on the asymptotic convergence of b spline wavelets to gabor functions a family of nonorthogonal polynomial spline wavelet transforms is considered these transforms are fully reversible and can be implemented efficiently the corresponding wavelet functions have a compact support it is proven that these b spline wavelets converge to gabor functions modulated gaussian pointwise and in all l sub p norms with 1

shift orthogonal wavelet bases shift orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation or shifts within one level but not with respect to dilations across scales we characterize these wavelets and investigate their main properties by considering two general construction methods in the first approach we start by specifying the analysis and synthesis function spaces and obtain the corresponding shift orthogonal basis functions by suitable orthogonalization in the second approach we take the complementary view and start from the digital filterbank we present several illustrative examples including a hybrid version of the battle lemarie 1987 1988 spline wavelets we also provide filterbank formulas for the fast wavelet algorithm a shift orthogonal wavelet transform is closely related to an orthogonal transform that uses the same primary scaling function both transforms have essentially the same approximation properties one experimentally confirmed benefit of relaxing the interscale orthogonality requirement is that we can design wavelets that decay faster than their orthogonal counterpart

fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

rational approximations to the zeta function this article describes a sequence of rational functions which converges locally uniformly to the numerators and denominators of these rational functions can be expressed as characteristic polyn

analytical footprints compact representation of elementary singularities in wavelet bases we introduce a family of elementary singularities that are point holder regular these singularities are self similar and are the green functions of fractional derivative operators i e by suitable fractional differentiation one retrieves a dirac function at the exact location of the singularity we propose to use fractional operator like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain we show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed form expression termed the analytical footprint practically this means that the dictionary of wavelet footprints is embodied in a single analytical form we show that the wavelet coefficients of the nonredundant decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity we propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

robust multigrid algorithms for the navier strokes equations abstract anisotropies occur naturally in computational fluid dynamics where the simulation of small scale physical phenomena such as boundary layers at high reynolds numbers causes the grid to be highly stretched leading to a slowdown in convergence of multigrid methods several approaches aimed at making multigrid a robust solver have been proposed and analyzed in the literature using the scalar diffusion equation however they have rarely been applied to solving more complicated models such as the incompressible navier stokes equations this paper contains the first published numerical results of the behavior of two popular robust multigrid approaches alternating plane smoothers combined with standard coarsening and plane implicit smoothers combined with semi coarsening for solving the 3 d incompressible navier stokes equations in the simulation of the driven cavity and a boundary layer over a flat plate on a stretched grid grid size grid stretching and reynolds number are the factors considered in evaluating the robustness of the multigrid methods both approaches yield large increases in convergence rates over cell implicit smoothers on stretched grids the combination of plane implicit smoothers and semi coarsening was found to be fully robust in the flat plate simulation up to reynolds numbers 10 6 and the best alternative in the driven cavity simulation for reynolds numbers above 10 3 the alternating plane approach exhibits a better behavior for lower reynolds numbers below 10 3 in the driven cavity simulation

on the shiftability of dual tree complex wavelet transforms the dual tree complex wavelet transform dt bbcwt is known to exhibit better shift invariance than the conventional discrete wavelet transform we propose an amplitude phase representation of the dt ibbc wt which among other things offers a direct explanation for the improvement in the shift invariance the representation is based on the shifting action of the group of fractional hilbert transform fht operators which extends the notion of arbitrary phase shifts from sinusoids to finite energy signals wavelets in particular in particular we characterize the shift ability of the dt bbcwt in terms of the shifting property of the fhts at the heart of the representation are certain fundamental invariances of the fht group namely that of translation dilation and norm which play a decisive role in establishing the key properties of the transform it turns out that these fundamental invariances are exclusive to this group next by introducing a generalization of the bedrosian theorem for the fht operator we derive an explicitly understanding of the shifting action of the fht for the particular family of wavelets obtained through the modulation of lowpass functions e g the shannon and gabor wavelet this in effect links the corresponding dual tree transform with the framework of windowed fourier analysis finally we extend these ideas to the multidimensional setting by introducing a directional extension of the fht the fractional directional hilbert transform in particular we derive a signal representation involving the superposition of direction selective wavelets with appropriate phase shifts which helps explain the improved shift invariance of the transform along certain preferential directions

construction of coarse grained order parameters in nonequilibrium systems we develop a renormalization group rg procedure that includes important system specific features the key ingredient is to systematize the coarse graining procedure that generates the rg flow the coarse graining technology comes from the control and operator theoretic model reduction the resulting generalized rg is a consistent generalization of the wilsonian rg we apply the procedure to a deterministic nonlinear wave equation nlwe with probabilistic initial conditions we derive the form of the projection operator from the dynamics of the nlwe and then use it to generate the rg flow for the distribution of initial conditions the probability density of the initial conditions is chosen to be a boltzmann weight that is quartic in the field variables in our calculation we find that in contrast to conventional implementations of the rg naive power counting breaks down we also show that the resulting rg equations are different from those derived from the conventional rg

basic smoothing procedures for the multigrid treatment of elliptic 3d operators from 2d multigrid it is a fundamental insight that pointwise relaxation combined with standard coarsening gives no reasonable smoothing effect if the given elliptic operator shows appreciably anisotropic behavior corresponding questions and practical consequences are discussed in this paper for the 3d case in the general anisotropic 3d case even line relaxation is not sufficient if standard coarsening is maintained instead plane relaxation is necessary in certain cases if plane relaxation is applied correctly and performed by use of appropriate 2d multigrid the resulting 3d multigrid method has an asymptotic complexity of o n where n number of 3d grid points and is highly efficient even for moderate values of n in contrast to the common opinion plane relaxation turns out to be a simple and general smoothing concept for standard elliptic 3d problems

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

a phase cell cluster expansion for a hierarchical o 3 4 model the formalism developed in a previous paper is applied to yield a phase cell cluster expansion for a hierarchical b model the field is expanded into modes with specific renormatization group scaling properties the present cluster expansion for a vacuum expectation value is formally the natural factorization of each term in the perturbation expansion into the contribution of modes connected to the variables in the expectation via interactions and that of the complementary set the expectation value is thus realized as a sum of contributions due to finite subsets of the modes we emphasize the following additional features 1 partitions of unity are not used 2 there are essentially no cut offs 3 the expansion is developed directly without an initial need to prove an ultraviolet stability bound the most difficult part of the traditional approach our main interest in the present phase cell cluster expansion is founded in the belief that it may be the right vehicle for proving the existence of a non trivial four dimensional field theory

modified multigrid for 3d elliptic equations with cross derivatives a portable fortran code that solves the general nonseparable three dimensional linear elliptic partial differential equation pde with cross derivative terms on a rectangular region is described boundary conditions can be any combination of periodic specified or mixed derivative a multigrid scheme modified to handle complexities introduced by cross derivative terms and nonnormal derivative components in boundary conditions is utilized successful application is illustrated with an analytically prescribed abstract example and simulation of an idealized geophysical fluid flow

spectral analysis and the riemann hypothesis the explicit formulas of riemann and guinand weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of riemann we recall alain connes spectral interpretation of the critical zeros of the riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable hilbert space we then give a spectral interpretation of the zeros of the dedekind zeta function of an algebraic number field k of degree n in an automorphic setting if k is a complex quadratic field the torical forms are the functions defined on the modular surface x such that the sum of this function over the gauss set of k is zero and eisenstein series provide such torical forms in the case of a general number field one can associate to k a maximal torus t of the general linear group g the torical forms are the functions defined on the modular variety x associated to g such that the integral over the subvariety induced by t is zero alternately the torical forms are the functions which are orthogonal to orbital series on x we show here that the riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms constructed from eisenstein series the torical wave packets furthermore we define a hilbert space and a self adjoint operator on this space whose spectrum equals the set of critical zeros of the dedekind zeta function of k

fourier analysis of gmres m preconditioned by multigrid this paper deals with convergence estimates of gmres m saad and schultz siam j sci statist comput 7 1986 pp 856 869 preconditioned by multigrid brandt math comp 31 1977 pp 333 390 hackbusch multi grid methods and applications springer berlin 1985 fourier analysis is a well known and useful tool in the multigrid community for the prediction of two grid convergence rates brandt math comp 31 1977 pp 333 390 stuben and trottenberg in multigrid methods lecture notes in math 960 k stuben and u trottenberg eds springer berlin pp 1 176 this analysis is generalized here to the situation in which multigrid is a preconditioner since it is possible to obtain the whole spectrum of the two grid iteration matrix a preconditioned krylov subspace acceleration method like gmres m implicitly builds up a minimal residual polynomial the determination of the polynomial coefficients is easily possible and can be done explicitly since from fourier analysis a simple block diagonal two grid iteration matrix results based on the gmres m polynomial sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix several numerical scalar test problems are computed in order to validate the theoretical predictions

wavelets and recursive filter banks it is shown that infinite impulse response iir filters lead to more general wavelets of infinite support than finite impulse response fir filters a complete constructive method that yields all orthogonal two channel filter banks where the filters have rational transfer functions is given and it is shown how these can be used to generate orthonormal wavelet bases a family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband butterworth filters when there is an odd number of zeros at pi it is shown that closed forms for the filters are available without need for factorization a still larger class of orthonormal wavelet bases having the same moment properties and containing the daubechies and butterworth filters as the limiting cases is presented it is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case and a constructive method is given it is also shown how compactly supported bases may be orthogonalized and bases for the spline function spaces are constructed

on the computation of battle lemarie s wavelets we propose a matrix approach to the computation of battle lemarie s wavelets the fourier transform of the scaling function is the prod uct of the inverse f x of a square root of a positive trigonometric polynomial and the fourier transform of a b spline of order m the polynomial is the symbol of a bi infinite matrix b associated with a b spline of order 2m we approximate this bi infinite matrix b2m by its finite section as a square ma trix of finite order we use as to compute an approximation s of x whose discrete fourier transform is f x we show that xs converges pointwise to x exponentially fast this gives a feasible method to compute the scaling func tion for any given tolerance similarly this method can be used to compute the wavelets

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

wavelets in wandering subspaces mallat s construction via a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions the connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in hilbert space is exploited in the general setting an example of multiresolution approximation generated by cardinal hermite b splines is constructed

pointwise convergence of wavelet expansions in this note we announce that under general hypotheses wavelet type ex pansions of functions in lp p in one or more dimensions converge pointwise almost everywhere and identify the lebesgue set of a function as a set of full measure on which they converge it is shown that unlike the fourier sum mation kernel wavelet summation kernels pj are bounded by radial decreasing l convolution kernels as a corollary it follows that best l spline approxi mations on uniform meshes converge pointwise almost everywhere moreover summation of wavelet expansions is partially insensitive to order of summation we also give necessary and su cient conditions for given rates of conver gence of wavelet expansions in the sup norm such expansions have order of convergence s if and only if the basic wavelet is in the homogeneous sobolev space h s d h we also present equivalent necessary and su cient conditions on the scaling function the above results hold in one and in multiple dimen sions

fast and robust sixth order multigrid computation for the three dimensional convection diffusion equation we present a sixth order explicit compact finite difference scheme to solve the three dimensional 3d convection diffusion equation we first use a multiscale multigrid method to solve the linear systems arising from a 19 point fourth order discretization scheme to compute the fourth order solutions on both a coarse grid and a fine grid then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equations with a high reynolds number we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact soc scheme compared with the previously published fourth order compact foc scheme

noise covariance properties in dual tree wavelet decompositions dual tree wavelet decompositions have recently gained much popularity mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy when the decomposition of a random process is performed which occurs in particular when an additive noise is corrupting the signal to be analyzed it is useful to characterize the statistical properties of the dual tree wavelet coefficients of this process as dual tree decompositions constitute over complete frame expansions correlation structures are introduced among the coefficients even when a white noise is analyzed in this paper we show that it is possible to provide an accurate description of the covariance properties of the dual tree coefficients of a wide sense stationary process the expressions of the cross covariance sequences of the coefficients are derived in the one and two dimensional cases asymptotic results are also provided allowing to predict the behavior of the second order moments for large lag values or at coarse resolution in addition the cross correlations between the primal and dual wavelets which play a primary role in our theoretical analysis are calculated for a number of classical wavelet families simulation results are finally provided to validate these results

rational filter wavelets abstract in this paper we introduce two types of rational filters for the construction of wavelets the rational filters are a natural development of the polynomial filters we use rational filters to derive a large family of the wavelets which can include daubechies wavelets and battle lemarie wavelets especially the ii type rational filter wavelets among the family have linear phases furthermore we analyze the regularity of the rational filter wavelets and estimate their regularity indices some examples are also given

a gmres based plane smoother in multigrid to solve 3d anisotropic fluid flow problems for a discretization of the 3d steady incompressible navier stokes equations a solution method is presented for solving flow problems on stretched grids the discretization is a vertex centered finite volume discretization with a flux splitting approach for the convective terms second order accuracy is obtained with the well known defect correction technique b koren j comput phys 87 25 1990 the solution method used is multigrid for which a plane smoother is presented for obtaining good convergence in flow domains with severely stretched grids a matrix is set up in a plane which is solved iteratively with a preconditioned gmres method here a stop criterion for gmres is tested which reduces the number of inner iterations compared to an exact plane solver without affecting the multigrid convergence rates the performance of the solution method is shown for a poisson model problem and for 3d incompressible channel flow examples

aspects of multigrid methods for problems in three dimensions smoothing methods are analyzed for use in multigrid methods applied to flow problems in three dimensions a three dimensional version of incomplete block factorization is described the anisotropic diffusion and convection diffusion equations are used as model problems unlike the two dimensional case none of the smoothing methods considered is suitable for all relevant values of the physical parameters

mudpack 2 multigrid software for approximating elliptic partial differential equations on uniform grids with any resolution we describe a multigrid software package mudpack 2 that approximates elliptic partial differential equations in two and three dimensions on uniform meshes without restrictions on the number of grid points in each direction mudpack 2 retains all the functionality of the original mudpack

wavelet packets of fractional brownian motion asymptotic analysis and spectrum estimation this paper provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional brownian motion it also discusses the convergence speed to the limit autocorrelation function when the input random process is either a fractional brownian motion or a wide sense stationary second order random process the analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the shannon paraunitary filters from this analysis we derive wavelet packet based spectrum estimation for fractional brownian motions and wide sense stationary random processes experimental tests show good results for estimating the spectrum of 1 f processes

periodic quasi orthogonal spline bases and applications to least squares curve fitting of digital images presents a new covariant basis dubbed the quasi orthogonal q spline basis for the space of n degree periodic uniform splines with k knots this basis is obtained analogously to the b spline basis by scaling and periodically translating a single spline function of bounded support the construction hinges on an important theorem involving the asymptotic behavior in the dimension of the inverse of banded toeplitz matrices the authors show that the gram matrix for this basis is nearly diagonal hence the name quasi orthogonal the new basis is applied to the problem of approximating closed digital curves in 2d images by least squares fitting since the new spline basis is almost orthogonal the least squares solution can be approximated by decimating a convolution between a resolution dependent kernel and the given data the approximating curve is expressed as a linear combination of the new spline functions and new control points another convolution maps these control points to the classical b spline control points a generalization of the result has relevance to the solution of regularized fitting problems

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

lattice and continuum wavelets and the block renormalization group the authors obtain a resolution of the identity operator for functions on a lattice var epsilon z sup d which is derived from the block renormalization group the authors use eigenfunctions of the terms of the decomposition to form a basis for l sub 2 var epsilon z sup d and show how the basis is generated from lattice wavelets the lattice spacing var epsilon is taken to zero and continuum wavelets are obtained 12 refs

introduction to wavelets in engineering the aim of this paper is to provide an introduction to the subject of wavelet analysis for engineering applications the paper selects from the recent mathematical literature on wavelets the results necessary to develop wavelet based numerical algorithms in particular we provide extensive details of the derivation of mallat s transform and daubechies wavelet coefficients since these are fundamental to gaining an insight into the properties of wavelets the potential benefits of using wavelets are highlighted by presenting results of our research in one and two dimensional data analysis and in wavelet solutions of partial differential equations

wavelet packet division multiplexing and wavelet packet design under timing error effects wavelet packet division multiplexing wpdm is a multiple signal transmission technique in which the message signals are waveform coded onto wavelet packet basis functions for transmission the overlapping nature of such waveforms in time and frequency provides a capacity improvement over the commonly used frequency division multiplexing fdm and time division multiplexing tdm schemes while their orthogonality properties ensure that the overlapping message signals can be separated by a simple correlator receiver the interference caused by timing offset in transmission is examined a design procedure that exploits the inherent degrees of freedom in the wpdm structure to mitigate the effects of timing error is introduced and a waveform that minimizes the energy of the timing error interference is designed an expression for the probability of error due to the presence of gaussian noise and timing error for the transmission of binary data is derived the performance advantages of the designed waveform over standard wavelet packet basis functions are demonstrated by both analytical and simulation methods the capacity improvement of wpdm its simple implementation and the possibility of having optimum waveform designs indicate that wpdm holds considerable promise as a multiple signal transmission technique

from differential equations to the construction of new wavelet like bases in this paper an approach is introduced based on differential operators to construct wavelet like basis functions given a differential operator l with rational transfer function elementary building blocks are obtained that are shifted replicates of the green s function of l it is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential b spline representation the corresponding b splines are entirely specified by their poles and zeros they are compactly supported have an explicit analytical form and generate multiresolution riesz bases moreover they satisfy generalized refinement equations with a scale dependent filter and lead to a representation that is dense in l sub 2 this allows us to specify a corresponding family of semi orthogonal exponential spline wavelets which provides a major extension of earlier polynomial spline constructions these wavelets are completely characterized and it is proven that they satisfy the following remarkable properties 1 they are orthogonal across scales and generate riesz bases at each resolution level 2 they yield unconditional bases of l sub 2 either compactly supported b spline type or with exponential decay orthogonal or dual type 3 they have n vanishing exponential moments where n is the order of the differential operator 4 they behave like multiresolution versions of the operator l from which they are derived and 5 their order of approximation is n m where n and m give the number of poles and zeros respectively last but not least the new wavelet like decompositions are as computationally efficient as the classical ones they are computed using an adapted version of mallat s filter bank algorithm where the filters depend on the decomposition level

wspm wavelet based statistical parametric mapping recently we have introduced an integrated framework that combines wavelet based processing with statistical testing in the spatial domain in this paper we propose two important enhancements of the framework first we revisit the underlying paradigm i e that the effect of the wavelet processing can be considered as an adaptive denoising step to improve the parameter map followed by a statistical detection procedure that takes into account the non linear processing of the data with an appropriate modification of the framework we show that it is possible to reduce the bias of the method with respect to the best linear estimate providing conservative results that are closer to the original data second we propose an extension of our earlier technique that compensates for the lack of shift invariance of the wavelet transform we demonstrate experimentally that both enhancements have a positive effect on performance in particular we present a reproducibility study for multi session data that compares wspm against spm with different amounts of smoothing the full approach is available as a toolbox named wspm for the spm2 software it takes advantage of multiple options and features of spm such as the general linear model

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

coherent structures in random media and wavelets abstract we construct low dimensional dynamical models for the motion of coherent structures such as those frequently observed in extended turbulent flows these models are derived from the wavelet based galerkin projection of the pde describing the flow and account for the local interaction of a small number of coherent structures we show that under rather general assumptions the wavelet projection is close to the proper orthogonal decomposition in the average energy sense in the specific case of the 1d kuramoto sivashinsky equation we show numerically that the dynamics of our model agrees qualitatively with that of the original ks equation

multi grid solution of three dimensional problems with discontinuous coefficients a three dimensional multi grid algorithm is developed and compared to an efficient three dimensional incomplete cholesky conjugate gradient iccg algorithm on a suite of test problems with discontinouous and or anisotropic coefficients the multi grid algorithm used plane relaxation and a definition of coarse grid operators which is an extension of that used for two dimensional problems with discontinuous coefficients numerical results show that the three dimensional results are not nearly as good as the two dimensional results on similar problems

mudpack multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations mudpack is a package of portable fortran subprograms which use multigrid iteration for solving real or complex elliptic partial differential equations the solution regions are rectangles in two dimensions and boxes in three dimensions any combination of periodic dirichlet and mixed derivative boundary conditions is allowed the equations are automatically discretized using second order finite differencing the package will vectorize on cray computers examples are given which demonstrate ease of use efficiency and applicability to a wide range of problems

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

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3d convection diffusion equation an improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection diffusion equation is presented the novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation the new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method wang and zhang 2010 16 to obtain high order solutions since the proposed method needs a fourth order compact scheme with unequal meshsize grids a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection diffusion equation numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure

ondelettes and phase cell cluster expansions a vindication y meyer has recently developed a particularly useful o n basis forl2 r d expansions using this basis i e expansions into ondelettes or wavelets have yielded important new results in soft and hard analysis the expansion into ondelettes of a boson scalar field naturally leads to phase cell cluster expansions a formalism already developed by the authors using other related bases adoption of ondelettes expansions into the phase cell program gives improvements of some extant results and excises an early error

a fourier domain formula for the least squares projection of a function onto a repetitive basis in n dimensional space a theorem concerning the least squares projection of an arbitrary function onto an infinite basis of translated function is given the theorem provides an explicit formula for the fourier transform of the projected function the formula has the advantage of being valid for least squares projections in any n dimensional space the expression for the projected function can be approximately inverted using the discrete fourier transform to find the actual basis coefficients

multigrid for high dimensional elliptic partial differential equations on non equidistant grids this work presents techniques theory and numbers for multigrid in a general d dimensional setting the main focus of this paper is the multigrid convergence for high dimensional partial differential equations on non equidistant grids such as may be encountered in a sparse grid solution as a model problem we have chosen the anisotropic stationary diffusion equation on a rectangular hypercube we present some techniques for building the general d dimensional adaptations of the multigrid components and propose grid coarsening strategies to handle anisotropies that are induced due to discretization on a non equidistant grid apart from the practical formulas and techniques we present in detail the smoothing analysis of the point omega red black jacobi method for a general multidimensional case we show how relaxation parameters may be evaluated efficiently and used for better convergence this analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as the second and the fourth order finite difference operators finally we present some results derived from numerical experiments based on the test problem

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

on the regularity of wavelets the regularity index alpha sub n of the scaling functions sub n phi n 2 3 of multiresolution analysis introduced by i daubechies 1988 is investigated it is shown that 0 51

a linear wavelet filter for parametric imaging with dynamic pet describes a new filter for parametric images obtained from dynamic positron emission tomography pet studies the filter is based on the wavelet transform following the heuristics of a previously published method that are here developed into a rigorous theoretical framework it is shown that the space time problem of modeling a dynamic pet sequence reduces to the classical one of estimation of a normal multivariate vector of independent wavelet coefficients that under least squares risk can be solved by straightforward application of well established theory from the study of the distribution of wavelet coefficients of pet images it is inferred that a james stein linear estimator is more suitable for the problem than traditional nonlinear procedures that are incorporated in standard wavelet filters this is confirmed by the superior performance of the james stein filter in simulation studies compared to a state of the art nonlinear wavelet filter and a nonstationary filter selected from literature finally the formal framework is interpreted for the practitioner s point of view and advantages and limitations of the method are discussed

determining significant connectivity by 4d spatiotemporal wavelet packet resampling of functional neuroimaging data an active area of neuroimaging research involves examining functional relationships between spatially remote brain regions when determining whether two brain regions exhibit significant correlation due to true functional connectivity one must account for the background spatial correlation inherent in neuroimaging data we define background correlation as spatiotemporal correlation in the data caused by factors other than neurophysiologically based functional associations such as scanner induced correlations and image preprocessing we develop a 4d spatiotemporal wavelet packet resampling method which generates surrogate data that preserves only the average background spatial correlation within an axial slice across axial slices and through each voxel time series while excluding the specific correlations due to true functional relationships we also extend an amplitude adjustment algorithm which adjusts our surrogate data to closely match the amplitude distribution of the original data our method improves upon existing wavelet based methods and extends them to 4d we apply our resampling technique to determine significant functional connectivity from resting state and motor task fmri datasets

a fourth order kernel free boundary integral method for implicitly defined surfaces in three space dimensions abstract the kernel free boundary integral kfbi method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains it evaluates boundary or volume integrals involved in the solution of boundary integral equations bies by solving equivalent but simple interface problems on regular grids so that the integral kernel or green function is never needed or computed this is the essential difference of the kfbi method from the traditional ones it takes advantage of the well conditioning property of discrete bies so that the number of krylov subspace iterations is essentially independent of discretization parameter or system dimension this paper presents a fourth order kernel free boundary integral method for second order elliptic partial differential equations on complex domains in three space dimensions whose boundaries are given by implicitly defined surfaces it represents the domain boundary and discretizes data on it by intersection points of the surface with cartesian grid lines the approach has a variety of advantages the current work solves simple interface problems with corrected 27 point compact finite difference schemes and calculates the discrete equations with a fast fourier transform based elliptic solver numerical examples show that the proposed method is efficient as well as accurate

local reproducible smoothing without shrinkage a simple local smoothing filter is defined for curves or surfaces combining the advantages of gaussian smoothing and fourier curve description unlike gaussian filters the filter described has no shrinkage problem repeated application of the filter does not yield a curve or surface smaller than the original but simply reproduces the approximate result that would have been obtained by a single application at the largest scale unlike fourier description the filter is local in space the wavelet transform of y meyer 1989 is also shown to have these properties

