postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

the error amended sharp edge ease scheme for image zooming this paper proposes a new interpolation method called the error amended sharp edge ease scheme which is a modified bilinear method in order to remove reduce interpolation artifacts such as image blur and the checkerboard effect ringing ease tries to amend the interpolation error by employing the classical interpolation error theorem in an edge adaptive fashion ease is applied for image zooming by both integer and noninteger magnification factors the new interpolation scheme has proved to result in high resolution images having clearer and sharper edges than linear interpolation methods for all synthetic and natural images we have tested ease can be implemented with ease it turns out to be similarly efficient as cubic interpolation schemes

regularity preserving image interpolation assumptions about image continuity lead to oversmoothed edges in common image interpolation algorithms a wavelet based interpolation method that imposes no continuity constraints is introduced the algorithm estimates the regularity of edges by measuring the decay of wavelet transform coefficients across scales and preserves the underlying regularity by extrapolating a new subband to be used in image resynthesis the algorithm produces visibly sharper edges than traditional techniques and exhibits an average peak signal to noise ratio psnr improvement of 2 5 db over bilinear and bicubic techniques

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

on the role of exponential splines in image interpolation a sobolev reproducing kernel hilbert space approach to image interpolation is introduced the underlying kernels are exponential functions and are related to stochastic autoregressive image modeling the corresponding image interpolants can be implemented effectively using compactly supported exponential b splines a tight l2 upper bound on the interpolation error is then derived suggesting that the proposed exponential functions are optimal in this regard experimental results indicate that the proposed interpolation approach with properly tuned signal dependent weights outperforms currently available polynomial b spline models of comparable order furthermore a unified approach to image interpolation by ideal and nonideal sampling procedures is derived suggesting that the proposed exponential kernels may have a significant role in image modeling as well our conclusion is that the proposed sobolev based approach could be instrumental and a preferred alternative in many interpolation tasks

subpixel edge localization and the interpolation of still images in this paper we present a nonlinear interpolation scheme for still image resolution enhancement the algorithm is based on a source model emphasizing the visual integrity of detected edges and incorporates a novel edge fitting operator that has been developed for this application a small neighborhood about each pixel in the low resolution image is first mapped to a best fit continuous space step edge the bilevel approximation serves as a local template on which the higher resolution sampling grid can then be superimposed where disputed values in regions of local window overlap are averaged to smooth errors the result is an image of increased resolution with noticeably sharper edges and in all tried cases lower mean squared reconstruction error than that produced by linear techniques

fuzzy decision modeling for supply chain management managing a supply chain sc is very difficult since various sources of uncertainty and complex interrelationships between various entities exist in the sc moreover the reducing product life cycle and the heightened expectations of customers have also made the sc even harder to manage especially for innovative products although the innovative products can enable a company to achieve higher profit margins they make demands for them unpredictable because no historical data is available this paper develops a fuzzy decision methodology that provides an alternative framework to handle sc uncertainties and to determine sc inventory strategies while there is lack of certainty in data or even lack of available historical data fuzzy set theory is used to model sc uncertainty a fuzzy sc model based on possibility theory is developed to evaluate sc performances based on the proposed fuzzy sc model a genetic algorithm approach is developed to determine the order up to levels of stock keeping units in the sc to minimize the sc inventory cost subject to the restriction of fulfilling the target fill rate of the finished product the proposed model allows decision makers to express their risk attitudes and to analyze the trade off between customer service level and inventory investment in the sc and better sc inventory strategies can be made a simulation approach is used to validate the concept developed

superconvergence analysis of finite element method for the time dependent schrodinger equation in this paper we consider the two dimensional time dependent schrodinger equation firstly we use the rectangular lagrange type finite element of order p to get a semi discrete scheme of the equation and discuss the superconvergence error estimate in the h 1 norm secondly we use the crank nicolson method in time to get a fully discrete scheme of the equation and the superconvergence estimate in the h 1 norm can be obtained in this scheme finally a numerical example with the order p 1 is provided to verify our theoretical results

curvature interpolation method for image zooming we introduce a novel image zooming algorithm called the curvature interpolation method cim which is partial differential equation pde based and easy to implement in order to minimize artifacts arising in image interpolation such as image blur and the checkerboard effect the cim first evaluates the curvature of the low resolution image after interpolating the curvature to the high resolution image domain the cim constructs the high resolution image by solving a linearized curvature equation incorporating the interpolated curvature as an explicit driving force it has been numerically verified that the new zooming method can produce clear images of sharp edges which are already denoised and superior to those obtained from linear methods and pde based methods of no curvature information various results are given to prove effectiveness and reliability of the new method

