the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

complete parameterization of piecewise polynomial interpolation kernels every now and then a new design of an interpolation kernel appears in the literature while interesting results have emerged the traditional design methodology proves laborious and is riddled with very large systems of linear equations that must be solved analytically we propose to ease this burden by providing an explicit formula that can generate every possible piecewise polynomial kernel given its degree its support its regularity and its order of approximation this formula contains a set of coefficients that can be chosen freely and do not interfere with the four main design parameters it is thus easy to tune the design to achieve any additional constraints that the designer may care for

locking free isogeometric collocation methods for spatial timoshenko rods in this work we present the application of isogeometric collocation techniques to the solution of spatial timoshenko rods the strong form equations of the problem are presented in both displacement based and mixed formulations and are discretized via nurbs based isogeometric collocation several numerical experiments are reported to test the accuracy and efficiency of the considered methods as well as their applicability to problems of practical interest in particular it is shown that mixed collocation schemes are locking free independently of the choice of the polynomial degrees for the unknown fields such an important property is also analytically proven

spline collocation for strongly elliptic equations on the torus we prove convergence and error estimates in sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus shaped domains in three or more dimensions for odd degree splines we prove convergence of nodal collocation for any strongly elliptic operator for even degree splines and midpoint collocation we find an additional condition for the convergence which is satisfied for the classical boundary integral operators our analysis is a generalization to higher dimensions of the corresponding analysis of arnold and wendland 4

on the convergence and representation of random fuzzy number integrals in this paper a new kind of convergence w s convergence is introduced for certain type of fuzzy numbers taking values in separable reflexive banach spaces the convergence theorem of random fuzzy number integrals in the w s sense is given and a condition under which a fuzzy number function can be represented by random fuzzy number integrals is obtained

numerical solution of linear volterra integral equations of the second kind with sharp gradients collocation methods are a well developed approach for the numerical solution of smooth and weakly singular volterra integral equations in this paper we extend these methods through the use of partitioned quadrature based on the qualocation framework to allow the efficient numerical solution of linear scalar volterra integral equations of the second kind with smooth kernels containing sharp gradients in this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel we illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations while keeping the solution approximation fixed the computational performance improvement introduced by our new method is examined through several test examples the final example we consider is the original problem that motivated this work the problem of calculating the probability density associated with a continuous time random walk in three dimensions that may be killed at a fixed lattice site to demonstrate how separating the solution approximation from quadrature approximation may improve computational performance we also compare our new method to several existing gregory sinc and global spectral methods where quadrature approximation and solution approximation are coupled

local error estimates for some petrov galerkin methods applied to strongly elliptic equations on curves on etudie les proprietes de convergence locales de quelques methodes de petrov galerkin appliquees aux equations fortement elliptiques sur des courbes regulieres du plan on etablit des estimations d erreurs locales pour une grande echelle de normes de sobolev

transformation between continuous and discrete representations of images a perceptual approach the transformation of image signals between the continuous domain of the real world and the discrete domain of modern data processing can have a significant effect on quality and efficiency analysis based on perceptual rather than mathematical considerations has been carried out a series of experiments based on the analysis has shown that substantial improvement over usual techniques can be achieved by the use of a cascade of a presharpening filter combined with gaussian presampling and interpolation filters the resulting sharpened gaussian filter although not exactly circularly symmetrical gives a high degree of isotropy each element in the cascade is separable so that computational efficiency is high a favorable tradeoff is achieved among sharpness smoothness and the effects of aliasing subjective testing in comparison with other commonly used filters has shown the clear superiority of this filter

a fully discrete bem fem for the exterior stokes problem in the plane we reformulate the johnson nedelec approach for the exterior two dimensional stokes problem taking advantage of the parameterization of the artificial boundary the main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem this one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals we give a complete error analysis of both the galerkin and fully discrete galerkin methods

