the use of pde centres in the local rbf hermitian method for 3d convective diffusion problems in this work an extension is proposed to the local hermitian interpolation lhi method a meshless numerical method based on interpolation with small and heavily overlapping radial basis function rbf systems this extension to the lhi method uses interpolation functions which themselves satisfy the partial differential equation pde to be solved in this way a much improved reconstruction of partial derivatives can be obtained resulting in significantly improved accuracy in many cases the implementation algorithm is described and is validated via three convection diffusion reaction problems for steady and transient situations a crank nicolson implicit time stepping technique is used for the time dependent problems in the proposed approach a form of analytical upwinding is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function which includes the desired information about the convective velocity field the implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one dimensional travelling front problem at peclet numbers of 500 1000 2000 5000 and infinity which corresponds to a shock front in the case of infinity in addition the accuracy of the numerical scheme is validated against a one dimensional steady state solution exhibiting strong boundary layer effects and also against a steady and a transient three dimensional convection diffusion problem on irregular datasets all the test cases are validated against the corresponding analytical solutions finally the effect of various interpolation stencil configurations is investigated and some important limitations on local data centre distribution are identified

an extension of a bound for functions in sobolev spaces with applications to m s spline interpolation and smoothing given a function f on a bounded open subset of mathbb r n with a lipschitz continuous boundary we obtain a sobolev bound involving the values of f at finitely many points of overline omega this result improves previous ones due to narcowich et al math comp 74 743 763 2005 and wendland and rieger numer math 101 643 662 2005 we then apply the sobolev bound to derive error estimates for interpolating and smoothing m s splines in the case of smoothing noisy data as well as exact data are considered

general constraint preconditioning iteration method for singular saddle point problems for the singular saddle point problems with nonsymmetric positive definite 1 1 block we present a general constraint preconditioning gcp iteration method based on a singular constraint preconditioner using the properties of the moore penrose inverse the convergence properties of the gcp iteration method are studied in particular for each of the two different choices of the 1 1 block of the singular constraint preconditioner a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the gcp iteration method moreover the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual gmres and quasi minimal residual qmr methods are also tested

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

error bounds for gmls derivatives approximations of sobolev functions this paper provides the error estimates for generalized moving least squares gmls derivatives approximations of a sobolev function in l p norms and extends them for local weak forms of dmlpg methods sometimes they are called diffuse or uncertain derivatives but precisely they are direct approximants of exact derivatives which possess the optimal rates of convergence gmls derivatives approximations are different from the standard derivatives of mls approximation while they are much easier to evaluate at considerably lower cost in this paper the same orders of convergence with comparison to the standard derivatives are obtained for them

a weak form rbf generated finite difference method abstract in this paper the idea of direct discretization via radial basis functions rbfs is applied on a local petrov galerkin test space of a partial differential equation pde this results to a weak based rbf generated finite difference rbf fd scheme that possesses some useful properties the error and stability issues are considered when the pde solution or the basis function has low smoothness the new method gives more accurate results than the already well established strong based collocation methods although the method uses a galerkin formulation it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over non connected independent and well shaped subdomains some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method

convergence behavior of generalized parameterized uzawa method for singular saddle point problems in this paper we will seek the least squares solution for singular saddle point problems the parameterized uzawa pu method is further studied and a generalized pu gpu proper splitting is proposed the convergence behavior of the corresponding gpu iteration is studied it is proved that the gpu iteration method can converge to the best least squares solutions of the singular saddle point problems in addition we prove that the gpu preconditioned gmres for singular saddle point problems will also determine the least squares solution at breakdown the eigenvalue distributions of the gpu preconditioned matrix are derived numerical experiments are presented which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the gpu iteration has better convergence behavior than the pu iteration both as a solver and a preconditioner of gmres

h 2 convergence of least squares kernel collocation methods the strong form asymmetric kernel based collocation method commonly referred to as the kansa method is easy to implement and hence is widely used for solving engineering problems and partial differential equations despite the lack of theoretical support the simple least squares ls formulation on the other hand makes the study of its solvability and convergence rather nontrivial in this paper we focus on general second order linear elliptic differential equations in omega subset mathbb r d under dirichlet boundary conditions with kernels that reproduce h m omega and some smoothness assumptions on the solution we provide conditions for a constrained ls method and a class of weighted ls algorithms to be convergent theoretically for max 2 lceil d 1 2 rceil leq nu leq m we identify some h nu omega convergent ls formulations that have an optimal error behavior like h m nu for d leq3 the proposed methods are optimal in h 2 omega we demonstrate the effect

