aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

an adaptive level set method for nondifferentiable constrained image recovery the formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals in previous years nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges they also arise naturally in minimax inconsistent set theoretic recovery problems at the same time the issue of developing reliable numerical algorithms to solve such convex programs in the context of image recovery applications has received little attention we address this issue and propose an adaptive level set method for nondifferentiable constrained image recovery the asymptotic properties of the method are analyzed and its implementation is discussed numerical experiments illustrate applications to total variation and minimax set theoretic image restoration and denoising problems

on global and local convergence of half quadratic algorithms this paper provides original results on the global and local convergence properties of half quadratic hq algorithms resulting from the geman and yang gy and geman and reynolds gr primal dual constructions first we show that the convergence domain of the gy algorithm can be extended with the benefit of an improved convergence rate second we provide a precise comparison of the convergence rates for both algorithms this analysis shows that the gr form does not benefit from a better convergence rate in general moreover the gy iterates often take advantage of a low cost implementation in this case the gy form is usually faster than the gr form from the cpu time viewpoint

potential integrals associated with quadratic distributions in a triangular domain exact expressions are presented for the potentials at an arbitrary point in space due to two quadratic distributions in a triangular disc these expressions are obtained by means of extensions of the analytical techniques previously developed for evaluating the potential integrals for constant and linear distributions it is shown that the potential due to any quadratic distribution in a triangle may be obtained by using the two expressions derived here together with an expression for the potential corresponding to a linear distribution this latter expression was presented in an earlier paper 1 numerical values obtained through the use of the two expressions are compared with those which result from the numerical evaluation of the potential integrals by triangle quadratures

cauchy noise removal using group based low rank prior abstract although the extensive research on gaussian noise removal few works consider the cauchy noise removal problem in this paper we propose a novel group based low rank method for cauchy noise removal by exploiting the nonlocal self similarity of natural images we consider a group of similar patches as an approximate low rank matrix and formulate the denoising of each group as a low rank matrix recovery problem meanwhile we develop the alternating direction method of multipliers algorithm to solve the proposed nonconvex model with guaranteed convergence experiments illustrate that our method has superior performance over the state of the art methods in terms of both visual and quantitative measures

a regularization parameter for nonsmooth tikhonov regularization in this paper we develop a novel rule for choosing regularization parameters in nonsmooth tikhonov functionals it is solely based on the value function and applicable to a broad range of nonsmooth models and it extends one known criterion a posteriori error estimates of the approximations are derived an efficient numerical algorithm for computing the minimizer is developed and its convergence properties are discussed numerical results for several common nonsmooth models are presented including deblurring natural images the numerical results indicate the rule can yield results comparable with those achieved with the discrepancy principle and the optimal choice and the algorithm merits a fast and steady convergence

a new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method the purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto plastic solids respectively by the direct boundary element method bem triangle polar co ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares the method was subsequently extended to the cubature of singular integrals over three dimensional internal cells as occur in applications of the bem to three dimensional elasto plasticity for this purpose so called tetrahedron polar co ordinates were introduced singular boundary integrals stretching over either linear triangular or quadratic quadilateral isoparametric boundry elements and singular volume integrals extending over either linear tetrahedral or quadratic hexahedral isoparametric internal cells are treated in case of higher order isoparametric boundary elements and internal cells division into a number of subelements and subcells respectively is necessary the analytical investigation is followed by a numerical study restricted to the use of quadratic quadrilateral isoparametric boundary elements this is justified by the fact that such elements as opposed to linear elements yield singular boundary integrals which cannot be integrated analytically the results of the numerical investigation demonstrate the potential of the developed concept

linear convergence for vector sequences and some applications abstract in this paper we first give the relation between some definitions of linearly convergent vector sequences proposed by nievergelt osada and sadok and we choose the first one in order to generalize a theorem which gives the relation between the asymptotic behaviour of the ratios of the errors and that of the differences for linear convergent scalar sequences some properties of the matrix used in the definition of linear convergence are studied and a convergence acceleration result which generalizes some results given by nievergelt and sadok is given

a new concept in near singular integral evaluation the continuation approach the functional homogeneity of green functions is exploited in the derivation of continuation formulae for the accurate evaluation of near singular integrals these formulae provide a means of systematically continuing such unconventional integrals as the principal value and finite part integrals of singular functions to conventional integrals of nonsingular functions a new concept the notion of continuum integral which is applicable to singular as well as to nonsingular integrals is introduced as a generalization of principal value and finite part integrals numerical examples are provided that illustrate the computational advantages of the continuation method and give a new perspective on the subtleties of singular and near singular integrals