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

on fatou s lemma and parametric integrals for set valued functions abstract in this paper we present new versions of the set valued fatou s lemma for sequences of measurable multifunctions and their conditional expectations then we use them to study the continuity and measurability properties of parameterized set valued integrals

measurability and integrability of the weak upper limit of a sequence of multifunctions we provide several properties of the weak upper limit of a sequence of subsets of a separable banach space such as a criterion of non vacuity of closedness etc we also examine the measurability of the multifunction defined as the weak upper limit of a sequence of multifunctions at last applications to the existence of a measurable and bochner integrable selector for this multifunction are presented

an a posteriori error estimate for a first kind integral equation in this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind the estimate is local and sharp for quasi uniform meshes and so improves earlier work of ours the mesh dependence of the constants is analyzed and shown to be weaker than expected from our previous work besides the galerkin boundary element method the collocation method and the qualocation method are considered a numerical example is given involving an adaptive feedback algorithm

convergence of conditional expectations for unbounded random sets integrands and integral functionals given a sequence of unbounded convex random sets we study under which conditions fatou s lemma for the weak upper limit of their conditional expectations holds we also give multivalued versions of dominated and monotone convergence theorems and we discuss the special case of the integral finally applications to epigraphic convergence of integrands and to mosco convergence of certain integral functionals are provided

color image resolution conversion in this paper we look at the problem of spatially scaling color images we focus on an approach that takes advantage of the human visual system s color spatial frequency sensitivity the algorithm performs an efficient least squares ls resolution conversion for the luminance channel and a low complexity pixel replication reduction in the chrominance channels the performance of the algorithm is compared to a ls method in srgb and cielab color spaces as well as standard bilinear interpolation in srgb space the comparisons are made in terms of computational cost and color error in scielab

comparison of interpolating methods for image resampling when resampling an image to a new set of coordinates for example when rotating an image there is often a noticeable loss in image quality to preserve image quality the interpolating function used for the resampling should be an ideal low pass filter to determine which limited extent convolving functions would provide the best interpolation five functions were compared a nearest neighbor b linear c cubic b spline d high resolution cubic spline with edge enhancement a 1 and e high resolution cubic spline a 0 5 the functions which extend over four picture elements c d e were shown to have a better frequency response than those which extend over one a or two b pixels the nearest neighbor function shifted the image up to one half a pixel linear and cubic b spline interpolation tended to smooth the image the best response was obtained with the high resolution cubic spline functions the location of the resampled points with respect to the initial coordinate system has a dramatic effect on the response of the sampled interpolating function the data are exactly reproduced when the points are aligned and the response has the most smoothing when the resampled points are equidistant from the original coordinate points thus at the expense of some increase in computing time image quality can be improved by resampled using the high resolution cubic spline function as compared to the nearest neighbor linear or cubic b spline functions

error analysis of a boundary element collocation method for a screen problem in we examine the numerical approximation of the first kind integral equation on a plane rectangle defined by the single layer potential of the threedimensional laplacian the solution is approximated by nodal collocation with piecewise bilinear trial functions on a rectangular grid we prove stability and convergence of this method in the sobolev space j1 2 a key ingredient in the proof is the observation that the collocation equations define symmetric positive definite toeplitz matrices

a nystrom flavored calderon calculus of order three for two dimensional waves time harmonic and transient in this paper we present and test a full discretization of all elements of the calderon calculus layer potentials and integral operators for the helmholtz equation in smooth closed curves in the plane the resulting integral equations provide approximations of order three for all variables involved tests are shown for a wide array of direct indirect and combined field integral equation at fixed frequency and for a convolution quadrature based approximation in the time domain

a note on convergence properties of interval valued capacity functionals and choquet integrals based on the classical theory of random sets feng and nguyen 2007 5 studied the convergence of capacity functionals of random sets in terms of choquet integrals in this paper we consider an interval valued capacity functional which is motivated by the goal to generalize a capacity functional and the choquet integral with respect to an interval valued capacity in particular we discuss some convergence theorems for interval valued capacity functionals and interval valued probability measures in the hausdorff metric and choquet weak sense