convergence of unsymmetric kernel based meshless collocation methods this paper proves convergence of variations of the unsymmetric kernel based collocation method introduced by kansa in 1986 since then this method has been very successfully used in many applications though it may theoretically fail in special situations and though it had no error bound or convergence proof up to now thus it is necessary to add assumptions or to make modifications our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates these rates improve with the smoothness of the solution the domain and the kernel providing the trial spaces but they are currently not yet optimal and deserve refinement they are based on rates of approximation to the residuals by nonstationary meshless kernel based trial spaces and they are independent of the type of differential operator the results are applicable to large classes of linear problems in strong form provided that there is a smooth solution and the test and trial discretizations are chosen with some care our analysis does not require assumptions like ellipticity and it can be extended to ill posed problems

a scalable and implicit meshless rbf method for the 3d unsteady nonlinear richards equation with single and multi zone domains the local hermitian interpolation lhi method is a strong form meshless numerical technique which uses radial basis function rbf interpolants to satisfy linear and nonlinear governing equations and boundary operators recent developments have shown that for linear transport problems applying the pde governing equation directly to the basis functions can greatly improve the accuracy and stability of the resulting solutions in this work the lhi formulation with local pde interpolation is extended to the nonlinear gravity driven richards equation in order to solve unsteady problems involving flow in unsaturated porous media the application of the linearised pde operator to the basis functions incorporates information such as the effective velocity field directly into the solution construction this results in a form of analytical upwinding which helps to stabilise the solution in addition the local interpolation itself satisfies the linearised governing equation allowing for more accurate reconstruction of partial derivatives and hence a more accurate solution the procedure is tested using a 3d infiltration problem with a known analytical solution the performance of the lhi method both with and without local pde interpolation is compared to the finite element method via the femwater software the lhi formulation with pde data centres shows consistent improvement over the femwater solutions reducing errors by several orders of magnitude in addition a procedure is introduced to model strongly heterogeneous and layered soils the physically correct matching conditions are applied over layer interfaces i e continuity of pressure and mass flux the double collocation property of the hermitian rbf method is exploited to enforce both matching conditions at the same set of locations on the layer interface this procedure allows the accurate capture of solutions across such interfaces replicating the required discontinuities in the first derivatives of the pressure profile the multi layer formulation is validated using a transient two layer infiltration problem with the analytical solution replicated to a high precision in a variety of configurations copyright 2010 john wiley sons ltd

multilevel sparse grids collocation for linear partial differential equations with tensor product smooth basis functions abstract radial basis functions have become a popular tool for approximation and solution of partial differential equations pdes the recently proposed multilevel sparse interpolation with kernels musik algorithm proposed in georgoulis et al 2013 shows good convergence in this paper we use a sparse kernel basis for the solution of pdes by collocation we will use the form of approximation proposed and developed by kansa 1986 we will give numerical examples using a tensor product basis with the multiquadric mq and gaussian basis functions this paper is novel in that we consider space time pdes in four dimensions using an easy to implement algorithm with smooth approximations the accuracy observed numerically is as good with respect to the number of data points used as other methods in the literature see langer 2016 and wang et al 2016

on unsymmetric collocation by radial basis functions solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by kansa in 1990 it rewrites the problem as a generalized interpolation problem and the solution is obtained by solving a possibly large linear system the method has been used successfully in a variety of applications but a proof of nonsingularity of the linear system was still missing this paper shows that a general proof of this fact is impossible however numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort

direct meshless kernel techniques for time dependent equations we provide a class of positive definite kernels that allow to solve certain evolution equations of parabolic type for scattered initial data by kernel based interpolation or approximation avoiding time intergation completely some numerical illustrations are given

approximate interpolation with applications to selecting smoothing parameters in this paper we study the global behavior of a function that is known to be small at a given discrete data set such a function might be interpreted as the error function between an unknown function and a given approximant we will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region we will apply these results to spline smoothing and show that a specific a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant this has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

