals with application to fractal modulation a potentially important family of self similar signals based upon a deterministic scale invariance characterization is introduced these signals which are referred to as dy homogeneous signals because they generalize the well known homogeneous functions have highly convenient representations in terms of orthonormal wavelet bases in particular wavelet representations can be exploited to construct orthonormal self similar bases for these signals the spectral and fractal characteristics of dy homogeneous signals make them appealing candidates for use in a number of applications as one potential example their use in a communications based context is considered specifically a strategy for embedding information into a dy homogeneous waveform on multiple time scales is developed this multirate modulation strategy called fractal modulation is potentially well suited for use with noisy channels of simultaneously unknown duration and bandwidth

multiscale multiresolution genetic algorithm with a golden sectioned population composition a new genetic algorithm ga strategy called the multiscale multiresolution ga is proposed for expediting solution convergence by orders of magnitude the motivation for this development was to apply gas to a certain class of large optimization problems which are otherwise nearly impossible to solve for the algorithm standard binary design variables are binary wavelet transformed to multiscale design variables by working with the multiscale variables evolution can proceed in multiresolution converged solutions at a low resolution are reused as a part of individuals of the initial population for the next resolution evolution it is shown that the best solution convergence can be achieved if three initial population groups having different fitness levels are mixed at the golden section ratio an analogy between cell division and the proposed multiscale multiresolution strategy is made the specific applications of the developed method are made in topology optimization problems

the multi symplectic fourier pseudospectral method for solving two dimensional hamiltonian pdes in this paper the multi symplectic fourier pseudospectral msfp method is generalized to solve two dimensional hamiltonian pdes with periodic boundary conditions using the fourier pseudospectral method in the space of the two dimensional hamiltonian pde 2d hpde the semi discrete system obtained is proved to have semi discrete multi symplectic conservation laws and a global symplecticity conservation law then the implicit midpoint rule is employed for time integration to obtain the msfp method for the 2d hpde the fully discrete multi symplectic conservation laws are also obtained in addition the proposed method is applied to solve the zakharov kuznetsov zk equation and the kadomtsev petviashvili kp equation numerical experiments on soliton solutions of the zk equation and the kp equation show the high accuracy and effectiveness of the proposed method

mapping equivalence for symbolic sequences theory and applications processing of symbolic sequences represented by mapping of symbolic data into numerical signals is commonly used in various applications it is a particularly popular approach in genomic and proteomic sequence analysis numerous mappings of symbolic sequences have been proposed for various applications it is unclear however whether the processing of symbolic data provides an artifact of the numerical mapping or is an inherent property of the symbolic data this issue has been long ignored in the engineering and scientific literature it is possible that many of the results obtained in symbolic signal processing could be a byproduct of the mapping and might not shed any light on the underlying properties embedded in the data moreover in many applications conflicting conclusions may arise due to the choice of the mapping used for numerical representation of symbolic data in this paper we present a novel framework for the analysis of the equivalence of the mappings used for numerical representation of symbolic data we present strong and weak equivalence properties and rely on signal correlation to characterize equivalent mappings we derive theoretical results which establish conditions for consistency among numerical mappings of symbolic data furthermore we introduce an abstract mapping model for symbolic sequences and extend the notion of equivalence to an algebraic framework finally we illustrate our theoretical results by application to dna sequence analysis

stable and efficient numerical schemes for two dimensional maxwell equations in lossy medium abstract in this paper some stable and efficient numerical methods are developed for two dimensional maxwell equations to avoid solving large scale algebraic equations we use local one dimensional lod technique and split the original equations into several lod equations then the resulted lod equations are discretized by wendroff scheme which is usually applied to simulate hyperbolic type equations the lie trotter composition and strang composition are employed to improve the computational efficiency and the convergent accurate the space direction is approximated by high order compact method some benchmark symbols such as stability conservation laws of the schemes are analyzed some numerical examples confirm the theoretical analysis numerical results show that the new schemes can not only accurately simulate the electromagnetic waves including profiles amplitude but also capture the energy structures exactly

preservation of physical properties of stochastic maxwell equations with additive noise via stochastic multi symplectic methods stochastic maxwell equations with additive noise are a system of stochastic hamiltonian partial differential equations intrinsically possessing the stochastic multi symplectic conservation law it is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic maxwell equations with additive noise preserves the divergence in the sense of expectation moreover we propose three novel stochastic multi symplectic methods to discretize stochastic maxwell equations in order to investigate the preservation of these properties numerically we make theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence meanwhile we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method especially the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case numerical experiments are performed to verify our theoretical results

a method of sieves for multiresolution spectrum estimation and radar imaging a method of sieves using splines is proposed for regularizing maximum likelihood estimates of power spectra this method has several important properties including the flexibility to be used at multiple resolution levels the resolution level is defined in terms of the support of the polynomial b splines used an expression for the optimal rate of growth of the sieve is derived using a discrepancy measure derived from the kullback leibler divergence of parameterized density functions while the sieves may be defined on nonuniform grids in the case of uniform grids the optimal sieve size corresponds to an optimal resolution iterative algorithms for obtaining the maximum likelihood sieve estimates are derived applications to spectrum estimation and radar imaging are proposed

a stochastic multi symplectic scheme for stochastic maxwell equations with additive noise abstract in this paper we investigate a stochastic multi symplectic method for stochastic maxwell equations with additive noise based on the stochastic version of variational principle we find a way to obtain the stochastic multi symplectic structure of three dimensional 3 d stochastic maxwell equations with additive noise we propose a stochastic multi symplectic scheme and show that it preserves the stochastic multi symplectic conservation law and the local and global stochastic energy dissipative properties which the equations themselves possess numerical experiments are performed to verify the numerical behaviors of the stochastic multi symplectic scheme

a split step approach for the 3 d maxwell s equations split step procedures have previously been used successfully in a number of situations e g for hamiltonian systems such as certain nonlinear wave equations in this study we note that one particular way to write the 3 d maxwell s equations separates these into two parts requiring only the solution of six uncoupled 1 d wave equations the approach allows arbitrary orders of accuracy in both time and space and features in many cases unconditional stability

conformal conservation laws and geometric integration for damped hamiltonian pdes conformal conservation laws are defined and derived for a class of multi symplectic equations with added dissipation in particular the conservation laws of energy and momentum are considered along with those that arise from linear symmetries numerical methods that preserve these conformal conservation laws are presented in detail providing a framework for proving a numerical method exactly preserves the dissipative properties considered the conformal methods are compared analytically and numerically to standard conservative methods which includes a thorough inspection of numerical solution behavior for linear equations damped klein gordon and sine gordon equations and a damped nonlinear schrodinger equation are used as examples to demonstrate the results

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

nonlinear multiresolution signal decomposition schemes ii morphological wavelets interest in multiresolution techniques for signal processing and analysis is increasing steadily an important instance of such a technique is the so called pyramid decomposition scheme this paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis the proposed theory is based on the following ingredients 1 the pyramid consists of a finite or infinite number of levels such that the information content decreases toward higher levels and 2 each step toward a higher level is implemented by an information reducing analysis operator whereas each step toward a lower level is implemented by an information preserving synthesis operator one basic assumption is necessary synthesis followed by analysis yields the identity operator meaning that no information is lost by these two consecutive steps several examples of pyramid decomposition schemes are shown to be instances of the proposed theory a particular class of linear pyramids morphological skeleton decompositions the morphological haar pyramid median pyramids etc furthermore the paper makes a distinction between single scale and multiscale decomposition schemes i e schemes without or with sample reduction finally the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks

two energy conserved splitting methods for three dimensional time domain maxwell s equations and the convergence analysis we devote the present paper to high accuracy energy preserving s avf 2 and s avf 4 schemes for the three dimensional time domain maxwell s equations based on the exponential operator splitting technique the fourier pseudospectral method and the averaged vector field method to obtain the present schemes the key is to propose the splitting methods for maxwell s equations in which all subsystems should hold the same hamiltonian the proposed schemes are energy preserving high order accurate and unconditionally stable while being implemented explicitly both schemes capture four energy invariants simultaneously rigorous error estimates of the schemes are established in the discrete l 2 norm the theoretical results show that the s avf 2 s avf 4 scheme converges with spectral accuracy in space and second order fourth order accuracy in time respectively numerical results support the theoretical analysis

energy dissipation splitting finite difference time domain method for maxwell equations with perfectly matched layers abstract in this paper we develop a novel kind of energy dissipation splitting finite difference time domain scheme for solving two dimensional maxwell equations with perfectly matched layers the discrete energy dissipation law convergence dispersion relation and stability are investigated for the scheme theoretical analysis shows that the scheme is unconditionally stable and of second order accuracy both in time and space numerical experiments confirm the theoretical results

local structure preserving algorithms for the good boussinesq equation in this paper we derive a series of local structure preserving algorithms for the good boussinesq equation including multisymplectic geometric structure preserving algorithms local energy preserving algorithms and local momentum preserving algorithms the outstanding advantage of the proposed algorithms is that they conserve these local structures in any time space region exactly for example the proposed local energy preserving algorithms preserve the local energy conservation law in any local domain therefore the local structure preserving algorithms overcome the shortage of global structure preserving algorithms on the boundary conditions especially with suitable boundary conditions such as periodic or homogeneous boundary conditions the local structure preserving algorithms will be global structure preserving algorithms numerical results verify the theoretical analysis

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

numerical dispersion analysis of a multi symplectic scheme for the three dimensional maxwell s equations in this paper we study a multi symplectic scheme for three dimensional maxwell s equations in a simple medium this is a system of pdes with multi symplectic structures we prove that this multi symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence furthermore we extend the discussion to several dispersion properties of the multi symplectic scheme including the numerical dispersion relation the numerical group velocity the effect of large time steps and the cfl condition

the multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this article we consider partitioned runge kutta prk methods for hamiltonian partial differential equations pdes and present some sufficient conditions for multi symplecticity of prk methods of hamiltonian pdes

semi implicit operator splitting pade method for higher order nonlinear schrodinger equations abstract we propose a semi implicit finite difference operator splitting pade ospd method for solving the higher order nonlinear schrodinger equation which describes the optical soliton wave propagation in fibers the method achieves fourth order of accuracy in space and has been proven to be stable by linear stability analysis numerical experiments and comparisons are investigated to show the advantages and effectivity of the ospd method and some interesting collision behaviors are also observed

sharpening holder s inequality abstract we strengthen holder s inequality the new family of sharp inequalities we obtain might be thought of as an analog of the pythagorean theorem for the l p spaces our treatment of the subject matter is based on bellman functions of four variables

development of a wavenumber preserving scheme for solving maxwell s equations in curvilinear non staggered grids abstract in this paper a compact finite difference solver for solving the maxwell s equations in curvilinear coordinates is presented the scheme formulated in time domain can theoretically preserve zero divergence condition and scaled wavenumber characteristics in non staggered grids the inherent local conservation laws are also retained discretely all the time the space and time derivative terms are approximated to yield an equal fourth order spatial and temporal accuracy in irregular physical domain maxwell s equations are recast in terms of the contravariant and covariant field variables so that the developed dual preserving solver can be directly implemented in addition in curvilinear coordinates the four components in the metric tensor have been calculated under the guideline that the determinant of the transformation matrix is computed exactly through the computational exercises it is demonstrated that the newly proposed solver with a fairly small numerical scaled wavenumber error in curvilinear coordinates is computationally efficient for use to get the long time accurate maxwell s solutions in irregular physical domain

computing linear transforms of symbolic signals signals that represent information may be classified into two forms numeric and symbolic symbolic signals are discrete time sequences that at any particular index have a value that is a member of a finite set of symbols set membership defines the only mathematical structure that symbolic sequences satisfy consequently symbolic signals cannot be directly processed with existing signal processing algorithms designed for signals having values that are elements of a field numeric signals or a group generalizing an approach due to stoffer see biometrika vol 85 p 201 213 1998 we extend time frequency and time scale analysis techniques to symbolic signals and describe a general linear approach to developing processing algorithms for symbolic signals we illustrate our techniques by considering spectral and wavelet analyses of dna sequences

energy evolution of multi symplectic methods for maxwell equations with perfectly matched layer boundary abstract in this paper we consider the energy evolution of multi symplectic methods for three dimensional maxwell equations with perfectly matched layer boundary and present the energy evolution laws of maxwell equations under the discretization of multi symplectic yee method and general multi symplectic runge kutta methods

application of the symplectic finite difference time domain scheme to electromagnetic simulation an explicit fourth order finite difference time domain fdtd scheme using the symplectic integrator is applied to electromagnetic simulation a feasible numerical implementation of the symplectic fdtd sfdtd scheme is specified in particular new strategies for the air dielectric interface treatment and the near to far field nff transformation are presented by using the sfdtd scheme both the radiation and the scattering of three dimensional objects are computed furthermore the energy conserving characteristic hold for the sfdtd scheme is verified under long term simulation numerical results suggest that the sfdtd scheme is more efficient than the traditional fdtd method and other high order methods and can save computational resources

energy conserved splitting fdtd methods for maxwell s equations in this paper two new energy conserved splitting methods ec s fdtdi and ec s fdtdii for maxwell s equations in two dimensions are proposed both algorithms are energy conserved unconditionally stable and can be computed efficiently the convergence results are analyzed based on the energy method which show that the ec s fdtdi scheme is of first order in time and of second order in space and the ec s fdtdii scheme is of second order both in time and space we also obtain two identities of the discrete divergence of electric fields for these two schemes for the ec s fdtdii scheme we prove that the discrete divergence is of first order to approximate the exact divergence condition numerical dispersion analysis shows that these two schemes are non dissipative numerical experiments confirm well the theoretical analysis results

constraint based design of two channel paraunitary filter banks of a given length over rm gf 2 r over the real field all degree j paraunitary pu multirate systems can be described by the multiplication of j degree 1 lattice blocks and a unitary matrix over the finite field gf 2r this degree 1 factorization is not complete i e it only describes a subset of all possible pu systems in the two channel case degree 2tau blocks are also required to completely describe all pu systems over gf 2 r therefore different factorizations can be considered each factorization generates a subset of pu systems it is interesting to consider if these different factorizations have distinct properties in this correspondence we specifically consider constraining the length of the filter bank to be equal to n 1 this is required in certain error control coding applications we contrast this factorization based method with an existing trial and error approach employing the berlekamp factoring algorithm a key advantage of the proposed method is the elimination of redundant polyphase factorizations further simplifications over gf 2 identified by this method are also discussed

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

on the failure of lower square function estimates in the non homogeneous weighted setting we show that the classical a infty condition is not sufficient for a lower square function estimate in the non homogeneous weighted l 2 space we also show that under the martingale a 2 condition an estimate holds true but the optimal power of the characteristic jumps from 1 2 to 1 even when considering the classical a 2 characteristic this is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete time non homogeneous martingale transforms last we give a sharp a infty estimate for the n adic homogeneous case growing with n

the numerical study of a nonlinear one dimensional dirac equation we study the convergence of a difference scheme for a nonlinear one dimensional dirac equation also we show some numerical experiments by using this scheme

near lossless lossless compression of error diffused images using a two pass approach the current international standard the joint bilevel image experts group jbig is a representative of a bilevel image compression algorithm it compresses bilevel images with high performance but it shows relatively low performance in compressing error diffused halftone images this paper proposes a new bilevel image compression for error diffused images which is based on bayes theorem the proposed coding procedure consists of two passes it groups 2 spl times 2 dots into a cell where each cell is represented by the number of black dots and the locations of the black dots in the cell the number of black dots in the cell is encoded in the first pass and their locations are encoded in the second pass the first pass performs a near lossless compression which can be refined to be lossless by the second pass experimental results show a high compression performance for the proposed method when it is applied to error diffused images

a binary wavelet decomposition of binary images we construct a theory of binary wavelet decompositions of finite binary images the new binary wavelet transform uses simple module 2 operations it shares many of the important characteristics of the real wavelet transform in particular it yields an output similar to the thresholded output of a real wavelet transform operating on the underlying binary image we begin by introducing a new binary field transform to use as an alternative to the discrete fourier transform over gf 2 the corresponding concept of sequence spectra over gf 2 is defined using this transform a theory of binary wavelets is developed in terms of two band perfect reconstruction filter banks in gf 2 by generalizing the corresponding real field constraints of bandwidth vanishing moments and spectral content in the filters we construct a perfect reconstruction wavelet decomposition we also demonstrate the potential use of the binary wavelet decomposition in lossless image coding

the splitting finite difference time domain methods for maxwell s equations in two dimensions in this paper we consider splitting methods for maxwell s equations in two dimensions a new kind of splitting finite difference time domain methods on a staggered grid is developed the corresponding schemes consist of only two stages for each time step which are very simple in computation the rigorous analysis of the schemes is given by the energy method it is proved that the scheme is unconditionally stable and convergent for the problems with perfectly conducting boundary conditions numerical dispersion analysis and numerical experiments are presented to show the efficient performance of the proposed methods furthermore the methods are also applied to solve a scattering problem successfully

an explicit fourth order staggered finite difference time domain method for maxwell s equations a new explicit fourth order accurate staggered finite difference time domain fdtd scheme is proposed and applied to electromagnetic wave problems it is fourth order accurate in both space and time conditionally stable and highly efficient with respect to yee s scheme and still retains much of the original simplicity of yee s scheme both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented numerical examples are shown to illustrate the efficiency of the method