asymptotic expansions of the error of spline galerkin boundary element methods in this paper we analyse the existence of asymptotic expansions of the error of galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels these expansions are obtained in terms of an interpolation operator and are useful for the application of richardson extrapolation and for obtaining sharper error bounds we also present and analyse a family of fully discrete spline galerkin methods for the solution of the same equations following the analysis of galerkin methods we show the existence of asymptotic expansions of the error

on the convergence of boundary element methods for initial neumann problems for the heat equation in this paper we study boundary element methods for initial neumann problems for the heat equation error estimates for some fully discrete methods are established numerical examples are presented

galerkin methods with splines for singular integral equations over 0 1 in this paper a convergence analysis of galerkin methods with splines for strongly elliptic singular integral equations over the interval 0 1 is given as trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions multiplied by a weight function using inequalities of garding type for singular integral operators in weightedl 2 spaces and the complete asymptotics of solutions at the endpoints we provide error estimates in certain sobolev norms

on the convergence of collocation methods for symm s integral equation on open curves on etablit la convergence des methodes de collocation pour l equation integrale de symm sur des courbes ouvertes on confirme experimentalement les resultats theoriques sur les vitesses de convergence asymptotiques

on a fem bem formulation for an exterior quasilinear problem in the plane we use a version of the fem bem method introduced by costabel boundary elements ix vol 1 c a brebbia et al eds springer verlag 1987 and han j comput math 8 1990 pp 223 232 to discretize an exterior quasilinear problem we provide error estimates for the galerkin method and propose a fully discrete scheme based on simple quadrature formulas furthermore we show that these numerical integration schemes preserve the optimal rates of convergence finally we present results of numerical experiments involving our discretization method

on the super approximation property of galerkin s method with finite elements for galerkin s method with finite elements as trial functions for strongly elliptic operator equations in the hilbert scaleh t the super approximation property and the optimal convergence rate are obtained by using the aubin nitsche lemma this applies in particular to spline collocation methods for a wide class of pseudodifferential equations

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

fatou s lemmas for the set of integrable selections abstract in this paper we prove fatou type results for the set of bochner integrable selections from a set valued function also we obtain some new dominated convergence theorems for the aumann integral our results have applications in economic theory

combinations of collocation and finite element methods for poisson s equation in this paper we provide a framework of combinations of collocation method cm with the finite element method fem the key idea is to link the galerkin method to the least squares method which is then approximated by integration approximation and led to the cm the new important uniformly v 0 h elliptic inequality is proved interestingly the integration approximation plays a role only in satisfying the uniformly v 0 h elliptic inequality for the combinations of the finite element and collocation methods fem cm the optimal convergence rates can be achieved the advantage of the cm is to formulate easily linear algebraic equations where the associated matrices are positive definite but nonsymmetric we may also solve the algebraic equations of fem and the collocation equations directly by the least squares method thus to greatly improve numerical stability numerical experiments are also carried for poisson s problem to support the analysis note that the analysis in this paper is distinct from the existing literature and it covers a large class of the cm using various admissible functions such as the radial basis functions the sinc functions etc

a representation theorem for aumann integrals in recent years the study of set valued functions has been developed extensively by many authors with applications to mathematical economics and control theory see refs 5 11 15 161 in those papers three approaches can be distinguished according to whether the range space values of setvalued functions is z a banach space or a locally convex topological space the purpose of this paper is to establish properties of aumann s integrals of set valued functions f t 2 whose values are nonempty subsets of a real separable reflexive banach space x and to continue the work due to aumann 2 and datko 7 81 while previous analysis has always treated the case of special finite nonatomic measure spaces we focus here on the case of general a finite nonatomic measure spaces in this last situation moreover the analogous results we establish hold under less stringent hypotheses more precisely all through the paper we consider a measure space t c p where p is supposed to be positive nonatomic and u finite and we give the following statements

the convergence of even degree spline collocation solution for potential problems in smooth domains of the plane in this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions it turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed 1 some numerical examples which conform the theoretical results are presented