extension of sampling inequalities to sobolev semi norms of fractional order and derivative data this paper devoted to sampling inequalities provides some extensions of previous results by arcangeli et al numer math 107 2 181 211 2007 j approx theory 161 198 212 2009 given a function u in a suitable sobolev space defined on a domain in mathbb r n sampling inequalities typically yield bounds of integer order sobolev semi norms of u in terms of a higher order sobolev semi norm of u the fill distance d between overline omega and a discrete set a subset overline omega and the values of u on a the extensions established in this paper allow us to bound fractional order semi norms and to incorporate if available values of partial derivatives on the discrete set both the cases of a bounded domain and omega mathbb r n are considered

unsymmetric meshless methods for operator equations a general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well posed linear operator equations is presented the results provide optimal convergence rates if the test and trial spaces satisfy a stability condition operators need not be elliptic and the problems can be posed in weak or strong form without changing the theory non stationary kernel based trial and test spaces are shown to fit into the framework disregarding the operator equation as a special case unsymmetric meshless kernel based methods solving weakly posed problems with distributional data are treated in some detail this provides a foundation of certain variations of the meshless local petrov galerkin technique of s n atluri and collaborators

a shooting reproducing kernel hilbert space method for multiple solutions of nonlinear boundary value problems in this work an iterative method is proposed to predict and demonstrate the existence and multiplicity of solutions for nonlinear boundary value problems in addition the proposed method is capable of calculating analytical approximations for all branches of solutions this method is a combination of reproducing kernel hilbert space method and a shooting like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems the formulation and implementation of this iterative method is discussed for nonlinear second order with two and three point boundary value problems also the convergence of the proposed method is proved to demonstrate the computational efficiency the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear bratu equation and nonlinear reaction diffusion equation it is also applied successfully to two nonlinear three point boundary value problems with unknown exact solutions in the last example a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method may be successful in cases where purely analytic methods are not

mixed meshless formulation for analysis of shell like structures abstract an efficient mixed meshless computational method based on the local petrov galerkin approach for analysis of plate and shell structures is presented a concept of a three dimensional solid is applied allowing the use of complete three dimensional constitutive equations and exact shell geometry can be described discretization is carried out by using both the moving least square approximation and the polynomial functions independent field variables are the strain and stress tensor components expressed in terms of the nodal values which are then replaced by the nodal displacements by using the independent displacement interpolation a closed global system of equations with only nodal displacements as unknown variables is derived the undesired locking phenomena are fully suppressed the proposed mixed formulation is numerically more efficient than the available meshless fully displacement approach as demonstrated by the numerical examples

rbf ps scheme for solving the equal width equation the equal width equation is solved by a rbf ps scheme this technique is meshless and there is no need to linearize the nonlinear terms the radial kernels are used to transform the pde into a system of odes a higher order ode solver is used to solve the ode system the method is tested for single solitary wave interaction of two and three solitary waves the development of an undular bore and the interaction of two solitary waves the results are compared with earlier work and other methods

sampling theorems for shift invariant spaces gabor frames and totally positive functions we study nonuniform sampling in shift invariant spaces and the construction of gabor frames with respect to the class of totally positive functions whose fourier transform factors as hat g xi prod j 1 n 1 2 pi i delta j xi 1 e c xi 2 for delta 1 ldots delta n in mathbb r c 0 in which case g is called totally positive of gaussian type in analogy to beurling s sampling theorem for the paley wiener space of entire functions we prove that every separated set with lower beurling density 1 is a sampling set for the shift invariant space generated by such a g in view of the known necessary density conditions this result is optimal and validates the heuristic reasonings in the engineering literature using a subtle connection between sampling in shift invariant spaces and the theory of gabor frames we show that the set of phase space shifts of g with respect to a rectangular lattice alpha mathbb z times beta mathbb z forms a frame if and only if alpha beta 1 this solves an open problem going back to daubechies in 1990 for the class of totally positive functions of gaussian type the proof strategy involves the connection between sampling in shift invariant spaces and gabor frames a new characterization of sampling sets without inequalities in the style of beurling new properties of totally positive functions and the interplay between zero sets of functions in a shift invariant space and functions in the bargmann fock space