linear interpolation revitalized we present a simple original method to improve piecewise linear interpolation with uniform knots we shift the sampling knots by a fixed amount while enforcing the interpolation property we determine the theoretical optimal shift that maximizes the quality of our shifted linear interpolation surprisingly enough this optimal value is nonzero and close to 1 5 we confirm our theoretical findings by performing several experiments a cumulative rotation experiment and a zoom experiment both show a significant increase of the quality of the shifted method with respect to the standard one we also observe that in these results we get a quality that is similar to that of the computationally more costly high quality cubic convolution

superconvergence in high order galerkin finite element methods in this paper we shall use local estimates to give the superconvergence of high order galerkin finite element method for the elliptic equation of second order with constant coefficients by using the symmetric technique and integral identity we get improved superconvergence on the inner locally symmetric mesh with respect to a point x0 for rectangular and triangular meshes

global superconvergence for maxwell s equations in this paper the global superconvergence is analysed on two schemes a mixed finite element scheme and a finite element scheme for maxwell s equations in r 3 such a supercovergence analysis is achieved by means of the technique of integral identity which has been used in the supercovergence analysis for many other equations and schemes on a rectangular mesh and then are generalized into more general domains and problems with the variable coefficients besides being more direct our analysis generalizes the results of monk

superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis abstract in this article we address adaptive methods for isogeometric analysis based on local refinement guided by recovery based a posteriori error estimates isogeometric analysis was introduced a decade ago and an impressive progress has been made related to many aspects of numerical methods and advanced applications however related to adaptive mesh refinement guided by a posteriori error estimators rather few attempts are pursued besides the use of classical residual based error estimators in this article we explore a feature common for isogeometric analysis iga namely the use of structured tensorial meshes that facilitates superconvergence behavior of the gradient in the galerkin discretization by utilizing the concept of structured mesh refinement using lr b splines our aim is to facilitate superconvergence behavior for locally refined meshes as well superconvergence behavior matches well with the use of recovery based a posteriori estimator in the superconvergent patch recovery spr procedure however to our knowledge so far the spr procedure has not been exploited in the iga community we start out by addressing the existence of derivative superconvergent points in the computed finite element solution based on b splines and lr b splines for an elliptic model problem 1d and 2d poisson then we present some recovery procedures for improving the derivatives or gradient of the isogeometric finite element solution where the spr procedure will be the main focus in particular we show that our spr procedure for the improvement of derivatives fulfills the desired consistency criteria at the end we develop a posteriori error estimator where the improved gradient obtained from the proposed recovery procedures is used numerical results are presented to illustrate the efficiency of using spr procedure for the improvement of derivatives or gradient of computed solution in isogeometric analysis then the proposed a posteriori error estimator based adaptive refinement methodology is tested to solve smooth and non smooth elliptic benchmark problems the focus is put on whether optimal convergence rates are obtained in the computed solution or not as well as the effectivity index of the proposed error estimators

segment adaptive gradient angle interpolation we introduce a new edge directed interpolator based on locally defined straight line approximations of image isophotes spatial derivatives of image intensity are used to describe the principal behavior of pixel intersecting isophotes in terms of their slopes the slopes are determined by inverting a tridiagonal matrix and are forced to vary linearly from pixel to pixel within segments image resizing is performed by interpolating along the approximated isophotes the proposed method can accommodate arbitrary scaling factors provides state of the art results in terms of psnr as well as other quantitative visual quality metrics and has the advantage of reduced computational complexity that is directly proportional to the number of pixels

a numerical study of some hessian recovery techniques on isotropic and anisotropic meshes spaces of continuous piecewise linear finite elements are considered to solve a poisson problem and several numerical methods are investigated to recover second derivatives numerical results on 2d and 3d isotropic and anisotropic meshes indicate that the quality of the results is strongly linked to the mesh topology and that no convergence can be insured in general

superconvergence of h div finite element approximations for the stokes problem by l2 projection methods a general superconvergence result of h div finite element approximations for the stokes equations is established by using l 2 projection method regularity assumptions for the stokes problem with regular partitions is required numerical experiments are given to verify the theoretical results