on the representation of choquet integrals of set valued functions and null sets in this paper we consider the definition of set valued choquet integrals we obtain some properties and the representation of set valued choquet integrals and also we discuss some fundamental relations between null sets and set valued choquet integrals

recursive regularization filters design properties and applications least squares approximation problems that are regularized with specified highpass stabilizing kernels are discussed for each problem there is a family of discrete regularization filters r filters which allow an efficient determination of the solutions these operators are stable symmetric lowpass filters with an adjustable scale factor two decomposition theorems for the z transform of such systems are presented one facilitates the determination of their impulse response while the other allows an efficient implementation through successive causal and anticausal recursive filtering a case of special interest is the design of r filters for the first and second order difference operators these results are extended for two dimensional signals and for illustration purposes are applied to the problem of edge detection this leads to a very efficient implementation 8 multiplies 10 adds per pixel of the optimal canny edge detector based on the use of a separable second order r filter

a collocation method for high frequency scattering by convex polygons we consider the problem of scattering of a time harmonic acoustic incident plane wave by a sound soft convex polygon for standard boundary or finite element methods with a piecewise polynomial approximation space the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave recently chandler wilde and langdon proposed a novel galerkin boundary element method for this problem for which by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency here we propose a related collocation method using the same approximation space for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the galerkin scheme but with a substantially reduced computational cost

the computation of potentials near and on the boundary by an extraction technique for boundary element methods here we present the two dimensional version of a bootstrapping algorithm for the computation of arbitrary derivatives of the cauchy data and for the extraction of potentials near and on the boundary for the laplacian if corresponding boundary integral equations are used the method employs the derivatives of the green representation formula in terms of cauchy singular or weakly singular integrals and also their compositions with derivatives it allows the robust and superconvergent numerical evaluation of potentials and their derivatives up to and on the boundary curve furthermore we determine a criterion whether to use the traditional green representation formula or our extraction technique for the point evaluation of the potentials in the domain

on the boundary element method for some nonlinear boundary value problems here we analyse the boundary element galerkin method for two dimensional nonlinear boundary value problems governed by the laplacian in an interior or exterior domain and by highly nonlinear boundary conditions the underlying boundary integral operator here can be decomposed into the sum of a monotoneous hammerstein operator and a compact mapping we show stability and convergence by using leray schauder fixed point arguments due to petryshyn and necas

the superconvergence of a galerkin collocation method for first kind boundary integral equations on smooth curves here we present a galerkin collocation method for the solution of first kind boundary integral equations with logarithmic kernel as principal part by using piecewise constant trial functions with uniform meshes we discover the phenomenon of superconvergence at midpoints for the numerical solution we shall prove this result in this paper

high order methods for linear functionals of solutions of second kind integral equations the authors show that linear functionals e g inner products or point functionals of solutions of second kind fredholm integral equations may be found within o 1 n 2r by very simple computational schemes using the nystrom or quadrature method based on piecewise polynomials of order r on an n 1 point mesh our theory covers a class of nonsmooth wiener hopf type equations and applications include the calculation of crack energy and stress intensity factors in certain linear elastic fracture problems and the use of the boundary integral method to compute various superconvergent solutions to boundary value problems in polygonal domains a proof of the uniform convergence of nystrom s method for certain boundary integral equations on polygonal domains is also included

spline collocation for convolutional parabolic boundary integral equations summary we consider spline collocation methods for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover for example the case of the single layer heat operator equation when the spatial domain is a disc

fatou s lemma in reflexive banach spaces a version of an approximate fatou lemma for a uniformly integrable sequence of functions with values in a reflexive banach space is proved the usual assumption that this sequence is pointwisely dominated in norm by a real valued integrable function is omitted

fully discrete galerkin methods for systems of boundary integral equations in this paper we analyze a family of full discretizations of spline galerkin methods for a class of systems of boundary integral equations of the first kind with logarithmic principal part we prove the existence of an asymptotic expansion of the error of the galerkin and the optimal order galerkin collocation method we finally derive asymptotic expansions for some common postprocessings of the solutions both exactly and under the effect of additional discretization some examples where these techniques apply are provided

on the comparison of interpolation methods a study of different cubic interpolation kernels in the frequency domain is presented that reveals novel aspects of both cubic spline and cubic convolution interpolation the kernel used in cubic convolution is of finite support and depends on a parameter to be chosen at will at the nyquist frequency the spectrum attains a value that is independent of this parameter exactly the same value is found at the nyquist frequency in the cubic spline interpolation if a strictly positive interpolation kernel is of importance in applications cubic convolution with the parameter value zero is recommended