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

improvement of accuracy in numerical methods for inverting laplace transforms based on the post widder formula abstract the paper proposes a new computational version for numerical inversion of laplace transforms at a point and on an interval based on the post widder formula in this version the original sequence of post widder approximants is calculated using operations on series and the approximate value of the sought function is constructed as a limit of this sequence using polynomial and rational extrapolation to the limit some procedures are proposed for improvement of accuracy in the problem of inversion at equidistant points of an interval without computing additional approximants an extension of the method to the two dimensional case is discussed the one dimensional version is illustrated with some probabilistic examples involving both explicit and implicit functions the accuracy achieved in these examples with using the standard double precision arithmetic evaluated in terms of relative errors is about 10 10 10 14 even in case of out of scale parameters computational aspects are discussed in comparison with previously known realizations of the post widder method

extended procedures for extrapolation to the limit in this paper new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed they are based on the notions of error estimates and annihilation operators we obtain generalizations of the discrete and confluent e transformation which are the most general sequence and function transformations known so far it is shown that many transformations studied in the literature are included in our formalism particular cases of these procedures are discussed then several extensions of the e algorithm are given finally the procedure q is applied to our procedures to produce new procedures for extrapolation to the limit

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

green s functions for the biharmonic equation bonded elastic media a comprehensive formulation of green s functions for biharmonic equations with respect to perfectly bonded bimaterial linear isotropic elastic media is presented this is accomplished by extending mindlin s nuclei of strain method which employs the method of images the analysis involves properly scaling the nuclei grouping the nuclei into different classes and introducing a matrix structure for each class the strength vectors of the nuclei required at the object and image points to represent the solution are obtained by matrix vector multiplications involving two invariant matrices the reflection matrix and the transmission matrix algebraic expressions for the elements of the reflection and transmission matrices which depend on the elastic properties of the materials are presented

total variation projection with first order schemes this article proposes a new algorithm to compute the projection on the set of images whose total variation is bounded by a constant the projection is computed through a dual formulation that is solved by first order non smooth optimization methods this yields an iterative algorithm that applies iterative soft thresholding to the dual vector field and for which we establish convergence rate on the primal iterates this projection algorithm can then be used as a building block in a variety of applications such as solving inverse problems under a total variation constraint or for texture synthesis numerical results are reported to illustrate the usefulness and potential applicability of our tv projection algorithm on various examples including denoising texture synthesis inpainting deconvolution and tomography problems we also show that our projection algorithm competes favorably with state of the art tv projection methods in terms of convergence speed

a convergent overlapping domain decomposition method for total variation minimization in this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint to our knowledge this is the first successful attempt of addressing such a strategy for the nonlinear nonadditive and nonsmooth problem of total variation minimization we provide several numerical experiments showing the successful application of the algorithm for the restoration of 1d signals and 2d images in interpolation inpainting problems respectively and in a compressed sensing problem for recovering piecewise constant medical type images from partial fourier ensembles

parameter selection for total variation based image restoration using discrepancy principle there are two key issues in successfully solving the image restoration problem 1 estimation of the regularization parameter that balances data fidelity with the regularity of the solution and 2 development of efficient numerical techniques for computing the solution in this paper we derive a fast algorithm that simultaneously estimates the regularization parameter and restores the image the new approach is based on the total variation tv regularized strategy and morozov s discrepancy principle the tv norm is represented by the dual formulation that changes the minimization problem into a minimax problem a proximal point method is developed to compute the saddle point of the minimax problem by adjusting the regularization parameter adaptively in each iteration the solution is guaranteed to satisfy the discrepancy principle we will give the convergence proof of our algorithm and numerically show that it is better than some state of the art methods in terms of both speed and accuracy

the kauffman model on small world topology abstract we apply kauffman s automata on small world networks to study the crossover between the short range and the infinite range case we perform accurate calculations on square lattices to obtain both critical exponents and fractal dimensions particularly we find an increase of the damage propagation and a decrease in the fractal dimensions when adding long range connections