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

on semi convergence of parameterized uzawa methods for singular saddle point problems abstract bai et al recently proposed an efficient parameterized uzawa method for solving the nonsingular saddle point problems z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 in this paper we further prove the semi convergence of this method when it is applied to solve the singular saddle point problems under suitable restrictions on the involved iteration parameters the optimal iteration parameters and the corresponding optimal semi convergence factor for the parameterized uzawa method are determined in addition numerical experiments are used to show the feasibility and effectiveness of the parameterized uzawa method for solving singular saddle point problems

a high order analytically divergence free discretization method for darcy s problem we develop and analyze a meshfree discretization method for darcy s problem our approximation scheme is based upon optimal recovery which leads to a collocation scheme using divergence free positive definite kernels besides producing analytically incompressible flow fields our method can be of arbitrary order works in arbitrary space dimension and for arbitrary geometries after deriving the scheme we investigate the approximation error for smooth target functions and derive optimal approximation orders finally we illustrate the method by giving numerical examples

stable and convergent unsymmetric meshless collocation methods in the theoretical part of this paper we introduce a simplified proof technique for error bounds and convergence of a variation of kansa s well known unsymmetric meshless collocation method for a numerical implementation of the convergent variation a previously proposed greedy technique is coupled with linear optimization this algorithm allows a fully adaptive on the fly data dependent meshless selection of test and trial spaces the new method satisfies the assumptions of the background theory and numerical experiments demonstrate its stability

an iterative reproducing kernel method in hilbert space for the multi point boundary value problems abstract in this paper an iterative method is proposed to solve the nonlinear bitsadze samarskii boundary value problems with multi point boundary conditions the algorithm is based on the reproducing kernel hilbert space method we use an iterative scheme to overcome the nonlinearity of the problem the convergence and error estimate of the iterative scheme are established the reproducing kernel hilbert space method is used to generate an approximation of the linearized problem in fact the reproducing kernel hilbert space method is combined with an iterative scheme to approximate the solution and an error estimate of the approximate solution is derived in order to show the efficiency and versatility of the proposed method some numerical results are reported the comparison of numerical results with the analytical solution and the best results reported in the literature confirms the good accuracy and applicability of the proposed method

analysis of moving least squares approximation revisited in this article the error estimation of the moving least squares approximation is provided for functions in fractional order sobolev spaces the analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details an application to galerkin method for partial differential equations is also supplied

an improved subspace selection algorithm for meshless collocation methods choosing data points is a common problem for researchers who employ various meshless methods for solving partial differential equations on the one hand high accuracy is always desired on the other ill conditioning problems of the resultant matrices which may lead to unstable algorithms prevent some researchers from using meshless methods for example the optimal placements of source points in the method of fundamental solutions or of the centers in the radial basis functions method are always unclear intuitively such optimal locations will depend on many factors the partial differential equations the domain the trial basis used i e the employed method itself 1pt the computational precisions some user defined parameters and so on such complexity makes the hope of having an optimal centers placement unpromising in this paper we provide a data dependent algorithm that adaptively selects centers based on all the other variables copyright 2009 john wiley sons ltd

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

kernel based adaptive approximation of functions with discontinuities one of the basic principles of approximation theory is that the quality of approximations increase with the smoothness of the function to be approximated functions that are smooth in certain subdomains will have good approximations in those subdomains and these sub approximations can possibly be calculated efficiently in parallel as long as the subdomains do not overlap this paper proposes an algorithm that first calculates sub approximations on non overlapping subdomains then extends the subdomains as much as possible and finally produces a global solution on the given domain by letting the subdomains fill the whole domain consequently there will be no gibbs phenomenon along the boundaries of the subdomains the method detects faults and gradient faults with good accuracy throughout the algorithm works for fixed scattered input data of the function itself not on spectral data and it does not resample

exponential basis functions in solution of static and time harmonic elastic problems in a meshless style in this paper exponential basis functions ebfs are used in a boundary collocation style to solve engineering problems whose governing partial differential equations pdes are of constant coefficient type complex valued exponents are considered for the ebfs two dimensional elasto static and time harmonic elasto dynamic problems are chosen in this paper the solution procedure begins with first finding a set of appropriate ebfs and then considering the solution as a summation of such ebfs with unknown coefficients the unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique the basis and various forms of the transformation have been addressed and discussed we shall propose several strategies for selection of ebfs with the aid of the basis explained for the transformation while using the transformation the number of ebfs should not necessarily be equal to or less than the number of boundary information data a library of ebfs has also been presented for further use the effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of ebfs a number of sample problems are solved to demonstrate the capabilities of the method it has been shown that the time harmonic problems with high wave number can be solved without much effort the method categorized in meshless methods can be applied to many other problems in engineering mechanics and general physics since ebfs can easily be found for almost all problems with constant coefficient pdes

constraint preconditioners for solving singular saddle point problems we study constraint preconditioners for solving singular saddle point problems we analyze properties of the preconditioned matrices in particular their eigenvalue distributions and prove that for solving singular saddle point problems by preconditioned gmres methods with constraint preconditioners gmres will determine the least squares solutions at breakdown in addition we present some numerical examples to demonstrate the convergence behavior of preconditioned gmres for solving singular saddle point problems

on the convergence of a general class of finite volume methods in this paper we investigate numerical methods for solving hyperbolic conservation laws based on finite volumes and optimal recovery these methods can for example be applied in certain eno schemes their approximation properties depend in particular on the reconstruction from cell averages hence this paper is devoted to prove convergence results for such reconstruction processes from cell averages

fuzzy rule based simple interpolation algorithm for discrete signal abstract this paper describes a fuzzy rule based simple interpolation algorithm for discrete data a simple interpolation algorithm between two noise free points in the euclidean space is proposed by taking into account the fuzzy effects of the surrounding points the advanced algorithm for the noisy points is proposed where interpolation is done by classifying the neighboring noisy points into some groups the validity and the effectiveness of these methods have been verified by computer simulations and by applications to the actual time series noisy data the experimental results have provided substantial proofs for practical use

direction dependent learning approach for radial basis function networks direction dependent scaling shaping and rotation of gaussian basis functions are introduced for maximal trend sensing with minimal parameter representations for input output approximation it is shown that shaping and rotation of the radial basis functions helps in reducing the total number of function units required to approximate any given input output data while improving accuracy several alternate formulations that enforce minimal parameterization of the most general radial basis functions are presented a novel directed graph based algorithm is introduced to facilitate intelligent direction based learning and adaptation of the parameters appearing in the radial basis function network further a parameter estimation algorithm is incorporated to establish starting estimates for the model parameters using multiple windows of the input output data the efficacy of direction dependent shaping and rotation in function approximation is evaluated by modifying the minimal resource allocating network and considering different test examples the examples are drawn from recent literature to benchmark the new algorithm versus existing methods

the meshless kernel based method of lines for parabolic equations using the heat equation as a simple example we give a rigorous theoretical analysis of the method of lines implemented as a meshless method based on spatial trial spaces spanned by translates of positive definite kernels the technique can be generalized to other parabolic problems and some numerical illustrations are given

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

a two stage adaptive scheme based on rbf collocation for solving elliptic pdes abstract in this paper we present a new adaptive two stage algorithm for solving elliptic partial differential equations via a radial basis function collocation method our adaptive meshless scheme is based at first on the use of a leave one out cross validation technique and then on a residual subsampling method each of phases is characterized by different error indicators and refinement strategies the combination of these computational approaches allows us to detect the areas that need to be refined also including the chance to further add or remove adaptively any points the resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures several numerical experiments support our study by illustrating the performance of our two stage scheme finally the latter is also compared with an efficient adaptive finite element method

inverse and saturation theorems for radial basis function interpolation while direct theorems for interpolation with radial basis functions are intensively investigated little is known about inverse theorems so far this paper deals with both inverse and saturation theorems for an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use in case of thin plate spline interpolation we also give certain saturation theorems

sobolev type approximation rates for divergence free and curl free rbf interpolants recently error estimates have been made available for divergence free radial basis function rbf interpolants however these results are only valid for functions within the associated reproducing kernel hilbert space rkhs of the matrix valued rbf functions within the associated rkhs also known as the native space of the rbf can be characterized as vector fields having a specific smoothness making the native space quite small in this paper we develop sobolev type error estimates when the target function is less smooth than functions in the native space

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

a least squares radial basis function partition of unity method for solving pdes recently collocation based radial basis function rbf partition of unity methods pums for solving partial differential equations have been formulated and investigated numerically and theoretically when combined with stable evaluation methods such as the rbf qr method high order convergence rates can be achieved and sustained under refinement however some numerical issues remain the method is sensitive to the node layout and condition numbers increase with the refinement level here we propose a modified formulation based on least squares approximation we show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling we derive theoretical error estimates both for the collocation and least squares rbf pums numerical experiments are performed for the poisson equation in two and three space dimensions for regular and irregular geometries the convergence experiments confirm the theoretical estimates and the least squares formulation is shown to

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

on the semi convergence of regularized hss iteration methods for singular saddle point problems abstract recently bai and benzi proposed a class of regularized hermitian and skew hermitian splitting rhss iteration methods for solving the nonsingular saddle point problem in this paper we apply this method to solve the singular saddle point problem in the process of the semi convergence analysis we get that the rhss method and the hss method are unconditionally semi convergent which has improved the previous results then some spectral properties of the corresponding preconditioned matrices and a class of improved preconditioned matrices are analyzed finally some numerical experiments on linear systems arising from the discretization of the stokes equation are presented to illustrate the feasibility and effectiveness of this method and the corresponding preconditioners

coupling three field formulation and meshless mixed galerkin methods using radial basis functions in this work we solve the elliptic partial differential equation by coupling the meshless mixed galerkin approximation using radial basis function with the three field domain decomposition method the formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

semi convergence analysis of uzawa methods for singular saddle point problems recently zheng bai and yang studied the parameterized uzawa method for solving singular saddle point problems b zheng z z bai x yang on semi convergence of parameterized uzawa methods for singular saddle point problems linear algebra appl 431 2009 808 817 in this paper we discuss the inexact uzawa method which covers the uzawa method the preconditioned uzawa method and the parameterized uzawa method to solve the singular saddle point problems we prove the semi convergence result under restrictions by verifying two necessary and sufficient conditions that is all elementary divisors associated with the eigenvalue 1 of its iterative matrix are linear and the pseudo spectral radius of the iterative matrix is less than 1 sufficient conditions for the semi convergence of several uzawa type methods are also provided in addition numerical examples are given to demonstrate the semi convergence of uzawa type methods

2 d discrete signal interpolation and its image resampling application using fuzzy rule based inference this paper describes an interpolation algorithm for two dimensional 2 d discrete signals using fuzzy rule based inference the original signal is estimated by the main surface function in the interpolation region and four sub plane functions surrounding the interpolation region the main surface is a bilinearly interpolated function passing through four signal samples in the interpolation region and the four sub planes reect the tendencies of pixels from the left right up and down of the interpolation region drawing fuzzy inferences about signals from these ve functions we can estimate original signals very well even when the signals are buried in noise we veried the method by computer simulations of some assumed 2 d signals and by resampling of the actual image data c 2000 elsevier science b v all rights reserved

block sor methods for rank deficient least squares problems many papers have discussed preconditioned block iterative methods for solving full rank least squares problems however very few papers studied iterative methods for solving rank deficient least squares problems miller and neumann 1987 proposed the 4 block sor method for solving the rank deficient problem here a 2 block sor method and a 3 block sor method are proposed to solve such problem the convergence of the block sor methods is studied the optimal parameters are determined comparison between the 2 block sor method and the 3 block sor method is given also

a generalization of parameterized inexact uzawa method for singular saddle point problems in this paper we further study the generalized parameterized inexact uzawa method for solving singular saddle point problems obtaining the generalized parameterized inexact uzawa gpiu method theoretical analysis shows that the semi convergence of this new method can be guaranteed by suitable choices of the iteration parameters numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact uzawa method for solving singular saddle point problems

all well posed problems have uniformly stable and convergent discretizations this paper considers a large class of linear operator equations including linear boundary value problems for partial differential equations and treats them as linear recovery problems for functions from their data well posedness of the problem means that this recovery is continuous discretization recovers restricted trial functions from restricted test data and it is well posed or stable if this restricted recovery is continuous after defining a general framework for these notions this paper proves that all well posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well posedness of the problem and independent of the computational discretization provided that sufficiently many test data are used the solutions of discretized problems converge when enlarging the trial spaces and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions this allows new and very simple proofs of convergence rates for generalized finite elements symmetric and unsymmetric kansa type collocation and other meshfree methods like meshless local petrov galerkin techniques it is also shown that for a fixed trial space weak formulations have a slightly better convergence rate than strong formulations but at the expense of numerical integration since convergence rates are reduced to those coming from approximation theory and since trial spaces are arbitrary this also covers various spectral and pseudospectral methods all of this is illustrated by examples

analysis of the relaxed deteriorated pss preconditioner for singular saddle point linear systems abstract the relaxed deteriorated pss rdpss preconditioner proposed by cao et al 13 is further generalized to solve singular saddle point linear systems properties of the rdpss splitting and convergence of the corresponding iteration method are studied by using the moore penrose inverse the rdpss iteration method and the rdpss preconditioned gmres are both proved to converge to the least squares solution of the singular saddle point linear system the rdpss preconditioned matrix is also analyzed results about eigenvalue distributions are derived moreover the rdpss preconditioner is generalized to a class of more general preconditioners the corresponding convergence and eigenvalue distribution results are analyzed numerical experiments are presented to illustrate the effectiveness of the rdpss preconditioner and its variants to accelerate gmres for solving singular saddle point linear systems

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

convergence analysis of general spectral methods abstract if a spectral numerical method for solving ordinary or partial differential equations is written as a biinfinite linear system b z a with a map z l 2 l 2 that has a continuous inverse this paper shows that one can discretize the biinfinite system in such a way that the resulting finite linear system b z a is uniquely solvable and is unconditionally stable i e the stability can be made to depend on z only not on the discretization convergence rates of finite approximations b of b then carry over to convergence rates of finite approximations a of a spectral convergence is a special case some examples are added for illustration

convergence analysis of modified pgss methods for singular saddle point problems abstract recently variants of shift splitting iteration method have been proposed for solving singular saddle point problems however these methods can only be proved to converge to one of the solutions of the consistent singular linear system not knowing any further information about this solution in this work we consider a modified preconditioned generalized shift splitting mpgss iteration method for solving both consistent and inconsistent singular saddle point problems this method is proved to converge to the best least squares solution moreover based on the iteration form a preconditioner is obtained to accelerate krylov subspace methods theoretical analysis shows that the preconditioned gmres method also converges to the best least squares solution of the consistent singular saddle point problem in addition numerical results are presented to show the effectiveness and robustness of the proposed iteration method and preconditioner

radial basis functions for the solution of hypersingular operators on open surfaces we analyze the approximation by radial basis functions of a hypersingular integral equation on an open surface in order to accommodate the homogeneous essential boundary condition along the surface boundary scaled radial basis functions on an extended surface and lagrangian multipliers on the extension are used we prove that our method converges quasi optimally approximation results for scaled radial basis functions indicate that for highly regular radial basis functions the achieved convergence rates are close to the one of low order conforming boundary element schemes numerical experiments confirm our conclusions

divergence free kernel methods for approximating the stokes problem we develop a new discretization scheme for the stokes equations using analytically divergence free approximation spaces our scheme is based upon a meshfree method using matrix valued kernels the scheme works in arbitrary space dimension and can produce arbitrarily smooth and hence high order approximations after deriving the scheme we analyze the discretization error in sobolev spaces and give a numerical example

approximation of uncertain functional relationships in this paper we consider methods for the linguistic modelling of vague functional relationships where the linguistic model is supposed to be given as a set of rules two approaches for deriving a fuzzy valued function from such models are proposed it is argued that both approaches can be interpreted as a generalization of ordinary methods for the interpolation and approximation of functions mathematical properties of fuzzy functions associated with a set of rules are investigated particularly for the second approach a universal approximation property is shown

construction of differentiable membership functions in many applications of fuzzy logic it is of special interest to have a transfer function with good properties regarding differentiability to that end it is desirable to have continuously differentiable membership functions with only few parameters in this paper we propose a class of symmetrical and asymmetrical membership functions of exponential order moreover we present a class of more flexible membership functions by construction