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

the numerical solution of symm s equation on smooth open arcs by spline galerkin methods abstract we consider the classical fredholm linear integral equation of the first kind with logarithmic kernel on a smooth jordan open arc applying the well known cosine change of variable the arc is reparametrized and the problem is transformed into a new integral equation we investigate the existence of an asymptotic expansion for the error of the galerkin method with splines on a uniform mesh as test trial functions we also analyse a full discretization of the method based on the galerkin collocation method using high order integration formulae to keep the optimal error estimates of the galerkin method in weak norms asymptotic expansions of the error for this method are provided finally we show how these expansions extend to the computation of the potential the expansions of the error in powers of the discretization parameter are useful to obtain a posteriori estimates of the error and to apply richardson extrapolation for acceleration of convergence 1999 elsevier science ltd all rights reserved

stability of a spline collocation method for strongly elliptic multidimensional singular integral equations we consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp gw au f on a cube 0 1 m utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside for classical strongly elliptic translation invariant pseudodifferential operators we verify the stability of the considered collocation method inl 2 afterwards form 2 and a right hand sidef h 8 s m 2 we prove an asymptotic convergence estimate

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

collocation methods for poisson s equation in this paper we provide an analysis on the collocation methods cm which uses a large scale of admissible functions such as orthogonal polynomials trigonometric functions radial basis functions and particular solutions etc the admissible functions can be chosen to be piecewise i e different functions are used in different subdomains the key idea is that the collocation method can be regarded as the least squares method involving integration approximation and optimal convergence rates can be easily achieved based on the traditional analysis of the finite element method the key analysis is to prove the uniformly vh elliptic inequality and some inverse inequalities used this paper explores the interesting fact that for the collocation methods given in this paper the integration rules only affect on the uniformly vh elliptic inequality but not on the solution accuracy the advantage of the cm is to formulate easily the associated algebraic equations which can be solved from the collocation equations directly by the least squares method thus to greatly reduce the condition number of the associated matrix moreover the new effective condition number is proposed to provide a better upper bound of condition number and to show a good stability for real problems solved by the collocation methods note that the boundary approximation method in li z c li combined methods for elliptic equations with singularities interfaces and infinities kluwer academic publishers boston london 1998 is a special case of the cm where the admissible functions satisfy the equations exactly numerical experiments are also carried for poisson s problem to support the analysis made

the boundary element spline collocation for nonuniform meshes on the torus the spline collocation method for a class of biperiodic strongly elliptic pseudodifferential operators is considered as trial functions tensor products of odd degree splines are used and the collocation is imposed at the nodal points of the tensor product mesh it is shown that the collocation problem is uniquely solvable if the maximum mesh length is small enough moreover the approximation is stable and quasioptimal with respect to a norm depending on the order of the operator and the degree of approximating splines some convergence results are given for general and quasiuniform meshes the results cover for example the single layer and the hypersingular operators

quadratic interpolation for image resampling nearest neighbor linear and various cubic interpolation functions are frequently used in image resampling quadratic functions have been disregarded largely because they have been thought to introduce phase distortions this is shown not to be the case and a family of quadratic functions is derived the interpolating member of this family has visual quality close to that of the catmull rom cubic yet requires only 60 of the computation time

subdifferential characterization of probability functions under gaussian distribution probability functions figure prominently in optimization problems of engineering they may be nonsmooth even if all input data are smooth this fact motivates the consideration of subdifferentials for such typically just continuous functions the aim of this paper is to provide subdifferential formulae of such functions in the case of gaussian distributions for possibly infinite dimensional decision variables and nonsmooth locally lipschitzian input data these formulae are based on the spheric radial decomposition of gaussian random vectors on the one hand and on a cone of directions of moderate growth on the other by successively adding additional hypotheses conditions are satisfied under which the probability function is locally lipschitzian or even differentiable