edge direction preserving image zooming a mathematical and numerical analysis we focus in this paper on some reconstruction restoration methods whose aim is to improve the resolution of digital images the main point here is to study the ability of such methods to preserve one dimensional 1d structures indeed such structures are important since they are often carried by the image edges first we focus on linear methods give a general framework to design them and show that the preservation of 1d structures pleads in favor of the cancellation of the periodization of the image spectrum more precisely we show that preserving 1d structures implies the linear methods to be written as a convolution of the sinc interpolation as a consequence we cannot cope linearly with gibbs effects sharpness of the results and the preservation of the 1d structure second we study variational nonlinear methods and in particular the one based on total variation we show that this latter permits us to avoid these shortcomings we also prove the existence and consistency of an approximate solution to this variational problem at last this theoretical study is highlighted by experiments both on synthetic and natural images which show the effects of the described methods on images as well as on their spectrum

local strong homogeneity of a regularized estimator this paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones where typically they are constant or linear or more generally satisfy a linear equation the estimate is defined as the minimizer of an objective function combining a quadratic data fidelity term and a regularization prior term the latter term is the sum of the values obtained by applying a potential function pf to each component called a difference of a linear transform of the signal minimizers of functions of this form arise in various settings in statistics and optimization the features exhibited by such an estimate are closely related to the shape of the pf our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data to this end we require that the strongly homogeneous zones recovered by the estimator be insensitive to any variation of th

efficient schemes for total variation minimization under constraints in image processing this paper presents new fast algorithms to minimize total variation and more generally l 1 norms under a general convex constraint such problems are standards of image processing the algorithms are based on a recent advance in convex optimization proposed by yurii nesterov depending on the regularity of the data fidelity term we solve either a primal problem or a dual problem first we show that standard first order schemes allow one to get solutions of precision epsilon in o frac 1 epsilon 2 iterations at worst we propose a scheme that allows one to obtain a solution of precision epsilon in o frac 1 epsilon iterations for a general convex constraint for a strongly convex constraint we solve a dual problem with a scheme that requires o frac 1 sqrt epsilon iterations to get a solution of precision epsilon finally we perform some numerical experiments which confirm the theoretical results on various problems of image processing

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

a high order 3d boundary integral equation solver for elliptic pdes in smooth domains we present a high order boundary integral equation solver for 3d elliptic boundary value problems on domains with smooth boundaries we use nystrom s method for discretization and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals the overall asymptotic complexity of our method is o n3 2 where n is the number of discretization points on the boundary of the domain and corresponds to linear complexity in the number of uniformly sampled evaluation points a kernel independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators we describe a high order accurate method for evaluating the solution at arbitrary points inside the domain including points close to the domain boundary we demonstrate how our solver combined with a regular grid spectral solver can be applied to problems with distributed sources we present numerical results for the stokes navier and poisson problems

a parallel alternating direction method with application to compound l1 regularized imaging inverse problems we derive a parallel alternating direction method of multipliers padmm and apply it to compound l1 regularized imaging inverse problems the proposed method is capable of locating the saddle point of large scale convex minimization problems with the sum of several nonsmooth but proximable terms using an operator splitting strategy the objective is decomposed into subproblems that are conveniently individually and simultaneously solved with the assistance of the moreau decomposition our method excludes auxiliary variables that exist in the admm and possesses a compacter structure thus the proposed method is preferable in distributed computation the convergence proof and convergence rate analysis are presented application to both image restoration and image compressed sensing demonstrates the effectiveness and efficiency of the proposed method

parallel proximal algorithm for image restoration using hybrid regularization regularization approaches have demonstrated their effectiveness for solving ill posed problems however in the context of variational restoration methods a challenging question remains namely how to find a good regularizer while total variation introduces staircase effects wavelet domain regularization brings other artefacts e g ringing however a tradeoff can be made by introducing a hybrid regularization including several terms not necessarily acting in the same domain e g spatial and wavelet transform domains while this approach was shown to provide good results for solving deconvolution problems in the presence of additive gaussian noise an important issue is to efficiently deal with this hybrid regularization for more general noise models to solve this problem we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms for spatial domain regularization isotropic or anisotropic total variation definitions using various gradient filters are considered an accelerated version of the parallel proximal algorithm is proposed to perform the minimization some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper numerical experiments performed in the context of poisson data recovery show the good behavior of the algorithm as well as promising results concerning the use of hybrid regularization techniques

the equivalence of half quadratic minimization and the gradient linearization iteration a popular way to restore images comprising edges is to minimize a cost function combining a quadratic data fidelity term and an edge preserving possibly nonconvex regularization term mainly because of the latter term the calculation of the solution is slow and cumbersome half quadratic hq minimization multiplicative form was pioneered by geman and reynolds 1992 in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization by promoting the idea of locally homogeneous image models with a continuous valued line process they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process hence the name half quadratic since then a large amount of papers were dedicated to hq minimization and important results including edge preservation along with convex regularization and convergence have been obtained in this paper we show that hq minimization multiplicative form is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step in fact both methods give exactly the same iterations furthermore connections of hq minimization with other methods such as the quasi newton method and the generalized weiszfeld s method are straightforward