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

superconvergence analysis of second and third order rectangular edge elements with applications to maxwell s equations abstract superconvergence for the second and third order edge elements is investigated on nonuniform rectangular meshes first we develop the explicit expression for the nedelec interpolation based on the hierarchical basis functions then we prove that the pointwise interpolation error estimates are one order higher at element gauss points than the standard analysis can provide using the superconvergence at gauss points we establish the discrete l 2 norm superconvergence for the solution of maxwell s equations solved by both the second and third order rectangular edge elements numerical results justifying our theoretical analysis are presented

correspondence analysis with fuzzy data the fuzzy eigenvalue problem this paper constitutes a first step towards an extension of correspondence analysis with fuzzy data fca at this stage our main objective is to lay down the algebraic foundations for this fuzzy extension of the usual correspondence analysis a two step method is introduced to convert the fuzzy eigenvalue problem to an ordinary one we consider a fuzzy matrix as the set of its cuts each such cut is an interval valued matrix viewed as a line segment in the matrix space in this way line segments of cut matrices are transformed into intervals of eigenvalues therefore the two step method is essentially a reduction of the fuzzy eigenvalue problem to an ordinary one we illustrate the fca fuzzy eigenvalue problem with a simple numerical example we hope upon the completion of this project in near future to be able to supply the necessary tools for the end user of the correspondence analysis with fuzzy data

the local superconvergence of the trilinear element for the three dimensional poisson problem abstract in this paper we shall combine the finite element theory of green s function presented in this paper the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three dimensional poisson equation

superconvergence results on mildly structured triangulations abstract many results on superconvergence for recovered gradients of piecewise linear galerkin approximations on triangular mesh partitions to the weak solutions of elliptic boundary value problems in two dimensions have been proved in recent years these were obtained first for regular fully structured partitions in which the mid points of the diagonals of all quadrilaterals formed by pairs of adjacent elements are coincident this condition was then relaxed to allow for strongly regular meshes in which the distance between the above mid points is o h2 h being the mesh size parameter in this paper these conditions are weakend still further to the case of globally mildly structured meshes where the mid point distance is o h1 0

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient

a parametric representation of fuzzy numbers and their arithmetic operators abstract direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex implementation of the extension principle is equivalent to solving a nonlinear programming problem to overcome this difficulty many applications limit the membership functions to certain shapes usually either triangular fuzzy numbers tfn or trapezoidal fuzzy numbers trfn then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers thus making the calculations trivial unfortunately the tfn shape is not closed under multiplication and division the result of these operators is a polynomial membership function and the triangular shape only approximates the actual result the linear approximation can be quite poor and may lead to incorrect results when used in engineering applications we analyze this problem and propose six parameters which define parameterized fuzzy numbers pfn of which tfns are a special case we provide the methods for performing fuzzy arithmetic and show that the pfn representation is closed under the arithmetic operations the new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as tfns the new method has better accuracy and similar computational speed to using tfns and appears to have benefits when used in engineering applications

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

superconvergence of h div finite element approximations for the stokes problem by local l 2 projection methods in this paper a general superconvergence result of h div finite element approximations for the stokes equations is established by using the local l 2 projection method local regularity assumptions for the stokes problem on regular partitions are required in order to derive superconvergence conclusions several numerical experiments are given to verify the theoretical results

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

superconvergence of quadratic finite volume method on triangular meshes abstract this paper is concerned with the superconvergence properties of the quadratic finite volume method fvm on triangular meshes for elliptic equations we proved the 3rd order superconvergence rate of the gradient approximation u h u i 1 o h 3 and the 4th order superconvergence rate of the function value approximation u h u i 0 o h 4 for the quadratic fvm on triangular meshes here u h is the fvm solution and u i is the piecewise quadratic lagrange interpolation of the exact solution it should be pointed out that the superconvergence phenomena of fvm strongly depends on the construction of dual mesh specially for quadratic fvms the scheme presented in this paper is the unique scheme which holds superconvergence

derivative superconvergent points in finite element solutions of harmonic functions a theoretical justification finite element derivative superconvergent points for harmonic functions under local rectangular mesh are investigated all superconvergent points for the finite element space of any order that is contained in the tensor product space and contains the intermediate family can be predicted in the case of the serendipity family results are given for finite element spaces of order below 6 the results justify the computer findings of babugka et al