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

on the spline collocation method for the single layer heat operator equation we consider a boundary element collocation method for the heat equation as trial functions we use the tensor products of continuous piecewise linear splines with collocation at the nodal points convergence and stability is proved in the case where the spatial domain is a disc moreover practical implementation is discussed in some detail numerical experiments confirm our results

condition of boundary integral equations arising from flow computations abstract stating flow boundary conditions for normal and tangential velocities leads to fredholm integral equations of the first and second kind approximating the vortex distribution on pressure and suction side by piecewise linear functions with equidistant knots collocation yields a commonly overdetermined system of linear equations with the vortex densities as unknowns there are several possibilities for collocation the question arises what kind of integral equation and what collocation points are to be preferred our main conclusions based on condition numbers of the resulting linear systems are the equation of the first kind with cauchy singular kernel does not appear to be worse conditioned than the equation of the second kind with regular kernel collocation points for the equation of the first kind are better placed in between the knots for the equation of the second kind at the knots the circulation is mainly determined by the equation of the first kind the mean velocity level by the equation of the second kind employing simultaneously equations of the first and second kind generally improves the solution and allows to cope with aerodynamic profiles of any thickness and loading

extrapolation methods for spline collocation solutions of pseudodifferential equations on curves in this article we consider extrapolation methods in approximating smooth linear functionals by means of collocation solutions of boundary integral equations we are able to derive an asymptotic expansion of the error in main applications this expansion covers arbitrary high powers of the discretization parameter if the boundary solution is smooth the expansion gives rise to richardson type extrapolation schemes which require only simple postprocessing of calculated numbers the results are applicable for example in calculating pointwise values in the space domain when solving boundary value problems by means of integral equations in such examples the extrapolation rapidly improves the original rate of the convergence numerical experiments conform our results

fast b spline transforms for continuous image representation and interpolation efficient algorithms for the continuous representation of a discrete signal in terms of b splines direct b spline transform and for interpolative signal reconstruction indirect b spline transform with an expansion factor m are described expressions for the z transforms of the sampled b spline functions are determined and a convolution property of these kernels is established it is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering fast computational algorithms based on the recursive implementations of these filters are proposed a b spline interpolator can also be characterized in terms of its transfer function and its global impulse response cardinal spline of order n the case of the cubic spline is treated in greater detail the present approach is compared with previous methods that are reexamined from a critical point of view it is concluded that b spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner

orthogonal collocation in the nonconforming boundary element method this paper outlines the use of non conforming discontinuous elements in the collocation boundary element method for solving two dimensional potential and poisson type problems the roots of an orthogonal polynomial shifted jacobi polynomial are used as the collocation points this results in increased accuracy due to the least square minimization property of the orthogonal polynomials the advantage of using non conforming elements is realized when the method is applied i to problems with singularities both due to geometry and boundary conditions and ii in conjunction with domain decomposition techniques also the collocation points can be relocated within an element by changing two user defined parameters in the shifted jacobi polynomial thus providing an error indicator which can be used for mesh refinement purposes this technique called the rh method is discussed and illustrated the results obtained by using non conforming boundary elements for standard test problems are shown to be accurate

asymptotic estimation for the condition numbers in bem we apply the boundary element methods bem to the interior dirichlet problem of the two dimensional laplace equation and its discretization is carried out with the collocation method using piecewise linear elements in this paper some precise asymptotic estimations for the discretization matrix l n where n denotes the division number are investigated we show that the maximum norm of l 1 n and the condition number of l n have the forms c 0 n o 1 and c 1 n o n respectively as n to infty where the constants c 0 and c 1 are explicitly given in the proof although these estimates indicate illconditionedness of this numerical computation the o n 2 convergence of this scheme with maximum norm is proved as an application of the main results