denoising methods for thermomechanical decomposition for quasi equilibrium molecular dynamics simulations in this work we applied robust denoising methods well established in the signal processing field for the thermomechanical decomposition of velocity data obtained from molecular dynamics md simulations in the decomposition the atomic velocity was assumed to be the sum of the mechanical velocity and the thermal velocity which can be linked to the stress and temperature field at the continuum scale respectively for the quasi equilibrium process with the thermal velocity treated as the gaussian distributed stationary noise with zero mean and a positive variance that is linearly proportional to temperature and mechanical velocity as the clean signal the velocity decomposition can be recasted into a denoising problem for which powerful denoising methods have been developed to estimate a clean signal from noisy data we investigated the widely used linear parametric real domain linear nonparametric fourier based and nonlinear nonparametric wavelet based denoising methods first on their theoretical properties and then made comparsion among them for denosing some synthetic noisy 1 dimensional 1d data generated from md simulations the nonlinear wavelet based thresholding estimator possessed better optimality properties than the other estimators and also outperformed the other estimators in the synthetic data test a further test comparing the various denoising methods for an adiabatic shear crack nucleation and propagation process simulated using md simulations showed better performance by the wavelet based denoisng method results from this work reveal good potential of applying wavelet based denoising method to the study of thermomechanical processes simulated using md simulations

quantitative fourier analysis of approximation techniques ii wavelets for pt i see ibid vol 47 no 10 p 2783 95 1999 in a previous paper we proposed a general fourier method that provides an accurate prediction of the approximation error irrespective of the scaling properties of the approximating functions here we apply our results when these functions satisfy the usual two scale relation encountered in dyadic multiresolution analysis as a consequence of this additional constraint the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter this is in particular true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero one of the contributions of this paper is the computation of sharp asymptotically optimal upper bounds for the least squares approximation error another contribution is the application of these results to b splines and daubechies 1988 1992 scaling functions which yields explicit asymptotic developments and upper bounds thanks to these explicit expressions we can quantify the improvement that can be obtained by using b splines instead of daubechies wavelets in other words we can use a coarser spline sampling and achieve the same reconstruction accuracy as daubechies specifically we show that this sampling gain converges to spl pi as the order tends to infinity

a new family of spline based biorthogonal wavelet transforms and their application to image compression in this paper we design a new family of biorthogonal wavelet transforms and describe their applications to still image compression the wavelet transforms are constructed from various types of interpolatory and quasiinterpolatory splines the transforms use finite impulse response and infinite impulse response filters that are implemented in a fast lifting mode

a short note on standard parallel multigrid algorithms for 3d problems we present some results concerning parallel multigrid mg algorithms for 3d problems for certain practically important anisotropic cases plane relaxation is needed for smoothing if standard mg coarsening etc is used parallel versions of these sophisticated mg algorithms are described and their time complexity and multiprocessor efficiency is studied

localisation property of battle lemarie wavelets sums abstract explicit formulae are given for a type of battle lemarie scaling functions and related wavelets compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to nikolskii besov spaces on r

a simple cell centered multigrid method for 3d interface problems abstract a multigrid algorithm is presented for cell centered discretizations of interface problems instead of constructing the coarse grid operators by means of the galerkin approximation the coarse grid operators are obtained by discretization on the coarse grids the advantage of this approach is that we obtain m matrices on all grids and that the sparsity pattern of the fine grid matrix is retained on all grids moreover the coarse grid operators are very easy to construct numerical results of several test problems are presented

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

multigrid smoothing factors for red black gauss seidel relaxation applied to a class of elliptic operators analytic formulae are obtained for the smoothing factors yielded by gauss seidel relaxation in two color ordering for a class of scalar elliptic operators block and point relaxations in conjunction with full or partial coarsening are encompassed for operators with general constant positive coefficients in general dimensions and for an arbitrary number of relaxation sweeps it is found that there is no direct dependence of the smoothing factors on the dimension and that the effect of the number of relaxation sweeps on the smoothing factor is usually independent of the operator coefficients and of the relaxation scheme the results are compared with computed results of two level analyses smoothing strategies implied by the formulae are discussed

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order