natural superconvergence points in three dimensional finite elements a systematic and analytic process is conducted to identify natural superconvergence points of high degree polynomial c 0 finite elements in a three dimensional setting this identification is based upon explicitly constructing an orthogonal decomposition of local finite element spaces derivative and function value superconvergence points are investigated for both the poisson and the laplace equations superconvergence results are reported for hexahedral pentahedral and tetrahedral elements up to certain degrees

addendum b spline interpolation in medical image processing analyzes b spline interpolation techniques of degree 2 4 and 5 with respect to all criteria that have been applied to evaluate various interpolation schemes in a recently published survey on image interpolation in medical imaging lehmann et al 1999 it is shown that high degree b spline interpolation has superior fourier properties smallest interpolation error and reasonable computing times therefore high degree b splines are preferable interpolators for numerous applications in medical image processing particularly if high precision is required if no aliasing occurs this result neither depends on the geometric transform applied for the tests nor the actual content of images

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

superconvergence of the h p version of the finite element method in one dimension in this paper we investigate the superconvergence properties of the h p version of the finite element method fem for two point boundary value problems a postprocessing technique for the h p finite element approximation is analyzed the analysis shows that the postprocess improves the order of convergence furthermore we obtain asymptotically exact a posteriori error estimators based on the postprocessing results numerical examples are included to illustrate the theoretical analysis

on approximations of zadeh s extension principle in this paper we provide several results concerning the approximation of zadeh s extension of a given function we provide several simple observations due to which we provide a general procedure allowing to approximate zadeh s extension of any continuous map among other things we demonstrated that the technique of f transform can also be used to approximate this extension further we studied how the choice of the metric on the space of fuzzy sets resp fuzzy numbers can affect the quality of the approximation and we clarified that in comparison to recent habits the choice of the levelwise metric need not be the most appropriate one and that the approximation process could be simplified our results are general in the sense that we mostly require nothing but the continuity of the original map

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

nonlinear image upsampling method based on radial basis function interpolation in this paper we present a novel edge directed upsampling method based on radial basis function rbf interpolation in order to remove artifacts such as blurred edges or blocking effects we suggest a nonlinear method capable of taking edge information into account the resampling evaluation is determined according to the edge orientation the proposed scheme is as simple to implement as linear methods but it demonstrates improved visual quality by preserving the edge features better than the classical linear interpolation methods the algorithm is compared with some well known linear schemes as well as recently developed nonlinear schemes the resulting images demonstrate the new algorithm s ability to magnify an image while preserving the edge features

shifting interpolation kernel toward orthogonal projection orthogonal projection offers the optimal solution for many sampling reconstruction problems in terms of the least square error in the standard interpolation setting where the sampling is assumed to be ideal however the projection is impossible unless the interpolation kernel is related to the sinc function and the input is bandlimited in this paper we propose a notion of shifting kernel toward the orthogonal projection for a given interpolation kernel we formulate optimization problems whose solutions lead to shifted interpolations that while still being interpolatory are closest to the orthogonal projection in the sense of the minimax regret the quality of interpolation is evaluated in terms of the average approximation error over input shift for the standard linear interpolation we obtain several values of optimal shift dependent on a priori information on input signals for evaluation we apply the new shifted linear interpolations to a gaussian signal an ecg signal a speech signal a two dimensional signal and three natural images significant improvements are observed over the standard and the 0 21 shifted linear interpolation proposed early

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

an estimate for the three dimensional discrete green s function and applications in this paper we first introduce definitions of the regularized green s function and the discrete green s function in three dimensions with complicated arguments estimates for these two functions are then derived finally we give an application of the estimate for the discrete green s function to the high accuracy analysis of the three dimensional block finite element approximation