boundary integral equation methods for solving laplace s equation with nonlinear boundary conditions the smooth boundary case a nonlinear boundary value problem for laplace s equation is solved numerically by using a reformulation as a nonlinear boundary integral equation two numerical methods are proposed and analyzed for discretizing the integral equation both using product integration to approximate the singu lar integrals in the equation the first method uses the product simpson s rule and the second is based on trigonometric interpolation iterative methods in cluding two grid methods for solving the resulting nonlinear systems are also discussed extensively numerical examples are included

quadrature methods for strongly elliptic equations of negative order on smooth closed curves in this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order the results cover all the classical pseudodifferential equations of this kind and hence the boundary integral equations appearing in applications the quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh and on varepsilon collocation they are proved to have the stability and the convergence with the maximal rate o h beta of convergence in general and o h beta 1 for some special values of varepsilon here beta is the order of the operator the method is applicable to systems of boundary integral equations as well in this connection the authors have conducted some numerical experiments which confirm their theoretical results

integration of correspondences on loeb spaces we study the bochner and gel fand integration of banach space valued correspondences on a general loeb space though it is well known that the lyapunov type result on the compactness and convexity of the integral of a correspondence and the fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space we show that exact versions of these two results hold in the case we study we also note that our results on a hyperfinite loeb space capture the nature of the corresponding asymptotic results for the large finite case but the unit lebesgue interval fails to provide such a framework

a spline trigonometric galerkin method and an exponentially convergent boundary integral method we consider a galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions we analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the dirichlet problem in the interior or exterior of an analytic plane curve in constrast to ordinary spline galerkin methods we show that the method is stable and so provides quasioptimal approximation in a large family of hilbert spaces including all the sobolev spaces of negative order as a consequence we prove that the approximate solution to the dirichlet problem and all its derivatives converge pointwise with exponential rate

a bem fem overlapping algorithm for the stokes equation in this work we study the numerical solution of the dirichlet problem for the stokes equation by overlapping boundary and finite elements the domain where we state our problem is the intersection of a polyhedron with the exterior of a strictly contained obstacle with smooth boundary the solution is then decoupled as a sum of an incident flow defined on the polyhedron domain plus the response of the interior obstacle expressed as a single layer potential and constructed only on the exterior of the obstacle the numerical algorithm follows closely this ansatz by replacing the continuous terms of this decomposition by a finite and a boundary element respectively we prove that under not very restrictive assumptions the method is well defined and converges to the exact solution with the same order as the best approximation of the solution by the discrete spaces in the natural norms of the problem finally in the last section we show how this method can be implemented overcoming some of the difficulties appearing in the implementation and demonstrating its applicability to practical problems

a collocated isogeometric finite element method based on gauss lobatto lagrange extraction of splines generalizing the concept of bezier extraction we introduce an extraction operator that links c0c0 gauss lobatto lagrange functions with smooth splines this opens the door for collocated isogeometric analysis that combines the accuracy of the galerkin method with collocation type formation and assembly procedures we present the key ingredients of the technology i e integration by parts and the weighted residual form the interaction of gauss lobatto lagrange extraction with gauss lobatto quadrature and symmetrization with the ultra weak formulation we compare the new method with standard isogeometric galerkin and isogeometric point collocation methods for spline discretizations in three dimensions

the spline collocation method for parabolic boundary integral equations on smooth curves we consider the spline collocation method for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators for the localization we need also some special spline approximation results in anisotropic sobolev spaces

random fuzzy number integrals in banach spaces abstract in this paper a new kind of random fuzzy number integral is introduced by the pettis aumann integral of the set valued functions in banach spaces the existence of the random fuzzy integral is also shown levi s convergence theorem of the integral is given a special mention

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

a fully discrete bem fem method for an exterior elasticity system in the plane abstract we present a modified version of the usual bem fem coupling for the exterior elasticity problem in the plane cf costabel and stephan siam j numer anal 27 1990 1212 1226 this new formulation allows us to take advantage of techniques from hsiao et al computing 25 1980 557 566 to compute the boundary integral terms using simple quadrature formulas we provide error estimates for the galerkin method and prove that the corresponding fully discrete scheme preserves the optimal rates of convergence

avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods in this work we study isogeometric collocation methods for the timoshenko beam problem considering both mixed and displacement based formulations in particular we show that locking free solutions are obtained for mixed methods independently on the approximation degrees selected for the unknown fields moreover several numerical tests are provided in order to support our theoretical results and to show the good behavior and the flexibility of isogeometric collocation methods in this context

an approximation method for the hypersingular heat operator equation abstract the original model problem is the two dimensional heat conduction problem with vanishing initial data and a given neumann type boundary condition in particular certain choices of the representation formula for the heat potential yield the hypersingular heat operator equation of the first kind in this paper we concentrate on the problem of solving this hypersingular integral equation our approximation method is a petrov galerkin method where we use collocation with respect to the space variable and galerkin method with respect to the time variable the trial functions are tensor products of piecewise cubic space and piecewise linear time smoothest splines stability and convergence of the resulting scheme is proved when the spatial domain of the original heat conduction problem is a disc

fatou s lemma for multifunctions with unbounded values for multifunctions having unbounded sets as values we give fatou type inclusions for the kuratowski limes superior in finite as well as infinite dimensions these are derived from similar known fatou type inequalities for single valued multifunctions i e ordinary functions that is from balder s unifying fatou lemma in case the image set is finite dimensional and from an update of related results by balder in the infinite dimensional case for this extension from the single valued to the multivalued situation a lemma due to hess used to prove earlier fatou type inclusions is of critical importance also an asymptotic correction term introduced here plays an important role the two man results thus obtained subsume and extend an entire class of comparable fatou lemmas

weak sequential convergence in lp μ x we provide some new results on the weak convergence of sequences or nets lying in lp t x lp x 1 p i e the space of equivalence classes of x valued x is a banach space bochner integrable functions on the finite measure space t our theorems generalize in several directions recent resuls on weak sequential convergence in l1 x obtained by m a khan and m majumdar j math anal appl 114 1986 569 573 and z artstein j math econ 6 1979 277 282 and they can be used to obtain dominated convergence results for the aumann integral our results have useful applications in economics and game theory

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

on the convergence of collocation methods for boundary integral equations on polygons the integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions convergence proofs and error estimates for these approximation methods have been only available in the following cases fredholm integral equations of the second kind 4 7 one dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves 2 3 17 26 29 and some special results on the classical neumann integral equation of potential theory for polygonal plane domains 5 8 9 here we give convergence proofs for collocation with piecewise linear trial functions for neumann s integral equation and symm s integral equation on plane curves with corners we derive asymptotic error estimates in sobolev norms and analyze the effect of graded meshes 0 introduction in this paper we give convergence proofs and asymptotic error estimates in sobolev norms for collocation with piecewise linear spline trial func tions applied to two basic integral equations of potential theory on plane polygons namely the integral equation of the second kind with the double layer potential neumann s integral equation and the integral equation of the first kind with the simple layer potential symm s integral equation we use an idea of arnold and wendland 2 namely considering dirac delta functions the test functions in the collocation method as second derivatives of piecewise linear functions therefore similar results as presented here should be possible for splines of higher odd order corresponding results for even order splines are not yet available thus for one of the simplest methods of numerically solving dirichlet s problem on a plane domain with corners the midpoint collocation with piecewise constant trial functions for the first kind integral equation with the simple layer potential convergence is still an open problem the method of fourier series that yields the convergence proof in the case of a smooth boundary 27 cannot be applied in the presence of corners we apply the method of local mellin transformation that has previously been used to derive error estimates for galerkin methods for a wide class of operators including those occurring in boundary element methods in acoustics electromag

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

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis