polynomials for infinite domain spectral elements in this note it is found that laguerre type elements may be used in conjunction with legendre type elements to resolve more accurately regions of outflow and infinite boundaries

asymptotic coefficients of hermite function series abstract by using complex variable methods steepest descent and residues to asymptotically evaluate the coefficient integrals the numerical analysis of hermite function series is discussed there are striking similarities and differences with the author s earlier work on chebyshev polynomial methods j comp phys 45 1982 45 49 for infinite or semi infinite domains like chebyshev series the hermite coefficients are asymptotically given by the sum of two types of terms i stationary point steepest descent contributions and ii residues at the poles of f z the function being expanded as a hermite series the stationary point term is determined solely by the asymptotic behavior of f z i e how rapidly f z decays as z along the real axis unlike chebyshev series however it is necessary to perform a separate analysis for functions which decay faster or slower than the gaussian function exp az2 singular functions too fall into two categories those that decay rapidly with z have asymptotic hermite coefficients which are dominated by the singularity but functions which decay as slowly as sech z or slower have hermite coefficients dominated by the stationary point terms and the singularity is irrelevant the end product of the analysis is the same as in the earlier work simple explicit formulas to optimize the efficiency of hermite methods and estimate a priori how many degrees of freedom are needed provided one knowns at least crudely i the asymptotic behavior of f z and ii its singularity nearest the real axis rather surprisingly one finds hermite functions superior to chebyshev polynomials for some classes of functions when the computational domain is infinite

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

the test of the finite size scaling relations of the ising models in seven and eight dimensions on the creutz cellular automaton abstract the ising models in seven and eight dimensions are simulated on the creutz cellular automaton using the finite size lattices with the linear dimension 4 l 8 three different finite size exponents for the order parameter near the curie point are computed to be 1 74 3 0 95 2 2 42 5 1 99 19 0 96 1 and 2 95 11 for d 7 and d 8 dimensions respectively the obtained results are in good agreement with the theoretical predictions 7 4 1 5 2 2 1 3 for d 7 and 8 dimensions respectively the exponent in the finite size scaling relation for the magnetic suscebtibility at the infinite lattice critical temperature is computed to be 4 03 9 using 4 l 8 which is in very good agreement with the theoretical prediction of 4 for d 8 dimension the finite size scaling relation for the magnetic susceptibility at the infinite lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite size lattices the finite size scaling plots of the order parameter and the magnetic susceptibility verify the finite size scaling relations about the infinite lattice critical temperature

a new generalized laguerre spectral approximation and its applications a new family of generalized laguerre polynomials is introduced various orthogonal projections are investigated some approximation results are established as an example of their important applications the mixed spherical harmonic generalized laguerre approximation is developed a mixed spectral scheme is proposed for a three dimensional model problem its convergence is proved numerical results demonstrate the high accuracy of this new spectral method

generalized sampling a variational approach i theory we consider the problem of reconstructing a multidimensional vector function f sub in spl ropf sup m spl rarr spl ropf sup n from a finite set of linear measures these can be irregularly sampled responses of several linear filters traditional approaches reconstruct in an a priori given space e g the space of bandlimited functions instead we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion j first we present the solution of the generalized interpolation problem later we also consider the approximation problem and we show that both lead to the same class of solutions imposing generally desirable properties on the reconstruction largely limits the choice of the criterion j linearity leads to a quadratic criterion based on bilinear forms specifically we show that the requirements of translation rotation and scale invariance restrict the form of the criterion to essentially a one parameter family we show that the solution can be obtained as a linear combination of generating functions we provide analytical techniques to find these functions and the solution itself

rational legendre pseudospectral approach for solving nonlinear differential equations of lane emden type lane emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models in this paper a pseudospectral technique is proposed to solve the lane emden type equations on a semi infinite domain the method is based on rational legendre functions and gauss radau integration the method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations the comparison of the results with the other numerical methods shows the efficiency and accuracy of this method

a complete orthonormal system of functions in l 2 infty infty space a new complete orthonormal system of functions in the l 2 infty infty space is introduced the system consists of two sequences composed of odd and even functions respectively unlike hermite and laguerre sets of functions which behave exponentially at infinity the new system exhibits asymptoticbehavior x 1 for the odd sequence and x 2 for the even one formulae representing products derivatives etc in series in the system are developed a nonlinear differential equation with a requirement for summability of the square of its solutions instead of boundary conditions is solved the example displays most of the features of the new method important for essentially nonlinear problems

a fast adaptive parameter estimation for total variation image restoration estimation of the regularization parameter which strikes a balance between the data fidelity and regularity is essential for successfully solving ill posed image restoration problems based on the classical total variation tv model and prevalent alternating direction method of multipliers we hammer out a fast algorithm being able to simultaneously estimate the regularization parameter and restore the degraded image by applying variable splitting technique to both the regularization term and data fidelity term we overcome the nondifferentiability of tv and achieve a closed form to update the regularization parameter in each iteration the solution is guaranteed to satisfy morozov s discrepancy principle furthermore we present a convergence proof for the proposed algorithm on the premise of a variable regularization parameter experimental results demonstrate that the proposed algorithm is superior in speed and competitive in accuracy compared with several state of the art methods besides the proposed method can be smoothly extended to the multichannel image restoration

image restoration via the adaptive tvp regularization abstract to keep structures in the restoration problem is very important via coupling the local information of the image with the proposed model in this paper we propose a local self adaptive l p regularization model for p 0 2 based on the total variation scheme where the choice of p depends on the local structures described by the eigenvalues of the structure tensor since the proposed model as the classic l p problem unifies two classes of optimization problems such as the nonconvex and nonsmooth problem when p 0 1 and the convex and smooth problem when p 1 2 it is generally challenging to find a ready algorithmic framework to solve it here we propose a new and robust numerical method via coupling with the half quadratic scheme and the alternating direction method of multipliers admm the convergence of the proposed algorithm is established and the numerical experiments illustrate the possible advantages of the proposed model and numerical methods over some existing variational based models and methods

generalized higher degree total variation hdtv regularization we introduce a family of novel image regularization penalties called generalized higher degree total variation hdtv these penalties further extend our previously introduced hdtv penalties which generalize the popular total variation tv penalty to incorporate higher degree image derivatives we show that many of the proposed second degree extensions of tv are special cases or are closely approximated by a generalized hdtv penalty additionally we propose a novel fast alternating minimization algorithm for solving image recovery problems with hdtv and generalized hdtv regularization the new algorithm enjoys a tenfold speed up compared with the iteratively reweighted majorize minimize algorithm proposed in a previous paper numerical experiments on 3d magnetic resonance images and 3d microscopy images show that hdtv and generalized hdtv improve the image quality significantly compared with tv

low dose cbct reconstruction using hessian schatten penalties cone beam computed tomography cbct has been widely used in radiation therapy for accurate patient setup and treatment target localization it is important to obtain high quality reconstruction images the total variation tv penalty has shown the state of the art performance in suppressing noise and preserving edges for statistical iterative image reconstruction but it sometimes leads to the so called staircase effect in this paper we proposed to use a new family of penalties the hessian schatten hs penalties for the cbct reconstruction consisting of the second order derivatives the hs penalties are able to reflect the smooth intensity transitions of the underlying image without introducing the staircase effect we discussed and compared the behaviors of several convex hs penalties with orders 1 2 and infty for cbct reconstruction we used the majorization minimization approach with a primal dual formulation for the corresponding optimization problem experiments on two digital phantoms and two physical phantoms demonstrated the proposed penalty family s outstanding performance over tv in suppressing the staircase effect and the hs penalty with order 1 had the best performance among the hs penalties tested

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

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

the sobolev orthogonality and spectral analysis of the laguerre polynomials l n k for positive integers k for k n we consider the analysis of the classical laguerre differential expression l k y x 1 x ke x x k 1e xy x rx ke xy x x 0 where r 0 is fixed in several nonisomorphic hilbert and hilbert sobolev spaces in one of these spaces specifically the hilbert space l2 0 x ke x it is well known that the glazman krein naimark theory produces a self adjoint operator a k generated by l k that is bounded below by rl where i is the identity operator on l2 0 x ke x consequently as a result of a general theory developed by littlejohn and wellman there is a continuum of left definite hilbert spaces h s k v s k s k s 0 and left definite self adjoint operators b s k s 0 associated with the pair l2 0 x ke x a k for a k and each of the operators b s k it is the case that the tail end sequence l n k n k of laguerre polynomials form a complete set of eigenfunctions in the corresponding hilbert spaces in 1995 kwon and littlejohn introduced a hilbert sobolev space w k 0 in which the entire sequence of laguerre polynomials is orthonormal in this paper we construct a self adjoint operator in this space generated by the second order laguerre differential expression l k having l n k n 0 as a complete set of eigenfunctions the key to this construction is in identifying a certain closed subspace of w k 0 with the kth left definite vector space v k k

the spectral method for high order problems with proper simulations of asymptotic behaviors at infinity in this paper we investigate new spectral and multidomain spectral methods for high order problems we introduce a family of new generalized laguerre functions which are mutually orthogonal with the weight function x a d x c d 0 a and c being arbitrary real numbers the corresponding quasi orthogonal approximation and laguerre gauss radau type interpolation are proposed the spectral and multidomain spectral schemes are provided for several model problems which not only fit the mixed inhomogeneous boundary conditions on the fixed boundary exactly but also match the asymptotic behaviors at infinity reasonably numerical results demonstrate the efficiency of suggested algorithms and confirm the analysis well

failure of universal finite size scaling above the upper critical dimension abstract previous theories have predicted that o n symmetric systems in a finite cubic geometry with periodic boundary conditions have universal finite size scaling functions near criticality in d 4 dimensions on the basis of exact results for the o n symmetric 4 model in the large n limit we show that universal finite size scaling does not hold in the predicted form because of significant cut off and lattice effects for d 4 it is shown that finite size scaling is valid with two reference lengths which turn out to be identical with the amplitudes of the bulk correlation length for the 4 field theory the finite size scaling functions are shown to be non universal i e to depend explicitly on the cut off and on the bare four point coupling constant whereas for a 4 lattice model the finite size scaling functions have a different form that is independent of the lattice spacing and the four point coupling constant

finite size effect in seven dimensional site percolation on a hypercubic lattice with l7 sites and l up to 21 the site percolation threshold for the spanning of a cluster from top to bottom is found to approach its asymptotic value with a 1 l law in contrast to the ising curie temperature in five dimensions

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

pseudospectral method for non isotropic heat transfer using mixed hermite legendre interpolation abstract in this paper we develop a new mixed pseudospectral method for heat transfer by using generalized hermite functions and legendre polynomials in unbounded domains fundamental to spectral methods for various unbounded domain problems we establish some basic results on the mixed generalized hermite legendre interpolation as an example a new mixed generalized hermite legendre pseudospectral scheme is provided for non isotropic heat transfer its convergence is proved numerical results demonstrate the spectral accuracy of this approach

higher degree total variation hdtv regularization for image recovery we introduce novel image regularization penalties to overcome the practical problems associated with the classical total variation tv scheme motivated by novel reinterpretations of the classical tv regularizer we derive two families of functionals involving higher degree partial image derivatives we term these families as isotropic and anisotropic higher degree tv hdtv penalties respectively the isotropic penalty is the mixed norm of the directional image derivatives while the anisotropic penalty is the separable norm of directional derivatives these functionals inherit the desirable properties of standard tv schemes such as invariance to rotations and translations preservation of discontinuities and convexity the use of mixed norms in isotropic penalties encourages the joint sparsity of the directional derivatives at each pixel thus encouraging isotropic smoothing in contrast the fully separable norm in the anisotropic penalty ensures the preservation of discontinuities while continuing to smooth along the line like features this scheme thus enhances the linenlike image characteristics analogous to standard tv we also introduce efficient majorize minimize algorithms to solve the resulting optimization problems the numerical comparison of the proposed scheme with classical tv penalty current second degree methods and wavelet algorithms clearly demonstrate the performance improvement specifically the proposed algorithms minimize the staircase and ringing artifacts that are common with tv and wavelet schemes while better preserving the singularities we also observe that anisotropic hdtv penalty provides consistently improved reconstructions compared with the isotropic hdtv penalty

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

finite size scaling for percolation above six dimensions we consider percolation in d dimensions for finite samples of linear size l theoretical arguments are presented to show that for d 6 the shift in the percolation threshold pc l pc behaves like a l2 b ld3 for periodic boundary conditions a 0 these predictions are consistent with recent simulations is seven dimensions

jacobi and laguerre quasi orthogonal approximations and related interpolations in this paper we investigate jacobi quasi orthogonal approximation and generalized jacobi gauss lobatto interpolation we also propose laguerre quasi orthogonal approximation and generalized laguerre gauss radau interpolation a series of sharp results on these approximations are established which are applicable to spectral and pseudospectral methods for mixed nonhomogeneous boundary value problems of high order

percolation in high dimensions is not understood abstract the number of spanning clusters in four to nine dimensions does not fully follow the expected size dependence for random percolation

essential imposition of neumann condition in galerkin legendre elliptic solvers a new galerkin legendre direct spectral solver for the neumann problem associated with laplace and helmholtz operators in rectangular domains is presented the algorithm differs from other neumann spectral solvers by the high sparsity of the matrices exploited in conjunction with the direct product structure of the problem the homogeneous boundary condition is satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the legendre polynomials and have a zero slope at the interval extremes a double diagonalization process is employed pivoting around the eigenstructure of the pentadiagonal mass matrices in both directions instead of the full stiffness matrices encountered in the classical variational formulation of the problem with a weak natural imposition of the derivative boundary condition nonhomogeneous neumann data are accounted for by means of a lifting numerical results are given to illustrate the performance of the proposed spectral elliptic solver the algorithm extends easily to the three dimensional problem

approximation of some diffusion evolution equations in unbounded domains by hermite functions spectral and pseudospectral approximations of the heat equation are analyzed the solution is represented in a suitable basis constructed with hermite polynomials stability and convergence estimates are given and numerical tests are discussed

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

generalized laguerre interpolation and pseudospectral method for unbounded domains in this paper error estimates for generalized laguerre gauss type interpolations are derived in nonuniformly weighted sobolev spaces weighted with omega alpha beta x x alpha e beta x alpha 1 beta 0 generalized laguerre pseudospectral methods are analyzed and implemented two model problems are considered the proposed schemes keep spectral accuracy and with suitable choice of basis functions lead to sparse and symmetric linear systems

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

hermite pseudospectral approximations an error estimate abstract we find an error bound for the pseudospectral approximation of a function in terms of hermite functions h n x e x 2 h n x in certain weighted sobolev spaces we use properties of hermite polynomials as well as an asymptotic expression for their largest zeroes to achieve the mentioned estimate

hessian based norm regularization for image restoration with biomedical applications we present nonquadratic hessian based regularization methods that can be effectively used for image restoration problems in a variational framework motivated by the great success of the total variation tv functional we extend it to also include second order differential operators specifically we derive second order regularizers that involve matrix norms of the hessian operator the definition of these functionals is based on an alternative interpretation of tv that relies on mixed norms of directional derivatives we show that the resulting regularizers retain some of the most favorable properties of tv i e convexity homogeneity rotation and translation invariance while dealing effectively with the staircase effect we further develop an efficient minimization scheme for the corresponding objective functions the proposed algorithm is of the iteratively reweighted least square type and results from a majorization minimization approach it relies on a problem specific preconditioned conjugate gradient method which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time we validate the overall proposed regularization framework through deblurring experiments under additive gaussian noise on standard and biomedical images

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

upper critical dimension of the negative weight percolation problem by means of numerical simulations we investigate the geometric properties of loops on hypercubic lattice graphs in dimensions d 2 through 7 where edge weights are drawn from a distribution that allows for positive and negative weights we are interested in the appearance of system spanning loops of total negative weight the resulting negative weight percolation nwp problem is fundamentally different from conventional percolation as we have seen in previous studies of this model for the two dimensional case here we characterize the transition for hypercubic systems where the aim of the present study is to get a grip on the upper critical dimension d u of the nwp problem for the numerical simulations we employ a mapping of the nwp model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms we characterize the loops via observables similar to those in percolation theory and perform finite size scaling analyses e g three dimensional hypercubic systems with side length up to l 56 sites in order to estimate the critical properties of the nwp phenomenon we find our numerical results consistent with an upper critical dimension d u 6 for the nwp problem

generalized hermite spectral method and its applications to problems in unbounded domains in this paper we develop a spectral method based on generalized hermite functions with weight chi x equiv1 we also establish some basic results on generalized hermite orthogonal approximations which play an important role in spectral methods as examples the generalized ginzburg landau equation in a population problem and an elliptic equation with a harmonic potential are considered related spectral schemes are proposed and their convergence is proved numerical results demonstrate the spectral accuracy of this approach

splines in higher order tv regularization splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces in this paper we show in a strictly discrete setting that splines of degree m 1 solve also a minimization problem with quadratic data term and m th order total variation tv regularization term in contrast to problems with quadratic regularization terms involving m th order derivatives the spline knots are not known in advance but depend on the input data and the regularization parameter more precisely the spline knots are determined by the contact points of the m th discrete antiderivative of the solution with the tube of width 2 around the m th discrete antiderivative of the input data we point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the sobolev space w 2 0 m from this point of view the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines

generalized hermite spectral method matching asymptotic behaviors in this paper we propose the generalized hermite spectral method by using a family of new generalized hermite functions which are mutually orthogonal with the weight function 1 x 2 c c being an arbitrary real number we establish the basic results on the corresponding orthogonal approximation and interpolation which simulate the asymptotic behaviors of approximated functions at infinity reasonably as examples of applications the spectral schemes are provided for two model problems numerical results demonstrate their spectral accuracy in space

combined hermite spectral finite difference method for the fokker planck equation the convergence of a class of combined spectral finite difference methods using hermite basis applied to the fokker planck equation is studied it is shown that the hermite based spectral methods are convergent with spectral accuracy in weighted sobolev space numerical results indicating the spectral convergence rate are presented a velocity scaling factor is used in the hermite basis and is shown to improve the accuracy and effectiveness of the hermite spectral approximation with no increase in workload some basic analysis for the selection of the scaling factors is also presented

the spectral methods for parabolic volterra integro differential equations in this paper we study the numerical solutions to parabolic volterra integro differential equations in one dimensional bounded and unbounded spatial domains in a bounded domain the given parabolic volterra integro differential equation is converted to two equivalent equations then a legendre collocation method is used to solve them and finally a linear algebraic system is obtained for an unbounded case we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain in both cases some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

exponentially accurate nonconforming least squares spectral element method for elliptic problems on unbounded domain abstract non conforming approximation methods are becoming increasingly popular because of the potential to apply to multi material and multi model analysis for both bounded and unbounded domains in this paper we present a least square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain the method gives exponential accuracy and shows superior performance when compared to other numerical methods differentiability estimates and the main stability estimate theorem using a non conforming spectral element method are also discussed the exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

approximation theory for the p version of the finite element method in three dimensions part ii convergence of the p version of the finite element method this paper is the second in a series devoted to the approximation theory of the p version of the finite element method in three dimensions in this paper we analyze the approximability of functions in the framework of the jacobi weighted besov and sobolev spaces in the three dimensional setting and approximation error of the p version of the finite element method on meshes containing tetrahedral hexahedral and prism elements for elliptic problems with homogeneous and nonhomogeneous dirichlet boundary condition the convergence rate is rigorously proved for solutions in h k omega the optimal convergence of the p version for elliptic problems on polyhedral domains will be analyzed in the forthcoming part iii b q guo approximation theory of the p version of the finite element method in three dimensions part iii optimal convergence for problems on polyhedral domains in preparation of the series

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

chebyshev rational spectral and pseudospectral methods on a semi infinite interval a weighted orthogonal system on the half line based on the chebyshev rational functions is introduced basic results on chebyshev rational approximations of several orthogonal projections and interpolations are established to illustrate the potential of the chebyshev rational spectral method a model problem is considered both theoretically and numerically error estimates for the chebyshev rational spectral and pseudospectral methods are established preliminary numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach

hessian schatten norm regularization for linear inverse problems we introduce a novel family of invariant convex and non quadratic functionals that we employ to derive regularized solutions of ill posed linear inverse imaging problems the proposed regularizers involve the schatten norms of the hessian matrix which are computed at every pixel of the image they can be viewed as second order extensions of the popular total variation tv semi norm since they satisfy the same invariance properties meanwhile by taking advantage of second order derivatives they avoid the staircase effect a common artifact of tv based reconstructions and perform well for a wide range of applications to solve the corresponding optimization problems we propose an algorithm that is based on a primal dual formulation a fundamental ingredient of this algorithm is the projection of matrices onto schatten norm balls of arbitrary radius this operation is performed efficiently based on a direct link we provide between vector projections onto norm balls and matrix projections onto schatten norm balls finally we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data

a laguerre legendre spectral method for the stokes problem in a semi infinite channel a mixed spectral method is proposed and analyzed for the stokes problem in a semi infinite channel the method is based on a generalized galerkin approximation with laguerre functions in the x direction and legendre polynomials in the y direction the well posedness of this method is established by deriving a lower bound on the inf sup constant numerical results indicate that the derived lower bound is sharp rigorous error analysis is also carried out