the arithmetic of the digital computer a new approach a new approach to the arithmetic of the digital computer is surveyed the methodology for defining and implementing floating point arithmetic is described shortcomings of elementary floating point arithmetic are revealed through sample problems the development of automatic computation with emphasis on the user control of errors is reviewed the limitations of conventional rule of thumb procedures for error control in scientific computation are demonstrated by means of examples computer arithmetic is extended so that the arithmetic operations in the linear spaces and their interval correspondents which are most commonly used in computation can be performed with maximum accuracy on digital computers a new fundamental computer operation the scalar product is introduced to develop this advanced computer arithmetic a process of automatic error control called validation which delivers high accuracy with guarantees for scientific computations is described validation of computations for a large class of nu

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

sinc galerkin method for solving biharmonic problems there are many techniques available to numerically solve the biharmonic equation in this paper we show that the sinc galerkin method is a very effective tool in numerically solving this equation hermite interpolation is used to treat the nonhomogeneous boundary conditions our method is tested on examples and comparisons with other methods are made it is shown that the sinc galerkin method yields good results even when singularities occur at the boundaries

a multiresolution study of two dimensional scattering by metallic cylinders the application of wavelet transforms in method of moments mom solutions for scattering problems is extended to cases involving metallic cylinders whose periphery contains a variety of length scale features ranging from smoothly varying large scale features characterized by a radius of curvature of several wavelengths to rapidly varying small scale ones characterized by a radius of curvature that is small compared with the wavelength the basic idea is to first consider a periodic extension of the equivalent current in the arc length variable with a period identical to the scatterer circumference and then to expand this representation using a set of periodic wavelets derived from a conventional basis of wavelets by a periodic extension using a galerkin method and subsequently applying a threshold operation a substantial reduction in the number of elements of the moment method matrix is attained without virtually affecting the solution accuracy the proposed extension is illustrated by a numerical analysis of tm transverse magnetic and te transverse electric scattering from a cylinder of elliptic cross section a thorough study is carried out showing how the solution accuracy improves with increasing resolution level and how this accuracy is affected by a thresholding process which renders the moment matrix sparsely populated

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

families of biorthogonal wavelets several families of biorthogonal wavelet bases are constructed with various properties in particular for a given filter fn of finite length 2n 1 a parametric family of dual filters fvn of length 2n 1 is constructed the parametric nature of the dual filters makes it possible to design the optimum dual filter fv0n corresponding to a fixed filter fn

reduction of multivariate mixtures and its applications abstract we consider fast deterministic algorithms to identify the best linearly independent terms in multivariate mixtures and use them to compute up to a user selected accuracy an equivalent representation with fewer terms importantly the multivariate mixtures do not have to be a separated representation of a function one algorithm employs a pivoted cholesky decomposition of the gram matrix constructed from the terms of the mixture to select what we call skeleton terms and the other uses orthogonalization for the same purpose both algorithms require o r 2 n p d r n operations where n is the initial number of terms in the multivariate mixture r is the number of selected linearly independent terms and p d is the cost of computing the inner product between two terms of a mixture in d variables for general gaussian mixtures p d d 3 since we need to diagonalize a d d matrix whereas for separated representations p d d there is no need for diagonalization due to conditioning issues the resulting accuracy is limited to about one half of the available significant digits for both algorithms we also describe an alternative algorithm that is capable of achieving higher accuracy but is only applicable in low dimensions or to multivariate mixtures in separated form we describe a number of initial applications of these algorithms to solve partial differential and integral equations and to address several problems in data science for data science applications in high dimensions we consider the kernel density estimation kde approach for constructing a probability density function pdf of a cloud of points a far field kernel summation method and the construction of equivalent sources for non oscillatory kernels used in both computational physics and data science and finally show how to use the new algorithm to produce seeds for subdividing a cloud of points into groups

conserved quantities and adaptation to the edge of chaos certain dynamical systems such as the shift map and the logistic map have an edge of chaos in their parameter spaces on one side of this edge the dynamics are chaotic for many parameter values on the other side of the edge they are periodic we find that discrete time dynamical systems with wavelet filtered feedback from the dynamical variable to the parameters are attracted to a narrow parameter range near the edge of chaos the periodic boundary regime we show that the migration from the chaotic regime to the periodic boundary regime can be attributed to a conserved quantity and find that such adaptation to the edge of chaos is accompanied by a depopulation of the chaotic regime we use this conserved quantity to determine the location of the periodic boundary regime and show that its size is proportional to the size of the feedback further we compute the dynamics of the probability density for the parameter for a specific example

a note on wave number dependence of wavelet matrix compression for integral equations with oscillatory kernel this paper analyzes the effect of large wave numbers on the wavelet method for integral equations arising in electromagnetic applications it is shown that the compression of the stiffness matrix deteriorates with increasing wave number a characteristic that has been reported before in the literature here however the exact dependence on the wave number is calculated analytically for the two dimensional helmholtz problem

spectral methods using legendre wavelets for nonlinear klein sine gordon equations klein sine gordon equations are very important in that they can accurately model many essential physical phenomena in this paper we propose a new spectral method using legendre wavelets as basis for numerical solution of klein sine gordon equations due to the good properties of wavelets basis the proposed method can obtain good spatial and spectral resolution moreover the presented method can save more memory and computation time benefit from save more computation time benefit from the hierarchical scale structure of legendre wavelets 1d and 2d examples are included to demonstrate the validity and applicability of the new technique numerical results show the exponential convergence property and error characteristics of presented method

wave atoms and time upscaling of wave equations we present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying two dimensional time independent periodic media the method consists in representing the time dependent green s function in wave atoms a tight frame of multiscale directional wave packets obeying a precise parabolic balance between oscillations and support size namely wavelength diameter 2 wave atoms offer a uniquely structured representation of the green s function in the sense that the resulting matrix is universally sparse over the class of c coefficients even for large times the matrix has a natural low rank block structure after separation of the spatial indices the parabolic scaling is essential for these properties to hold as a result it becomes realistic to accurately build the full matrix exponential in the wave atom frame using repeated squaring up to some time typically of the form delta t sim sqrt delta x which is bigger than the standard cfl timestep once the expensive precomputation of the green s function has been carried out it can be used to perform unusually large upscaled cheap time steps the algorithm is relatively simple in that it does not require an underlying geometric optics solver we prove accuracy and complexity results based on a priori estimates of sparsity and separation ranks on a n by n grid the expensive precomputation takes somewhere between o n 3log n and o n 4log n steps depending on the separability of the acoustic medium the complexity of upscaled timestepping however beats the o n 3log n bottleneck of pseudospectral methods on an n by n grid for a wide range of physically relevant situations in particular we show that a naive version of the wave atom algorithm provably runs in o n 2 operations for arbitrarily small but for the final algorithm we had to slightly increase the exponent in order to reduce the large constant as a result we get estimates between o n 2 5 log n and o n 3 log n for upscaled timestepping we also show several numerical examples in practice the current wave atom solver becomes competitive over a pseudospectral method in regimes where the wave equation should be solved hundreds of times with different initial conditions as in reflection seismology in academic examples of accurate propagation of bandlimited wavefronts if the precomputation step is factored out then the wave atom solver is indeed faster than a pseudospectral method by a factor of about 3 5 at n 512 and a factor 10 20 at n 1024 for the same accuracy very similar gains are obtained in comparison versus a finite difference method

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

on computational proofs of the existence of solutions to nonlinear parabolic problems abstract this paper is an extension of the preceding study nakao this journal 1991 in which we described a numerical verification method of the solution for one space dimensional parabolic problems to the several space dimensional case here numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer we reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed point problem of a compact operator on certain function spaces as in the preceding study based upon a simple c 0 finite element approximation and its constructive a priori error estimates a numerical verification procedure is presented with some numerical examples

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

a new class of biorthogonal wavelet systems for image transform coding we construct general biorthogonal coifman wavelet systems a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair a time domain design method is employed and closed form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived the resulting filter coefficients are all dyadic fractions which is an attractive feature in the realization of multiplication free discrete wavelet transform even ordered systems in this family are symmetric which correspond to linear phase dual filters in particular three filterbanks fbs in this family are systematically verified to have competitive compression potential to the 9 7 tap biorthogonal wavelet fb by cohen et al 1992 which is currently the most widely used one in the field of wavelet transform coding in addition the proposed fb s have much smaller computational complexity in terms of floating point operations required in transformation and therefore indicate a better tradeoff between compression performance and computational complexity

galerkin method with new quadratic spline wavelets for integral and integro differential equations abstract the paper is concerned with the wavelet galerkin method for the numerical solution of fredholm linear integral equations and second order integro differential equations we propose a construction of a quadratic spline wavelet basis on the unit interval such that the wavelets have three vanishing moments and the shortest support among such wavelets we prove that this basis is a riesz basis in the space l 2 0 1 we adapt the basis to homogeneous dirichlet boundary conditions and using a tensor product we construct a wavelet basis on the hyperrectangle we use the wavelet galerkin method with the constructed bases for solving integral and integro differential equations and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices we present numerical examples and compare the results with the galerkin method using other quadratic spline wavelet bases and other methods

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

packing algorithms for arborescences and spanning trees in capacitated graphs in a digraph with real valued edge capacities we pack the greatest number of arborescences in time o n 3m log n 2 m the packing uses at mostm distinct arborescences heren andm denote the number of vertices and edges in the given graph respectively similar results hold for integral packing we pack the greatest number of arborescences in time o min n logn n 2m log n 2 using at mostm n 2 distinct arborescences heren denotes the largest integral capacity of an edge these resuts improve the best previous bounds for capacitated digraphs the algorithm extends to several related problems including packing spanning trees in an undirected capacitated graph and covering such graphs by forests the algorithm provides a new proof of edmonds theorem for arborescence packing for both integral and real capacities based on a laminar family of sets derived from the packing 1998 the mathematical programming society inc published by elsevier science b v

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

simulation of a waveguide filter using wavelet based numerical homogenization we apply wavelet based numerical homogenization to the simulation of an optical waveguide filter we use the method to derive approximate one dimensional models and subgrid models of the filter numerical examples of the technique are presented and the computational gains are investigated

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

improving christofides algorithm for the s t path tsp we present a deterministic 1 5 2 approximation algorithm for the s t path tsp for an arbitrary metric given a symmetric metric cost on n vertices including two prespecified endpoints the problem is to find a shortest hamiltonian path between the two endpoints hoogeveen showed that the natural variant of christofides algorithm is a 5 3 approximation algorithm for this problem and this asymptotically tight bound in fact has been the best approximation ratio known until now we modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the held karp relaxation rather than a minimum spanning tree we prove this simple but crucial modification leads to an improved approximation ratio surpassing the 20 year old ratio set by the natural christofides algorithm variant our algorithm also proves an upper bound of 1 5 2 on the integrality gap of the path variant held karp relaxation the techniques devised in this article can be applied to other optimization problems as well these applications include improved approximation algorithms and improved lp integrality gap upper bounds for the prize collecting s t path problem and the unit weight graphical metric s t path tsp

a domain decomposition method for biharmonic equation abstract the domain decomposition method proposed by schwarz 1 is gaining significance in view of the possibilities for parallel implementation in this paper we apply the domain decomposition method to the biharmonic equation in two overlapping discs in each subdomain a boundary integral formula is used to obtain the solution unlike the finite element or finite difference methods where the iterations have to be performed on the entire domain the iterations of the boundary integral formulation need to be done only at the discretized points of the boundaries another aspect of the paper is that in order to handle the operations with dense matrices arising from the boundary integral formulation the discrete wavelet transform is used to compress the matrices which results in reduced computations without loss of accuracy examples involving a laplace equation in an l shaped region and a domain bounded by a cardioid and a circle are also illustrated

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

solving nonlinear parabolic problems with result verification part i one space dimensional case abstract we propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension it also means that one can obtain a posteriori error bounds for the approximate solutions of the problems based upon schauder s fixed point theorem a verification condition is formulated and by the use of finite element approximation and its error estimates for a simple parabolic problem we present a numerical verification algorithm of exact solutions in a computer some numerical examples which are verified by the method are illustrated

optimal attack and reinforcement of a network in a nonnegative edge weighted network the weight of an edge represents the effort required by an attacker to destroy the edge and the attacker derives a benefit for each new component created by destroying edges the attacker may want to minimize over subsets of edges the difference between or the ratio of the effort incurred and the benefit received this idea leads to the definition of the strength of the network a measure of the resistance of the network to such attacks efficient algorithms for the optimal attack problem the problem of computing the strength and the problem of finding a minimum cost reinforcement to achieve a desired strength are given these problems are also solved for a different model in which the attacker wants to separate vertices from a fixed central vertex

wavelet simultaneous approximation from samples affected by noise the authors recently introduced a pth order wavelet regularization method together with the gcv criterion for approximating a function from a finite sample affected by noise convergence results of the method were proven for the l2 norm the present paper addresses the problem of simultaneous approximation which is of interest in applications where derivatives are useful for extracting several features from a signal it is proved that it is not needed to devise a special method to this purpose since the convergence of the method devised by the authors also works for the hq norm 0 q p

on the smoothing property of the galerkin method for parabolic equations we analyze the galerkin approximation of the general second order parabolic initial boundary value problem for the second order continuous time method with initial data only in l 2 we prove an l 2 error estimate of order o h 2 t analogous results are shown to hold for the error in negative sobolev norms and for the time derivative of the error our analysis uses only elementary energy techniques and the same techniques are shown to give a simple analysis of the backward euler time discretization

a posteriori error bounds for two point boundary value problems consider a general two point boundary value problem tpbvp begin gathered y t f t y hfill b 1 y a b 2 y b w quad hfill end gathered a leqq t leqq b where f r n 1 to r n f in c 2 b 1 and b 2 are n times n matrices and w in r n it is shown how one can bound a posteriori the error made in the numerical solution of the tpbvp the error bounds obtained are rigorous and include the truncation and the roundoff error in addition the computations establish the existence of solutions to the tpbvp numerical schemes are developed for the case where f t y is a polynomial in t and y examples are given of computational existence proofs for problems where analytical existence proofs are not known

numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element the verifications of solutions to weakly nonlinear elliptic equations by the method described e g by nakao 1988 1989 etc are sometimes hardly accomplished when the right hand sides of the equations are very large to overcome such difficulties a residual iteration technique with approximate solution was introduced by nakao 1993 in the present paper we propose an a posteriori method for the residual iteration and show that a remarkable improvement in efficiency and in accuracy of the verification can be obtained when we use a higher order finite element

construction of orthonormal wavelet like bases a general method for constructing wavelet like bases in a hilbert space h starting from any orthonormal basis in h and any periodic orthonormal wavelet basis is presented with this method we can take advantage of the characteristics of both types of bases to obtain orthonormal wavelet like bases that are suitable to represent functions and operators efficiently

detection of crack location and size in structures using wavelet finite element methods abstract a methodology to detect crack location and size is presented which takes advantage of wavelet finite element methods wfem in the modal analysis for singularity problems like a cracked beam first the beam is discretized into a set of wavelet finite elements and then the natural frequencies of the beam with various crack locations and sizes are accurately obtained the frequency response functions function of crack location and size are approximated by means of surface fitting techniques measured natural frequencies are used in a crack detection process and the crack location and size can be identified by finding the point of intersection of three frequency contour lines the experimental data from a free free beam studied by silva are employed to verify the accuracy of the method the present method is general and can be easily extended to complex structures with multiple cracks

wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra we derive minimax rates for estimation in anisotropic smoothness classes this rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension it is shown that this basis is superior to its one scale multiresolution analog if different degrees of smoothness in different directions are present as an important application we introduce a new adaptive wavelet estimator of the time dependent spectrum of a locally stationary time series using this model which was resently developed by dahlhaus we show that the resulting estimator attains nearly the rate which is optimal in gaussian white noise simultaneously over a wide range of smoothness classes moreover by our new approach we overcome the difficulty of how to choose the right amount of smoothing i e how to adapt to the appropriate resolution for reconstructing the local structure of the evolutionary spectrum in the time frequency plane

the h2 wavelet method abstract in the present paper we introduce the h 2 wavelet method for the fast solution of nonlocal operator equations on unstructured meshes on the given mesh we construct a wavelet basis which provides vanishing moments with respect to the traces of polynomials in the space with this basis at hand the system matrix in wavelet coordinates is compressed to o n log n relevant matrix coefficients where n denotes the number of boundary elements the compressed system matrix is computed with nearly linear complexity by using the h 2 matrix approach numerical results in three spatial dimensions validate that we succeeded in developing a fast wavelet galerkin scheme on unstructured triangular or quadrangular meshes

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

fractional covers for forests and matchings we give polynomial algorithms for the fractional covering problems for forests andb matchings min 1 y ya w y 0 wherea is a matrix whose rows are the incidence vectors of forests b matchings respectively it is shown that each problem can be solved by a series of max flow min cut calculations and hence the use of the ellipsoid algorithm to guarantee a polynomial algorithm can be avoided

a schwarz alternating algorithm for a three dimensional exterior harmonic problem with prolate spheroid boundary in this paper we investigate a schwarz alternating algorithm for a three dimensional exterior harmonic problem with prolate spheroid boundary based on natural boundary reduction the algorithm is constructed and its convergence is discussed the finite element method and the natural boundary element method are alternatively applied to solve the problem in a bounded subdomain and a typical unbounded subdomain the convergence rate is analyzed in detail for a typical domain two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method

on the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems we consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial boundary conditions our estimation technique uses a full discrete numerical scheme which is based on the galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space in our technique the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role combining these estimates with an argument for the discretized inverse operator and a contraction property of the newton type formulation we derive an a posteriori estimate of the norm for the infinite dimensional operator in numerical examples we show that the proposed method should be more efficient than the existing method moreover as an application we give some prototype results for numerical verification of solutions of nonlinear parabolic problems which confirm the actual usefulness of our technique

a wavelet optimized very high order adaptive grid and order numerical method wavelets detect information at different scales and at different locations throughout a computational domain furthermore wavelets can detect the local polynomial content of computational data numerical methods are most efficient when the basis functions of the method are similar to the data present by designing a numerical scheme in a completely adaptive manner around the data present in a computational domain one can obtain optimal computational efficiency this paper extends the numerical wavelet optimized finite difference wofd method to arbitrarily high order so that one obtains in effect an adaptive grid and adaptive order numerical method which can achieve errors equivalent to errors obtained with a spectrally accurate numerical method

comparison of fast boundary element methods on parametric surfaces we compare fast black box boundary element methods on parametric surfaces in r3 these are the adaptive cross approximation the multipole method based on interpolation and the wavelet galerkin scheme the surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels nonetheless parametric surface representations are easily accessible from computer aided design cad and are recently topic of the studies in isogeometric analysis especially we can apply two dimensional interpolation in the multipole method a main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry this results in a superior compression of the far field compared to other cluster methods

gauss type quadrature rule with complex nodes and weights for integrals involving daubechies scale functions and wavelets this paper deals with derivation of a gauss type quadrature rule named as gauss daubechies quadrature rule for numerical evaluation of integrals involving product of integrable function and daubechies scale functions wavelets some of the nodes and weights of the quadrature formula may be complex and appear with their conjugates this is in contrast with the standard gauss type quadrature rule for integrals involving products of integrable functions and non negative weight functions this quadrature rule has accuracy as good as the standard gauss type quadrature rule and is also found to be stable the efficacy of the quadrature rule derived here has been tested through some numerical examples

restoration of chopped and nodded images by framelets in infrared astronomy an observed image from a chop and nod process can be considered as the result of passing the original image through a high pass filter here we propose a restoration algorithm which builds up a tight framelet system that has the high pass filter as one of the framelet filters our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition the framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image a proof of the convergence of the algorithm based on convex analysis is also provided simulated and real images are tested to illustrate the efficiency of our method over the projected landweber method

wavelet bases in h div and h curl some years ago compactly supported divergence free wavelets have been constructed which also give rise to a stable biorthogonal wavelet splitting of h div these bases have successfully been used both in the analysis and numerical treatment of the stokes and navier stokes equations in this paper we construct stable wavelet bases for the stream function spaces h curl moreover curl free vector wavelets are constructed and analysed the relationship between h div and h curl are expressed in terms of these wavelets we obtain discrete orthogonal hodge decompositions our construction works independently of the space dimension but in terms of general assumptions on the underlying wavelet systems in l 2 omega that are used as building blocks we give concrete examples of such bases for tensor product and certain more general domains omega subset er n as an application we obtain wavelet multilevel preconditioners in h div and h curl email urban dragon ian pv cnr it keywords h div h curl stream function spaces wavelets

improving the performance of the boundary element method with time dependent fundamental solutions by the use of a wavelet expansion in the time domain the object of this paper is a wavelet based formulation of the boundary element method bem for diffusion problems characterized by time dependent fundamental solution while the bem is a well known and often used technique its time dependent formulation for diffusion problems is very rarely used in practical applications due to the high computational cost which characterizes it here a new formulation is proposed which through the use of the wavelet expansion of the time behaviour of the boundary elements is characterized by a lower cpu time consumption when compared with the standard bem diffusion formulation the problem to be solved is transformed into an algebraic system of higher dimension and its solution gives the time domain behaviour of the desired quantities in this way the time stepping procedure is avoided together with the formulation the analysis of the computational cost and two examples are given in the paper copyright 2006 john wiley sons ltd

variants of algebraic wavelet based multigrid methods application to shifted linear systems abstract in this paper we describe some new variants and applications of the wavelet algebraic multigrid method this method combines the algebraic multigrid method a well known family of multilevel techniques for solving linear systems without use of knowledge of the underlying problem and the discrete wavelet transform these two techniques can be combined in several ways obtaining different methods for solution of linear systems these can be used alone or as preconditioners for krylov iterative methods these methods can be applied for solution of linear systems with shifted matrices of the form a hi whose efficient solution is very important for implicit ode methods unsteady pdes computation of eigenvalues of large sparse matrices and other important problems

computation of wavelet coefficients from average samples there exist efficient methods to compute the wavelet coefficients of a function f t from its point samples f t email protected email protected n however in many applications the available samples are average samples of the type f t email protected u t dt where the averaging function u t reflects the characteristic of the acquisition device in this work methods to compute the coefficients in a biorthogonal wavelet system from average samples are studied error estimations are obtained and using them the optimal values for the parameters in the proposed approximation rules are calculated the obtained error estimations can also be applied to the rules that compute the coefficients from point samples and thus these estimations can be used to compare and to choose between the different methods proposed in the literature the methods proposed here also allow us to compute the biorthogonal wavelet coefficients from the coefficients in another biorthogonal wavelet system

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 fast bie iteration method for an arbitrary body in a flow of incompressible inviscid fluid the paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics the stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid with a discrete numerical treatment we approximate the exact matrix by a certain toeplitz one and then apply a fast algorithm for this matrix on each iteration step we illustrate the convergence of this iteration scheme by a number of numerical examples both for hard and soft boundary conditions it appears that the method is highly efficient for hard boundaries being much less efficient for soft boundaries

solution of large dense complex matrix equations utilizing wavelet like transforms this paper presents the wavelet like transforms which are quite different from the wavelet transform for the solution of large dense complex matrix equations from a purely numerical standpoint these wavelet like transforms are not true orthogonal transforms as the condition number of the resulting matrix changes after the thresholding these effects are illustrated through examples

numerical solution of fredholm integral equations by using cas wavelets a numerical method for solving the fredholm integral equations is presented the method is based upon cas wavelet approximations the properties of cas wavelet are first presented cas wavelet approximations method are then utilized to reduce the fredholm integral equations to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

densities matchings and fractional edge colorings given a multigraph g v e with a positive rational weight w e on each edge e the weighted density problem wdp is to find a subset u of v with u ge 3 and odd that maximizes 2w

adaptive multiscale moment method ammm for analysis of scattering from three dimensional perfectly conducting structures the adaptive multiscale moment method ammm is presented for the analysis of scattering from three dimensional 3d perfectly conducting bodies this algorithm employs the conventional moment method mm using the subsectional triangular patch basis functions and a special matrix transformation which is derived from solving the fredholm equation of the first kind by the multiscale technique this methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time the widely used triangular patch vector basis functions developed by rao et al 1982 are used for expansion and testing functions in the conventional mm the objective here is to compress the unknowns in existing mm codes which solves for the currents crossing the edges of the triangular patch basis functions by use of a matrix transformation the currents source terms and impedance matrix can be arranged in the form of different scales from one scale to another scale the initial guess for the solution can be predicted according to the properties of the multiscale technique ammm can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional mm the basic difference between this methodology and the other wavelet based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown two numerical results are presented which demonstrate that the ammm is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3d perfectly conducting bodies

combining kronecker product approximation with discrete wavelet transforms to solve dense function related linear systems a new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory this technique combines kronecker product approximation and wavelet sparsification for the kronecker product factors the user is only required to supply a procedure for computation of each entry of the given matrix the main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic skeleton method of the second author and data sparse preconditioners the incomplete lu decomposition with dynamic choice of the fill in structure with a prescribed threshold and the inverse kronecker product preconditioner constructed for the sum of kronecker products of sparsified finger like matrices computed by the discrete wavelet transform in some model but quite representative examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 gbyte operative memory

error analysis of coarse graining for stochastic lattice dynamics the coarse grained monte carlo cgmc algorithm was originally proposed in the series of works m a katsoulakis a j majda and d g vlachos j comput phys 186 2003 pp 250 278 m a katsoulakis a j majda and d g vlachos proc natl acad sci usa 100 2003 pp 782 787 m a katsoulakis and d g vlachos j chem phys 119 2003 pp 9412 9427 in this paper we further investigate the approximation properties of the coarse graining procedure and provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long time simulations we demonstrate that the numerical accuracy of the cgmc algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse graining ratio and that the natural small parameter is the coarse graining ratio over the range of particle particle interactions the error estimate is shown to hold in the weak convergence sense we employ the derived analytical results to guide cgmc algorithms and demonstrate a cpu speed up in demanding computational regimes that involve nucleation phase transitions and metastability

packing and covering with integral feasible flows integral supply demand networks polynomial time algorithms are presented for solving combinatorial packing and covering problems defined from the integral feasible flows in an integral supply demand network these algorithms are also shown to apply to packing and covering problems defined by the minimal integral solutions to general totally unimodular systems analogous problems arising from matroid bases are also discussed and it is demonstrated that a means for solving such problems is provided by recent work of cunningham

tomographic reconstruction and estimation based on multiscale natural pixel bases we use a natural pixel type representation of an object originally developed for incomplete data tomography problems to construct nearly orthonormal multiscale basis functions the nearly orthonormal behavior of the multiscale basis functions results in a system matrix relating the input the object coefficients and the output the projection data which is extremely sparse in addition the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system we exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well conditioned and sparse matrix this enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions in case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability map multiscale reconstruction estimates based on a certain self similar prior on the multiscale object coefficients the framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete irregular and noisy and in object feature recognition directly from projection data

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

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

numerical solution of the schrodinger equation in a wavelet basis for hydrogen like atoms an iterative method is proposed to solve the schrodinger eigenvalue problem in a wavelet framework orthonormal wavelets are used to represent the corresponding operator as a sparse band matrix this representation called the nonstandard ns form is obtained by means of the beylkin coifman rokhlin bcr algorithm and simplifies the numerical calculations problems due to the one dimensional mathematical model and to the discretization process receive special attention

integral operators on sparse grids in this paper we are concerned with the construction and use of wavelet approximation spaces for the fast evaluation of integral expressions the spaces are based on biorthogonal anisotropic tensor product wavelets we introduce sparse grid hyperbolic cross approximation spaces which are adapted not only to the smoothness of the kernel but also to the norm in which the error is measured furthermore we introduce compression schemes for the corresponding discretizations numerical examples for the laplace equation with dirichlet boundary conditions and an additional integral term with a smooth kernel demonstrate the validity of our theoretical results