computational methods for solving fully fuzzy linear systems abstract since many real world engineering systems are too complex to be defined in precise terms imprecision is often involved in any engineering design process fuzzy systems have an essential role in this fuzzy modelling which can formulate uncertainty in actual environment in addition this is an important sub process in determining inverse eigenvalue and some other useful matrix computations too one of the most practicable subjects in recent studies is based on lr fuzzy numbers which are defined and used by dubois and prade with some useful and easy approximation arithmetic operators on them recently dehghan et al m dehghan m ghatee b hashemi some computations on fuzzy matrices submitted for publication extended some matrix computations on fuzzy matrices where a fuzzy matrix appears as a rectangular array of fuzzy numbers in continuation to our previous work we focus on fuzzy systems in this paper it is proved that finding all of the real solutions which satisfy in a system with interval coefficients is np hard the same result can similarly be derived for fuzzy systems so we employ some heuristics based methods on dubois and prade s approach finding some positive fuzzy vector x which satisfies a x b where a and b are a fuzzy matrix and a fuzzy vector respectively we propose some new methods to solve this system that are comparable to the well known methods such as the cramer s rule gaussian elimination lu decomposition method doolittle algorithm and its simplification finally we extend a new method employing linear programming lp for solving square and non square over determined fuzzy systems some numerical examples clarify the ability of our heuristics

ultraconvergence of zz patch recovery at mesh symmetry points the ultraconvergence property of the zienkiewicz zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even order finite elements the result is valid at all rectangular mesh symmetry points different smoothing strategies are discussed and numerical examples are demonstrated

finite element superconvergence on shishkin mesh for 2 d convection diffusion problems in this work the bilinear finite element method on a shishkin mesh for convection diffusion problems is analyzed in the two dimensional setting a superconvergence rate o n 2 ln2 n en 1 5lnn in a discrete e weighted energy norm is established under certain regularity assumptions this convergence rate is uniformly valid with respect to the singular perturbation parameter e numerical tests indicate that the rate o n 2ln2 n is sharp for the boundary layer terms as a by product an e uniform convergence of the same order is obtained for the l2 norm furthermore under the same regularity assumption an e uniform convergence of order n 3 2ln5 2n en 1ln1 2n in the l norm is proved for some mesh points in the boundary layer region

enlargement or reduction of digital images with minimum loss of information the purpose of this paper is to derive optimal spline algorithms for the enlargement or reduction of digital images by arbitrary noninteger scaling factors in our formulation the original and rescaled signals are each represented by an interpolating polynomial spline of degree n with step size one and spl delta respectively the change of scale is achieved by determining the spline with step size spl delta that provides the closest approximation of the original signal in the l sub 2 norm we show that this approximation can be computed in three steps i a digital prefilter that provides the b spline coefficients of the input signal ii a resampling using an expansion formula with a modified sampling kernel that depends explicitly on spl delta and iii a digital postfilter that maps the result back into the signal domain we provide explicit formulas for n 0 1 and 3 and propose solutions for the efficient implementation of these algorithms we consider image processing examples and show that the present method compares favorably with standard interpolation techniques finally we discuss some properties of this approach and its connection with the classical technique of bandlimiting a signal which provides the asymptotic limit of our algorithm as the order of the spline tends to infinity

superconvergence of discontinuous galerkin method for scalar nonlinear hyperbolic equations in this paper we study the superconvergence behavior of the semi discrete discontinuous galerkin dg method for scalar nonlinear hyperbolic equations in one spatial dimension superconvergence results for problems with fixed and alternating wind directions are established on the one hand we prove that if the wind direction is fixed i e the derivative of the flux function is bounded away from zero both the cell average error and numerical flux error at cell interfaces converge at a rate of 2k 1 when upwind fluxes and piecewise polynomials of degree k are used moreover we also prove that the function value approximation of the dg solution is superconvergent at interior right radau points and the derivative value approximation is superconvergent at interior left radau points with an order of k 2 and k 1 respectively as a byproduct we show a k 2 th order superconvergence of the dg solution towards the gauss radau projection of the exact solution on the other hand superconvergence resu

superconvergence of numerical gradient for weak galerkin finite element methods on nonuniform cartesian partitions in three dimensions abstract a superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak galerkin finite element scheme on the uniform and non uniform cubic partitions due to the loss of the symmetric property from two dimensions to three dimensions this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions li et al 0000 from rectangular partitions to cubic partitions the error estimate for the numerical gradient in the l 2 norm arrives at a superconvergence order of o h r 1 5 r 2 when the lowest order weak galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed a series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions

a simple approximation of productivity scores of fuzzy production plans this paper suggests a simple approximation procedure for the assessment of productivity scores with respect to fuzzy production plans the procedure has a clear economic interpretation and all the necessary calculations can be performed in a spreadsheet making it highly operational

image zooming method using edge directed moving least squares interpolation based on exponential polynomials this paper presents a nonlinear image interpolation algorithm the suggested method is based on the moving least squares mls projection technique but introduces a fundamental modification the algebraic polynomial based mls methods provide very satisfactory results however the associated approximation space is shift and scale invariant so that it cannot be adjusted according to the characteristic of a given data as a result when upsampling images it has a limitation in producing sharp edges such that edges are often blurred in the magnified images to recover sharper edges we need to reduce smoothing parameter or adapt a new parameter sharpening the edges motivated by this observations we propose a novel mls method governed by a set of exponential polynomials with tension parameters such that they can be tuned to the characteristic of given data moreover for a better match to the local structures around the edges the suggested algorithm uses weights which consider the edge orientation numerical results are presented and compared visually and by using some quantitative fidelity measures psnr epsnr ssim and fsim to the bicubic spline interpolation and other recently developed nonlinear methods the results demonstrate the new algorithm s ability to magnify an image while preserving edge features

derivative superconvergent points in finite element solutions of poisson s equation for the serendipity and intermediate families a theoretical justification finite element derivative superconvergent points for the poisson equation under local rectangular mesh in the two dimensional case and local brick mesh in the three dimensional situation are investigated all superconvergent points for the finite element space of any order that is contained in the tensor product space and contains the intermediate family can be predicted in case of the serendipity family the results are given for finite element spaces of order below 7 any finite element space that contains the complete polynomial space will have at least all superconvergent points of the related serendipity family

explicit error bounds in a conforming finite element method the goal of this paper is to define a procedure for bounding the error in a conforming finite element method the new point is that this upper bound is fully explicit and can be computed locally numerical tests prove the efficiency of the method it is presented here for the case of the poisson equation and a first order finite element approximation

local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients abstract assume that u x satisfies the problem l u x x i a i j u x j f x x u x 0 x in this article using interpolation postprocessing technique we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u x using piecewise polynomials of degrees bi k k 3 over a rectangular partition assume that k 3 is odd and x 0 is an interior vertex satisfying x 0 c using the new interpolation postprocessing formula presented in this study we show that the primal variable and the derivative of the post processed finite element solution using piecewise of degrees bi k k 3 at x 0 converge to the primal variable and the derivative of the exact solution with order o h k 3 ln h under suitable regularity and mesh conditions respectively finally we use numerical experiments to illustrate our theoretical findings

is 2k conjecture valid for finite volume methods this paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes our main result is to prove the 2k conjecture at each vertex of the underlying rectangular mesh the bi k degree finite volume solution approximates the exact solution with an order o h 2k where h is the mesh size as byproducts superconvergence properties for finite volume discretization errors at lobatto and gauss points are also obtained all theoretical findings are confirmed by numerical experiments

superconvergence for the gradient of finite element approximations by l 2 projections a gradient recovery technique is proposed and analyzed for finite element solutions which provides new gradient approximations with high order of accuracy the recovery technique is based on the method of least squares surface fitting in a finite dimensional space corresponding to a coarse mesh it is proved that the recovered gradient has a high order of superconvergence for appropriately chosen surface fitting spaces the recovery technique is robust efficient and applicable to a wide class of problems such as the stokes and elasticity equations

ultraconvergence of high order fems for elliptic problems with variable coefficients in this paper we investigate local ultraconvergence properties of the high order finite element method fem for second order elliptic problems with variable coefficients under suitable regularity and mesh conditions we show that at an interior vertex which is away from the boundary with a fixed distance the gradient of the post precessed kth k ge 2 k 2 order finite element solution converges to the gradient of the exact solution with order mathcal o h k 2 mathrm ln h 3 o hk 2 lnh 3 the proof of this ultraconvergence property depends on a new interpolating operator some new estimates for the discrete green s function a symmetry theory derived in 26 and the richardson extrapolation technique in 20 numerical experiments are performed to demonstrate our theoretical findings

finite element differential forms on cubical meshes we develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions the family contains elements of all polynomial degrees and all form degrees in two dimensions these include the serendipity finite elements and the rectangular bdm elements in three dimensions they include a recent generalization of the serendipity spaces and new h curl and h div finite element spaces spaces in the family can be combined to give finite element subcomplexes of the de rham complex which satisfy the basic hypotheses of the finite element exterior calculus and hence can be used for stable discretization of a variety of problems the construction and properties of the spaces are established in a uniform manner using finite element exterior calculus

local ultraconvergence of linear and bilinear finite element method for second order elliptic problems abstract in this article local ultraconvergence in gradient of the bilinear and linear finite element solutions to two dimensional second order elliptic problems has been investigated special interpolation postprocessing algorithms of numerical solutions by richardson extrapolation have been developed the local symmetry theory and estimates of discrete green functions are used in the analysis numerical experiments are provided to confirm the theoretical findings

superconvergence of finite volume methods for the second order elliptic problem abstract this paper derives a superconvergence result for finite volume approximations of the second order elliptic problem by using a l2 projection post processing technique the superconvergence result is applicable to different kind of finite volume methods and to general quasi uniform meshes

approximate fuzzy arithmetic operations using monotonic interpolations abstract we suggest the use of piecewise monotonic interpolations to approximate and represent a fuzzy number or interval and to derive a procedure to control the absolute error associated to the arithmetic operations between fuzzy numbers in order to reduce the distance between the true result of the operation and its approximation the monotonic functions are then used to define a parametric representation of a large class of fuzzy numbers having general shapes of the membership function and a simple and accurate procedure is introduced for the fuzzy arithmetic several computational experiments are given to show the good performance of the proposed procedure

survey interpolation methods in medical image processing image interpolation techniques often are required in medical imaging for image generation e g discrete back projection for inverse radon transform and processing such as compression or resampling since the ideal interpolation function spatially is unlimited several interpolation kernels of finite size have been introduced this paper compares 1 truncated and windowed sine 2 nearest neighbor 3 linear 4 quadratic 5 cubic b spline 6 cubic g lagrange and 7 gaussian interpolation and approximation techniques with kernel sizes from 1 spl times 1 up to 8 spl times 8 the comparison is done by 1 spatial and fourier analyses 2 computational complexity as well as runtime evaluations and 3 qualitative and quantitative interpolation error determinations for particular interpolation tasks which were taken from common situations in medical image processing for local and fourier analyses a standardized notation is introduced and fundamental properties of interpolators are derived successful methods should be direct current dc constant and interpolators rather than dc inconstant or approximators each method s parameters are tuned with respect to those properties this results in three novel kernels which are introduced in this paper and proven to be within the best choices for medical image interpolation the 6 spl times 6 blackman harris windowed sinc interpolator and the c2 continuous cubic kernels with n 6 and n 8 supporting points for quantitative error evaluations a set of 50 direct digital x rays was used they have been selected arbitrarily from clinical routine in general large kernel sizes were found to be superior to small interpolation masks except for truncated sine interpolators all kernels with n 6 or larger sizes perform significantly better than n 2 or n 3 point methods p spl lt 0 005 however the differences within the group of large sized kernels were not significant summarizing the results the cubic 6 spl times 6 interpolator with continuous second derivatives as defined in 24 can be recommended for most common interpolation tasks it appears to be the fastest six point kernel to implement computationally it provides eminent local and fourier properties is easy to implement and has only small errors the same characteristics apply to b spline interpolation but the 6 spl times 6 cubic avoids the intrinsic border effects produced by the b spline technique however the goal of this study was not to determine an overall best method but to present a comprehensive catalogue of methods in a uniform terminology to define general properties and requirements of local techniques and to enable the reader to select that method which is optimal for his specific application in medical imaging

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

the optimal convergence of finite element methods for the three dimensional second order elliptic boundary value problem with singularity abstract let x 0 assume that g x 0 x satisfies the elliptic boundary value problem l g x 0 x x j a i j x g x 0 x x i x x 0 in g x 0 x 0 on where 3 is a convex polyhedral domain in this article we propose specially graded meshes t h furthermore we obtain an l 2 error estimate of order h k 1 ln h 3 2 for approximations with a p k finite element method

superconvergence points of polynomial spectral interpolation in this work we study superconvergence properties for some high order orthogonal polynomial interpolations at the superconvergent points the error is smaller by a factor of n alpha with alpha 0 where n is the polynomial degree the results are two fold when interpolating function values we identify those points where the first and second derivatives of the interpolant converge faster when interpolating the first derivative we locate those points where the function value of the interpolant superconverges for the earlier case we use various chebyshev polynomials and for the later case we also include the counterpart legendre polynomials

fuzzy point estimation and its application on fuzzy supply chain analysis this work describes fuzzy point estimation that reflects the subjective measurement of a real number or an interval of real numbers and a new set of definitions for arithmetic operations on fuzzy points in today s changing business world there is a growing need to manage the supply chains of products to allow businesses to offer competitive prices to their customers managing the supply chains is becoming more complex because of market uncertainties this paper provides a new approach to model the uncertainties involved in the supply chain management using the fuzzy point estimation this work provides a clearer description and analysis framework mathematically for supply chain management based on the stationary strategy established by the author the order up to levels may be calculated by means of formulae directly under a certain stationary condition and then the optimal solution may be gotten for a stationary supply chain this paper is intended to open a window toward the complete results for the analysis of non stationary supply chains

analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation abstract triangular and trapezoidal fuzzy numbers are commonly used in many applications it is well known that the operators used for the non linear operations such as multiplication division and inverse are approximations to the actual operators it is also commonly assumed that the error introduced by the approximations is small and acceptable this paper examines the error of approximation for repeated use of the multiplication operand and shows it can be sufficiently large in simple circumstances to produce erroneous results the computational complexity of the multiplication operation is analyzed and shown to be sufficiently complex that a computationally simpler approximation is needed as a consequence the error produced by the approximation for the multiplication operation is analyzed and a new approximation developed that is accurate for a large range of problems an error expression is developed for the new approximation that can be used to determine when it is producing unacceptable results

an interior estimate of superconvergence for finite element solutions for second order elliptic problems on quasi uniform meshes by local projections this paper establishes some superconvergence estimates for finite element solutions of second order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution the projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares in particular some local superconvergence estimates in the l2 and l infty norms are derived for the local projections of the galerkin finite element solution the results have two prominent features first they are established for any quasi uniform meshes which are of practicalimportance in scientific computation second they are derived on the basis of local properties of the domain and the solution for the second order elliptic problem therefore the result of this paper can be employed to provide useful a posteriori error estimators in practical computing

averaging for superconvergence verification and application of 2d edge elements to maxwell s equations in metamaterials abstract in this paper a simple averaging technique for edge elements is justified to improve the standard convergence rate hence achieving superconvergence when they are used to solve the maxwell s equations for simplicity here we focus on the lowest order triangular edge element which is widely used in practice though there exists no natural superconvergence points for the numerical solution obtained on such an edge element superconvergence can be obtained after a simple average of solutions over the neighboring elements a comprehensive analysis for the lowest order triangular edge element is carried out and one order higher convergence rate than the standard interpolation error estimate is proved for the averaged solution at midpoints of those interior edges of parallelograms formed by two triangles i e superconvergence happens at the parallelogram centers we also provide detailed analysis to explain why several cases do not have superconvergence extensive numerical results consistent with our analysis are presented

high accuracy analysis of two nonconforming plate elements we prove the superconvergence of morley element and the incomplete biquadratic nonconforming element for the plate bending problem under uniform rectangular meshes we obtain a superconvergence property at the symmetric points of the elements and a global superconvergent result by a proper postprocessing method

superconvergence and gradient recovery of linear finite elements for the laplace beltrami operator on general surfaces superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the laplace beltrami equation on general surfaces with mildly structured triangular meshes for a large class of practically useful grids the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the laplace beltrami equation and as a result various postprocessing gradient recovery including simple and weighted averaging local and global l 2 projections and zienkiewicz and zhu z z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution numerical experiments are presented to confirm the theoretical results

some error estimates for the lumped mass finite element method for a parabolic problem we study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous dirichlet boundary conditions improving earlier results we show that known optimal order smooth initial data error estimates for the standard galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation we also discuss the application to time discretization by the backward euler and crank nicolson methods

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

edge forming methods for color image zooming this paper introduces edge forming schemes for image zooming of color images by general magnification factors in order to remove reduce artifacts arising in image interpolation such as image blur and the checkerboard effect an edge forming method is suggested to be applied as a postprocess of standard interpolation methods the method is based on nonconvex nonlinear partial differential equations the equations are carefully discretized incorporating numerical schemes of anisotropic diffusion to be able to form reliable edges satisfactorily the alternating direction implicit adi method is employed for an efficient simulation of the model it has been numerically verified that the resulting algorithm can form clear edges in 2 to 3 adi iterations various results are given to show th effectiveness and reliability of the algorithm

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives