imposing boundary and interface conditions in multi resolution wavelet galerkin method for numerical solution of helmholtz problems abstract in this paper we propose a new formulation for numerical solution of boundary value problems using mesh free multi resolution wavelet galerkin wg method a difficulty with the wg method is imposing boundary condition constraints single scale wavelet basis functions together with a jump function approach using cubic polynomial functions have been used in solving some boundary problems in the literature ramp and cubic polynomial functions have been also used in the jump function approach described in element free galerkin context and it is proven ramp functions exactly satisfy the desirable property for the derivative of jump but the cubic polynomial functions do not in this paper a multi resolution wg method is employed and both ramp and cubic polynomial jump functions are applied obtained results in one dimension are compared with analytical solutions results obtained with ramp jump functions in two dimensions are also reported the results obtained with the wg method manifest boundary and interface conditions are accurately imposed when the ramp jump functions are applied numerical results are in good agreement with accurate solutions

solution of inverse problems in image processing by wavelet expansion we describe a wavelet based approach to linear inverse problems in image processing in this approach both the images and the linear operator to be inverted are represented by wavelet expansions leading to a multiresolution sparse matrix representation of the inverse problem the constraints for a regularized solution are enforced through wavelet expansion coefficients a unique feature of the wavelet approach is a general and consistent scheme for representing an operator in different resolutions an important problem in multigrid multiresolution processing this and the sparseness of the representation induce a multigrid algorithm the proposed approach was tested on image restoration problems and produced good results

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

solution of atomic orbitals in an interpolating wavelet basis the schrodinger equation of hydrogenic atoms and the hartree fock equations of some many electron atoms are solved using interpolating wavelets as basis functions the nonstandard operator form is used to compute operators in basis sets including multiple resolution levels we introduce an algorithm for converting matrices from nonstandard operator form to standard operator form we also consider the different components of the hamiltonian and fock operators separately and derive analytic formulas for their evaluation extension to many electron atoms is done within the hartree fock formalism convergence of atomic parameters such as orbital eigenvalues with respect to the number of resolution levels is inspected numerically for hydrogenlike atoms ions and some light many electron atoms helium lithium beryllium neon sodium magnesium and argon

a new image flux conduction model and its application to selective image smoothing a discrete image flux conduction equation which is completely new in this field is proposed the new approach starts with formulating a discrete image flux conduction equation based on the concept of heat conduction theory based on this discrete equation the status change at a time point can be directly computed from its spatial neighborhood to more accurately estimate an image flux we have used an orthogonal wavelet basis to approximate the gradient of the intensity at each point since the proposed approach is discrete by nature it is not necessary to formulate a continuous pde to fit the discrete image data set furthermore introduction of different numerical methods to solve the pde can also be avoided since the proposed approach does not require that a pde be solved it is therefore more efficient and accurate than the conventional methods experimental results obtained using both synthetic signals and real images have demonstrated that the proposed model could effectively handle the selective image smoothing problem

partial models and logics programming abstract three extensions of the standard prolog fixpoint semantics are presented called sat strong and weak using partial models models which may fail to assign truth values to all formulas each of these semantics takes negation and quantification into account all thee are conservative they agree with the conventional semantics on pure horn clause programs the sat and the strong semantics incorporate the domain closure assumption but differ on whether to assign a truth value to a classically valid formula some part of which lacks a truth value the weak semantics is similar to the strong semantics but abandons the domain closure condition and consequently all programs give rise to continuous operators in this semantics for the weak semantics a sound and complete proof procedure is given based on semantics tableaus or equivalently gentzen sequents

numerical study of time splitting and space time adaptive wavelet scheme for schrodinger equations in this paper we present a time splitting and space time adaptive wavelet scheme for solving schrodinger equations with a small planck constant the scheme is stable and it can overcome high frequency pseudo oscillation of the solution the concrete steps and numerical experiments are also given

wavelet solutions for the dirichlet problem a modified classical penalty method for solving a dirichlet boundary value problem is presented this new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains it is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast approximate calculation of boundary integral the compact support and orthonormality of the basis are essential for representing the boundary measure numerically and therefore for implementing this methodology

log polar wavelet energy signatures for rotation and scale invariant texture classification classification of texture images is important in image analysis and classification this paper proposes an effective scheme for rotation and scale invariant texture classification using log polar wavelet signatures the rotation and scale invariant feature extraction for a given image involves applying a log polar transform to eliminate the rotation and scale effects but at same time produce a row shifted log polar image which is then passed to an adaptive row shift invariant wavelet packet transform to eliminate the row shift effects so the output wavelet coefficients are rotation and scale invariant the adaptive row shift invariant wavelet packet transform is quite efficient with only o n spl middot log n complexity a feature vector of the most dominant log polar wavelet energy signatures extracted from each subband of wavelet coefficients is constructed for rotation and scale invariant texture classification in the experiments we employed a mahalanobis classifier to classify a set of 25 distinct natural textures selected from the brodatz album the experimental results based on different testing data sets for images with different orientations and scales show that the proposed classification scheme using log polar wavelet signatures outperforms two other texture classification methods its overall accuracy rate for joint rotation and scale invariance being 90 8 percent demonstrating that the extracted energy signatures are effective rotation and scale invariant features concerning its robustness to noise the classification scheme also performs better than the other methods

an efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the daubechies wavelet basis an efficient numerical quadrature is proposed for the approximate calculation of the potential energy in the context of pseudo potential electronic structure calculations with daubechies wavelet and scaling function basis sets our quadrature is also applicable in the case of adaptive spatial resolution our theoretical error estimates are confirmed by numerical test calculations of the ground state energy and wavefunction of the harmonic oscillator in one dimension with and without adaptive resolution as a byproduct we derive a filter which upon application on the scaling function coefficients of a smooth function renders the approximate grid values of this function this also allows for a fast calculation of the charge density from the wavefunction

linear cellular automata with boundary conditions the main results of the paper concern graphs of linear cellular automata with boundary conditions we show that the connected components of such graphs are direct sums of trees and cycles and we provide a complete characterization of the trees as well as enumerate the cycles of various lengths our work generalizes and clarifies results obtained previously in special cases

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

patch ordering based wavelet frame and its use in inverse problems in our previous work 1 we have introduced a redundant tree based wavelet transform rtbwt originally designed to represent functions defined on high dimensional data clouds and graphs we have further shown that rtbwt can be used as a highly effective image adaptive redundant transform that operates on an image using orderings of its overlapped patches the resulting transform is robust to corruptions in the image and thus able to efficiently represent the unknown target image even when it is calculated from its corrupted version in this paper we utilize this redundant transform as a powerful sparsity promoting regularizer in inverse problems in image processing we show that the image representation obtained with this transform is a frame expansion and derive the analysis and synthesis operators associated with it we explore the use of this frame operators to image denoising and deblurring and demonstrate in both these cases state of the art results

on the efficacy of the wavelet decomposition for high frequency vibration analyses abstract this paper reports the extraordinary ability of the wavelet decomposition for vibration analyses under the framework of rayleigh ritz method using a beam as an example daubechies wavelet scale functions are used as admissible functions for decomposing the flexural displacement of the structure along with the artificial springs at the boundary to predict vibration of an euler bernoulli beam in an extremely large frequency range it is shown that the use of wavelet basis allows reaching very high frequencies typically covering more than 1000 modes using conventional computational facility within the available numerical dynamics of the computers with no particular care needed for round off errors as a side benefit the use of spring boundary also allows handling any elastic boundary conditions through a dynamic contribution in the hamiltonian of the beam the wavelet decomposed approach combines the flexibility of the global methods and the accuracy of local methods by inheriting the versatility of the rayleigh ritz approach and the superior fitting ability of the wavelets numerical results on both free and forced vibrations are given in excellent agreement with predictions of classical methods

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

image coding using translation invariant wavelet transforms with symmetric extensions in this correspondence we address the problem of translation sensitivity of conventional wavelet transforms for two dimensional 2 d signals we propose wavelet transform algorithms that achieve the following desirable properties simultaneously i translation invariance ii reduced edge effects and iii size limitedness we apply this translation invariant biorthogonal wavelet transform with symmetric extensions to image coding applications with good results

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

multiscale adjoint waveform difference tomography using wavelets abstractfull waveform seismic inversions based on minimizing the distance between observed and predicted seismograms are in principle able to yield better resolved earth models than those minimizing misfits derived from traveltimes alone adjoint based methods provide an efficient way of calculating the gradient of the misfit function via a sequence of forward modeling steps which using spectral element codes can be carried out in realistically complex media convergence and stability of full waveform difference adjoint schemes are greatly improved when data and synthetics are progressively presented to the algorithms in a constructive multiscale approximation using a bi orthogonal wavelet transform wavelets provide the nonredundant spectral decomposition that paves the way for the inversion to proceed successively from long wavelength fitting to detailed exploration of the phases in the seismogram the choice of wavelet class and type the initial depth of the multiscale decomposition and the min

on the design of highly accurate and efficient iir and fir filters we describe a systematic method for designing highly accurate and efficient infinite impulse response iir and finite impulse response fir filters given their specifications in our approach we first meet the specifications by constructing an iir filter with possibly a large number of poles we then construct for any given accuracy an optimal iir version of such filter with a minimal number of poles finally also for any given accuracy we convert the iir filter to an efficient fir filter cascade either serial or parallel since in this fir approximation the non causal part of the iir filter only introduces an additional delay as a function of the desired accuracy our iir construction does not have to enforce causality thus we obtain a simple method for constructing linear phase filters if the specifications so require all of these procedures are accomplished via robust fast algorithms we provide several illustrative examples of our method

the applications of partial integro differential equations related to adaptive wavelet collocation methods for viscosity solutions to jump diffusion models this paper presents adaptive wavelet collocation methods for the numerical solutions to partial integro differential equations pides arising from option pricing in a market driven by jump diffusion process the first contribution of this paper lies in the formulation of the wavelet collocation schemes the integral and differential operators are formulated in the collocation setting exactly and efficiently in both adaptive and non adaptive wavelet settings the wavelet compression technique is employed to replace the full matrix corresponding to the nonlocal integral term by a sparse matrix an adaptive algorithm is developed which automatically obtains the solution on a near optimal grid the second contribution of this paper is the theoretical analysis of the wavelet collocation schemes due to the possible degeneracy of the parabolic operators classical solutions of the jump diffusion models may not exist in this paper we first prove the convergence and stability of the proposed numerical schemes under the framework of viscosity solution theory and then the numerical experiments demonstrate the accuracy and computational efficiency of the methods we developed

wavelet based schemes for linear advection dispersion equation abstract in this paper two wavelet based adaptive solvers are developed for linear advection dispersion equation the localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock like behaviour the first framework is based on wavelet galerkin and the second is based on multiscale decomposition of finite element method coiflet wavelet filter is incorporated in both the methods the main advantage of both the adaptive methods is the elimination of spurious oscillations at very high peclet number

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

convergence of cascade algorithms in sobolev spaces and integrals of wavelets the cascade algorithm with mask a and dilation m generates a sequence phi n n 1 2 ldots by the iterative process

a wavelet based multiresolution approach to large eddy simulation of turbulence the wavelet based multiresolution analysis mra technique is used to develop a modelling approach to large eddy simulation les and its associated subgrid closure problem the les equations are derived by projecting the navier stokes n s equations onto a hierarchy of wavelet spaces a numerical framework is then developed for the solution of the large and the small scale equations this is done in one dimension for the burgers equation and in three dimensions for the n s problem the proposed methodology is assessed in a priori tests on an atmospheric turbulent time series and on data from direct numerical simulation a posteriori dynamic tests are also carried out for decaying and force driven burgers turbulence

a fast adaptive wavelet collocation algorithm for multidimensional pdes a fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed the computational cost of the algorithm is independent of the dimensionality of the problem and is o n where n is the total number of collocation points the method can handle general boundary conditions the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where singularities or sharp transitions occur numerical results demonstrate the ability of the method to resolve localized structures such as shocks which change their location and steepness in space and time the present results indicate that the method has clear advantages in comparison with well established numerical algorithms

symplectic wavelet collocation method for hamiltonian wave equations this paper introduces a novel symplectic wavelet collocation method for solving nonlinear hamiltonian wave equations based on the autocorrelation functions of daubechies compactly supported scaling functions collocation method is conducted for the spatial discretization which leads to a finite dimensional hamiltonian system then appropriate symplectic scheme is employed for the integration of the hamiltonian system under the hypothesis of periodicity the properties of the resulted space differentiation matrix are analyzed in detail conservation of energy and momentum is also investigated various numerical experiments show the effectiveness of the proposed method

improved transient signal detection using a wavepacket based detector with an extended translation invariant wavelet transform this paper presents the theory of m band extended translation invariant eti wavelet transforms the eti generalizes the translation invariant wavelet transform of weiss 1933 it is shown that iteration of the eti in a tree structure provides a signal decomposition into an orthonormal wavepacket basis other properties such as translation invariance and invertibility of the transform are proven the theory is then applied to transient signal detection through development of a family of translation invariant wavepacket based detectors this family of detectors provides improved the performance over previously defined wavepacket based detectors a performance analysis is conducted roc curves generated by monte carlo simulation are presented indicating the detector performance the detector performance is demonstrated to be independent of the signal translation

a dynamic multiscale lifting computation method using daubechies wavelet an important property of wavelet multiresolution analysis is the capability to represent functions in a dynamic multiscale manner so the solution in the wavelet domain enables a hierarchical approximation to the exact solution the typical problem that arises when using daubechies wavelets in numerical analysis especially in finite element analysis is how to calculate the connection coefficients an integral of products of wavelet scaling functions or derivative operators associated with these the method to calculate multiscale connection coefficients for stiffness matrices and load vectors is presented for the first time and the algorithm of multiscale lifting computation is developed the numerical examples are given to verify the effectiveness of such a method

translation invariant data visualization with orthogonal discrete wavelets orthogonal discrete wavelet transforms can be made translation invariant by adding redundant wavelet coefficients through repeated shifting operations orthogonality is lost but isometry and compact time support can be preserved the practical application to data visualization of scalograms based on such transforms is discussed and illustrated with measured transient signals

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

interpolating wavelet collocation method of time dependent maxwell s equations characterization of electrically large optical waveguide discontinuities biorthogonal interpolating wavelets have been applied to electromagnetic field modeling through the wavelet collocation method in time domain yielding a versatile first principle algorithm for the solution of time dependent maxwell s equations with inhomogeneous media the resulting scheme maintains high accuracy while by virtue of its sub gridding capability significant reduction of the computational expenditure has been obtained the proposed method has been applied to the analysis of two dimensional dielectric waveguide discontinuities particularly for the modeling of electrically large optical waveguides where the dimension of the analyzed structure is much larger than the wavelength of the highest frequency content of the transmitted signal the proposed method has been proven to be highly efficient compared to the standard finite difference method

divergence free wavelets and high order regularization expanding on a wavelet basis the solution of an inverse problem provides several advantages first of all wavelet bases yield a natural and efficient multiresolution analysis which allows defining clear optimization strategies on nested subspaces of the solution space besides the continuous representation of the solution with wavelets enables analytical calculation of regularization integrals over the spatial domain by choosing differentiable wavelets accurate high order derivative regularizers can be efficiently designed via the basis s mass and stiffness matrices more importantly differential constraints on vector solutions such as the divergence free constraint in physics can be nicely handled with biorthogonal wavelet bases this paper illustrates these advantages in the particular case of fluid flow motion estimation numerical results on synthetic and real images of incompressible turbulence show that divergence free wavelets and high order regularizers are particularly relevant in this context

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

testing for homogeneity of variance in time series long memory wavelets and the nile river 1 we consider the problem of testing for homogeneity of variance in a time series with long memorystructure wedemonstratethat atest whose nullhypothesis isdesigned tobewhite noisecan in fact be applied on a scale by scale basis to the discrete wavelet transform of long memory processes in particular we show that evaluating a normalized cumulative sum of squares test statistic using critical levels for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractionally differenced process the point at which the test statistic using a nondecimated version of the discrete wavelet transform achieves its maximum value can be used to estimate the time of the unknown variance change we apply our proposed test statistic on five time series derived from the historical record of nile river yearly minimum water levels covering 622 1922 a d each series exhibiting various degrees of serial correlation including long memory in the longest subseries spanning 622 1284 a d the test confirms an inhomogeneity of variance at short time scales and identifies the change point around 720 a d which coincides closely with the construction of a new device around 715 a d for measuring the nile river the test also detects a change in variance for a record of only 36 years index terms 1829 hydrology groundwater hydrology 1869 hydrology stochastic processes 3299 mathematical geophysics general or miscellaneous

graph properties for normal logic programs abstract the completion of a program introduced by clark is important for giving a declarative semantics to the negation as failure rule and to sldnf resolution essential for having a completeness result is the consistency of the completion the aim of this work is to give a unifying presentation of several known classes of programs that have consistent completion the consistency of the completion of a program p is guaranteed by some properties of the graph which represents the dependency relation between the predicate symbols of p global dependencies or the ground atoms of the herbrand basis of p local dependencies a taxonomy of program classes which are discriminated by these properties is presented together with a uniform presentation of the mostly well known techniques of model construction for the programs of each class in this perspective it is also easier to understand the concept of perfect model and the relation between models of the completion and circumscription

multiscale adjoint waveform tomography for surface and body waves abstractwe have developed a wavelet multiscale adjoint scheme for the elastic full waveform inversion of seismic data including body waves bws and surface waves sws we start the inversion on the sw portion of the seismograms to avoid cycle skipping and reduce the dependence on the initial model of these dispersive waves we commence by minimizing an envelope based misfit function subsequently we proceed to the minimization of a waveform difference wd metric applied to the sws only after that we fit bws and sws indiscriminately using a wd misfit metric in each of these three steps we guide the iterative inversion through a sequence of nested subspace projections in a wavelet basis sw analysis preserves a wealth of near surface features that would be lost in conventional bw tomography we used a toy model to illustrate the dispersive and cycle skipping behavior of the sws and to introduce the two ways by which we combat the nonlinearity of waveform inversions involving sws the first is the

the fuzzy rule base solution of differential equations abstract the equivalence between differential equation models and fuzzy logic models is demonstrated for a certain family of fuzzy systems those which use fuzzy spline wavelets as membership functions the universal approximation property of fuzzy systems built with spline wavelets is exploited for replacing operational representations of differential equations with sparse matrix equations the solution of the matrix equations has a direct interpretation as a set of fuzzy rules the fuzzy rule base thus generated provides an approximate solution to the original differential equation while retaining the explanatory power of fuzzy systems the proposed method enjoys the excellent numerical and computational characteristics of the fast wavelet transform

applications of circumscription to formalizing common sense knowledge we present a new and more symmetric version of the circumscription method of nonmonotonic reasoning first described in mccarthy 1980 and some applications to formalizing common sense knowledge the applications in this paper are mostly based on minimizing the abnormality of different aspects of various entities included are nonmonotonic treatments of is a hierarchies the unique names hypothesis and the frame problem the new circumscription may be called formula circumscription to distinguish it from the previously defined domain circumscription and predicate circumscription a still more general formalism called prioritized circumscription is briefly explored

meshfree methods and boundary conditions summary we perform here some meshfree methods to inhomogeneous laplace equations we prove the efficiency of those methods compared with classical ones for one or two dimensional case for numerics and for one dimensional for theoretical results copyright 2006 john wiley sons ltd

a new framework for complex wavelet transforms although the discrete wavelet transform dwt is a powerful tool for signal and image processing it has three serious disadvantages shift sensitivity poor directionality and lack of phase information to overcome these disadvantages we introduce two stage mapping based complex wavelet transforms that consist of a mapping onto a complex function space followed by a dwt of the complex mapping unlike other popular transforms that also mitigate dwt shortcomings the decoupled implementation of our transforms has two important advantages first the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy this allows us to create a directional non redundant complex wavelet transform with potential benefits for image coding systems to the best of our knowledge no other complex wavelet transform is simultaneously directional and non redundant the second advantage of our approach is the flexibility to use any dwt in the transform implementation as an example we can exploit this flexibility to create the complex double density dwt cddwt a shift insensitive directional complex wavelet transform with a low redundancy of 3 sup m 1 2 sup m 1 in m dimensions to the best of our knowledge no other transform achieves all these properties at a lower redundancy

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

solution of time dependent diffusion equations with variable coefficients using multiwavelets a new numerical algorithm is developed for the solution of time dependent differential equations of diffusion type it allows for an accurate and efficient treatment of multidimensional problems with variable coefficients nonlinearities and general boundary conditions for space discretization we use the multiwavelet bases introduced by alpert 1993 siam j math anal 24 246 262 and then applied to the representation of differential operators and functions of operators presented by alpert beylkin and vozovoi representation of operators in the multiwavelet basis in preparation an important advantage of multiwavelet basis functions is the fact that they are supported only on non overlapping subdomains thus multiwavelet bases are attractive for solving problems in finite non periodic domains boundary conditions are imposed with a penalty technique of hesthaven and gottlieb 1996 siam j sci comput 579 612 which can be used to impose rather general boundary conditions the penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem this penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains consequently the matrices representing the differential operators on the finest scale have block diagonal structure for a fixed order of multiwavelets i e a fixed number of vanishing moments the computational complexity of the present algorithm is proportional to the number of subdomains the time discretization method of beylkin keiser and vozovoi 1998 pam report 347 is used in view of its favorable stability properties numerical results are presented for evolution equations with variable coefficients in one and two dimensions