a property of the minimum vectors of a regularizing functional defined by means of the absolute norm we consider a regularizing functional defined by means of the l sub 1 norm where the regularization is obtained using first differences as is well known such a functional can be put in relation with recursive median filters of appropriate window length we show that at least one of the minima is reached at a vector whose components have values over the same discrete set of the given signal this suggests a simple method to refine the approximate solution to the regularization problem which can be obtained with recursive median filters of increasing order we also report an example of application where the refinement method is employed for a signal detection problem

summability of hermite expansions ii we study the summability of n dimensional hermite expansions where n 1 we prove that the critical index for the riesz summability is n 1 2 we also prove analogues of the fejer lebesgue theorem and riemann s localisation principle when the index of the riesz means is 3n 2 6

analysis and evaluation of singular integrals by the invariant imbedding approach the invariant imbedding formulae for evaluating singular integrals on flat domains are used to analyse the singular integrals which arise in the boundary element method although all orders of singularity are considered the primary objective of the paper is the evaluation of cauchy principal value integrals cpv conditions for boundedness of the cpv are derived and evaluation techniques are presented for situations in which the singular point lies on or is in close proximity to an element boundary specific cases analysed include straight and curved element boundaries and corners other numerical aspects of the formulae are also discussed by way of example the analysis techniques are applied to the volume integrals which arise in continuum inelastic problems

multiplicative noise and blur removal by framelet decomposition and l_ 1 based l curve method this paper proposes a framelet based convex optimization model for multiplicative noise and blur removal problem the main idea is to employ framelet expansion to represent the original image and use the variable decomposition to solve the problem because of the nature of multiplicative noise we decompose the observed data into the original image variable and the noise variable to obtain the resulting model the original image variable is represented by framelet and it is determined by using l 1 norm in the selection and shrinkage of framelet coefficients the noise variable is measured by using the mean and the variance of the underlying probability distribution this framelet setting can be applied to analysis synthesis and balanced approaches and the resulting optimization models are convex such that they can be solved very efficiently by the alternating direction of a multiplier method an another contribution of this paper is to propose to select the regularization parameter by using the l 1 based l curve method for these framelet based models numerical examples are presented to illustrate the effectiveness of these models and show that the performance of the proposed method is better than that by the existing methods

an extended multistep shanks transformation and convergence acceleration algorithm with their convergence and stability analysis the molecule solution of an extended discrete lotka volterra equation is constructed from which a new sequence transformation is proposed a convergence acceleration algorithm for implementing this sequence transformation is found it is shown that our new sequence transformation accelerates some kinds of linearly convergent sequences and factorially convergent sequences with good numerical stability some numerical examples are also presented

can the nonlocal characterization of sobolev spaces by bourgain et al be useful for solving variational problems we question whether the recent characterization of sobolev spaces by bourgain brezis and mironescu 2001 could be useful to solve variational problems on w 1 p omega to answer this we introduce a sequence of functionals so that the seminorm is approximated by an integral operator involving a differential quotient and a radial mollifier then for the approximated formulation we prove existence uniqueness and convergence of the solution to the unique solution of the initial formulation we show that these results can also be extended in the bv case interestingly this approximation leads to a unified implementation for sobolev spaces including with high p values and for the bv space finally we show how this theoretical study can indeed lead to a numerically tractable implementation and we give some image diffusion results as an illustration

precompact convergence of the nonconvex primal dual hybrid gradient algorithm abstract the primal dual hybrid gradient pdhg algorithm is a powerful algorithm used quite frequently in recent years for solving saddle point optimization problems the classical application considers convex functions and it is well studied in literature in this paper we consider the convergence of an alternative formulation of the pdhg algorithm in the nonconvex case under the precompact assumption the proofs are based on the kurdyka ojasiewic functions that cover a wide range of problems a simple numerical experiment illustrates the convergence properties

accurate numerical integration of singular boundary element kernels over boundaries with curvature use of the green function for the solution of boundary value problems frequently results in singular integral equations algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method bem they are based upon the use of appropriately weighted gaussian quadrature formulae together with numerical geometrical transformations of the region of integration the use of high order subdomain expansion functions for interpolation over nonplanar elements allows boundary curvature to be accommodated in particular the handling of green functions with logarithmic and r 1 behaviour are detailed volume integrals with r 2 singularity are outlined operations are performed on a simplex thus resulting in generality and ease of automation this scheme has been incorporated into boundary element method software and successfully applied to a variety of problems

advanced boundary element analysis of two and three dimensional problems of elasto plasticity an advanced formulation of the boundary element method has been developed for inelastic analysis based on an initial stress approach the iterative solution algorithm makes use of an accelerated initial stress approach in which the past history of initial stresses are used to obtain an initial estimate for the current increment in the present analysis the geometry and functions are represented by higher order quadratic shape functions to model complex geometries and rapid functional variations accurately the methods of numerical integration of the kernels particularly the singular type are substantially improved by devising suitable automatic sub segmentation routines that incorporate the recent developments in mapping procedures the formulations have been implemented for two dimensional plane stress plane strain and three dimensional elasto plasticity problems

some properties of the sobolev laguerre polynomials abstract in this paper we study the case 0 of the sobolev laguerre polynomials we determine a generating function for the polynomials and an expansion formula

existence and uniqueness for a class of cauchy problems with characteristic initial manifolds on donne un theoreme d existence unicite pour une grande classe d equations d ordre 2 quasilineaires a donnees de cauchy sur une variete qui est caracteristique au sens ou l equation est singuliere d ordre 1 le long de la variete initiale

an invariant imbedding method for singular integral evaluation on finite domains the invariant imbedding technique is used to develop versatile and powerful integration formulae for the evaluation of singular integrals on finite domains encountered in continuum problems in engineering and applied science this method is considerably simpler than the several ad hoc integral evaluation methods presented in the recent literature moreover the method provides new insight into the central role played by the residue in the evaluation of principal value integrals this relationship is embodied as a gauge condition the main emphasis of the paper is on developing a general approach from which specific cases follow several examples chosen from different areas of engineering computation are provided in order to show the advantage of this approach

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

asymptotic summation of hermite series a new method for the numerical evaluation of slowly convergent or even divergent series involving the hermite functions is presented the author considers series with either of the forms where cm decays algebraically as m to infinity the first series is a fourier hermite series while the second arises in the representation of green functions for problems whose eigenfunctions involve the hermite functions by use of the poisson summation formula the author derives rapidly convergent asymptotic expansions for the remainders of these series after a sufficiently large number of terms the series can then be evaluated as a partial sum plus an asymptotic approximation to its remainder the asymptotic expansion for the remainder of g x y also reveals the nature of the possible singular behaviour of this series near x y

contrast invariant snr and isotonic regressions we design an image quality measure independent of contrast changes which are defined as a set of transformations preserving an order between the level lines of an image this problem can be expressed as an isotonic regression problem depending on the definition of a level line the partial order between adjacent regions can be defined through chains polytrees or directed acyclic graphs we provide a few analytic properties of the minimizers and design original optimization procedures together with a full complexity analysis the methods worst case complexities range from o n for chains to o n log n for polytrees and o frac n 2 sqrt epsilon for directed acyclic graphs where n is the number of pixels and epsilon is a relative precision the proposed algorithms have potential applications in change detection stereo vision image registration color image processing or image fusion a c implementation with matlab headers is available at https github com pierre weiss contrast invariant snr

asymptotics on laguerre or hermite polynomial expansions and their applications in gauss quadrature abstract in this paper we present asymptotic analysis on the coefficients of functions expanded in forms of laguerre or hermite polynomial series which shows the decay of the coefficients and derives new error bounds on the truncated series moreover by applying the asymptotics new estimates on the errors for gauss laguerre radau laguerre and gauss hermite quadrature are deduced these results show that gauss laguerre type and gauss hermite type quadratures are nearly of same convergence rates

some special interpolation formulae for triangular and quadrilateral elements presented herein are methods to generate special interpolation formulae of the following type i c0 continuous interpolation over triangular elements which induces a given type of singularity for the first derivatives at one of the vertices and yet preserves rigid and constant strain motions ii c1 continuous interpolation over triangular elements which induces a given type of singularity for second derivatives at one of the vertices

the epsilon algorithm and related topics the epsilon algorithm is recommended as the best all purpose acceleration method for slowly converging sequences it exploits the numerical precision of the data to extrapolate the sequence to its limit we explain its connections with pade approximation and continued fractions which underpin its theoretical base then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector valued sequences and some related topics in this paper we consider the class of methods based on using generalised inverses of vectors and the formulation specifically includes the complex case wherever possible

singular and near singular integrals in the bem a global approach a unified approach for analyzing and evaluating singular and near singular integrals relevant to the boundary element method bem is obtained the original n dimensional integral is mapped to one that is performed exclusively on the boundary of the integration domain therefore the dimensionality of the integration is reduced by l and integrand evaluations close to the singularity are avoided the new formulae are obtained through a straightforward one dimensional analytical integration regardless of the dimensionality of the problem and can be used generally for numerical integration the formulae for evaluating near singular integrals are a generalization of those obtained by the invariant imbedding technique for singular integrals and reduce to the singular integral formulae in the limit in this sense the singular integrals are treated as limits or continuations of nonsingular ones this leads to a general interpretation that includes the singular integrals as a special case in particular t

reconstruction of wavelet coefficients using total variation minimization we propose a model to reconstruct wavelet coefficients using a total variation minimization algorithm the approach is motivated by wavelet signal denoising methods where thresholding small wavelet coefficients leads to pseudo gibbs artifacts by replacing these thresholded coefficients by values minimizing the total variation our method performs a nearly artifact free signal denoising in this paper we detail the algorithm based on a subgradient descent combining a projection on a linear space the convergence of the algorithm is established and numerical experiments are reported

a parallel primal dual splitting method for image restoration we develop a parallel primal dual splitting method to solve large scale image restoration problems which involve the sum of several linear operator coupled nonsmooth but proximable terms with the proposed method the objective function is decomposed into pieces that can be processed individually no inverse operator is involved in our method and the highly parallel structure makes it preferable in distributed computation the convergence is proven and the convergence rate is analyzed besides its equivalence to the relaxed parallel linearized alternating direction method of multipliers pladmm is addressed applications to image restoration problems with compound l1 regularizer and comparisons with state of the art methods are detailed to show the superiority of the proposed method

descent gradient methods for nonsmooth minimization problems in ill posed problems descent gradient methods are the most frequently used algorithms for computing regularizers of inverse problems they are either directly applied to the discrepancy term which measures the difference between operator evaluation and data or to a regularized version incorporating suitable penalty terms in its basic form gradient descent methods converge slowly we aim at extending different optimization schemes which have been recently introduced for accelerating these approaches by addressing more general penalty terms in particular we use a general setting in infinite hilbert spaces and examine accelerated algorithms for regularization methods using total variation or sparsity constraints to illustrate the efficiency of these algorithms we apply them to a parameter identification problem in an elliptic partial differential equation using total variation regularization

sparse occlusion detection with optical flow we tackle the problem of detecting occluded regions in a video stream under assumptions of lambertian reflection and static illumination the task can be posed as a variational optimization problem and its solution approximated using convex minimization we describe efficient numerical schemes that reach the global optimum of the relaxed cost functional for any number of independently moving objects and any number of occlusion layers we test the proposed algorithm on benchmark datasets expanded to enable evaluation of occlusion detection performance in addition to optical flow

the continuation approach a general framework for the analysis and evaluation of singular and near singular integrals the continuation approach presented in this paper is a simple and unified framework for analyzing and computing a large class of singular and near singular integrals such as those that arise in the boundary element method the analysis formulates singular and near singular integrals consistently as instances of a single phenomenon with all types of algebraic singularities treated in a similar way for domains of arbitrary dimension singular integrals are viewed merely as continuations of nonsingular but perhaps near singular ones by placing the singularity of the integrand outside the integration domain and taking the limit as it approaches the domain the technique exploits the functional homogeneity shared by the green s functions of many physical problems for flat surfaces this allows the integral to be mapped to the contour of the integration domain curved surfaces are handled by projecting the integration domain to a tangent hyperplane where the singular components of the integral can be

subspace correction methods for total variation and ell_1 minimization this paper is concerned with the numerical minimization of energy functionals in hilbert spaces involving convex constraints coinciding with a seminorm for a subspace the optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces on each subspace an iterative proximity map algorithm is implemented via oblique thresholding which is the main new tool introduced in this work we provide convergence conditions for the algorithm in order to compute minimizers of the target energy analogous results are derived for a parallel variant of the algorithm applications are presented in domain decomposition methods for degenerate elliptic pdes arising in total variation minimization and in accelerated sparse recovery algorithms based on ell 1 minimization we include numerical examples which show efficient solutions to classical problems in signal and image processing

numerical analysis 2000 vol ii interpolation and extrapolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the rst 75 years of this century including a pioneering paper by kronecker in the 19th century c 2000 elsevier science b v all rights reserved