an operator splitting radial basis function method for the solution of transient nonlinear poisson problems abstract this paper presents an operator splitting radial basis function os rbf method as a generic solution procedure for transient nonlinear poisson problems by combining the concepts of operator splitting radial basis function interpolation particular solutions and the method of fundamental solutions the application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit adams bashforth time marching for half the time step this leaves a nonhomogeneous modified helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step the resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions radial basis functions are used to construct approximate particular solutions and a grid free dimension independent method with high computational efficiency is obtained this method is demonstrated for some prototypical nonlinear poisson problems in heat and mass transfer and for a problem of transient convection with diffusion the results obtained by the os rbf method compare very well with those obtained by other traditional techniques that are computationally more expensive the new os rbf method is useful for both general irregular two and three dimensional geometry and provides a mesh free technique with many mathematical flexibilities and can be used in a variety of engineering applications

a gradient reproducing kernel collocation method for boundary value problems summary the earlier work in the development of direct strong form collocation methods such as the reproducing kernel collocation method rkcm addressed the domain integration issue in the galerkin type meshfree method such as the reproducing kernel particle method but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence in this work we intend to address the computational complexity in rkcm while achieving optimal convergence by introducing a gradient reproduction kernel approximation the proposed gradient rkcm reduces the order of differentiation to the first order for solving second order pdes with strong form collocation we also show that different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence the same number of collocation points and source points can be used in gradient rkcm we also show that the same order of convergence rates in the primary unknown and its first order derivative is achieved owing to the imposition of gradient reproducing conditions the numerical examples are given to verify the analytical prediction copyright 2012 john wiley sons ltd

algorithms for cardinal interpolation using box splines and radial basis functions we describe an algorithm for bivariate cardinal interpolation which can be applied to translates of basis functions which include box splines or radial basis functions the algorithm is based on a representation of the fourier transform of the fundamental interpolant hence fast fourier transform methods are available in numerical tests the 4 directional box spline transformed to the characteristical submodule of 2 the thin plate spline and the multiquadric case give comparably equal and good results

theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions multivariate interpolation of smooth data using smooth radial basis functions is considered the behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically explicit criteria for different types of limits are given using the results for the limits the dependence of the error on the shape parameter of the radial basis function is investigated the mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

element free galerkin method convergence of the continuous and discontinuous shape functions abstract we consider numerical solutions of second order elliptic partial differential equations such as laplace s equation or linear elasticity in two dimensional non convex domains by the element free galerkin method efg this is a meshless method in which the shape functions are constructed by using weight functions of compact support for non convex domains two approaches to the determination of whether a node affects approximation at a particular point are used a contained path criterion and the visibility criterion we show that for non convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions the resulting approximation is no longer conforming and its convergence must be established by inspection of the so called consistency term we show that the variant of the element free galerkin method which uses the discontinuous shape functions is convergent and that in the practically important case of linear shape functions the convergence rate is not affected by the discontinuities the convergence of the discontinuous approximation is first established by the classical and generalized patch test as these tests do not provide an estimate of the convergence rate the rate of convergence in the energy norm is examined for both the continuous and discontinuous efg shape functions and for smooth and non smooth solutions by a direct inspection of the error terms

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

geometry and chaos near resonant equilibria of 3 dof hamiltonian systems in this paper we study the dynamics near resonant elliptic equilibria in three degree of freedom hamiltonian systems the resonances we consider have multiplicity two and the corresponding local normal form for the equilibrium is integrable at cubic order we prove the existence of families of 3 tori and whiskered 2 tori with nearby chaotic dynamics in the quartic normal form the whiskers of the 2 tori intersect in a non trivial way giving rise to multi pulse homoclinic and heteroclinic connections these connections survive in the full system as orbits homoclinic to invariant 3 spheres

applying multiquadric quasi interpolation to solve burgers equation abstract in this paper we develop a kind of univariate multiquadric mq quasi interpolation and use it to solve burgers equation with viscosity at first we construct the mq quasi interpolation which possesses the properties of linear reproducing and preserving monotonicity next we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable then we verify our method for two examples with distinguishing initial value condition one example is tested for three reynolds number that is r 10 r 100 and r 10 000 from the numerical experiments we see that the presented method in this paper is valid although the accuracy of the technique is not higher than hon and mao s one another example is used to examine the travelling of the shock we can improve the accuracy by selecting an appropriate shape parameter and using a higher accurate mq quasi interpolation the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

on the application of multiquadric bases in conjunction with the ltdrm method to solve nonlinear diffusion equations as transient diffusion equations with nonlinear source terms have broad engineering applications many numerical solution approaches have been proposed in the past one of the numerically efficient and accurate methods was the laplace transform dual reciprocity method ltdrm as a result of a series of studies to further improve the accuracy and efficiency of the ltdrm we shall present in this paper the ltdrm in conjunction with the multiquadric bases which were shown to possess a better convergence rate than other interpolation functions such as linear interpolation functions or augmented thin plate splines atps three test examples are included in this paper to support the previous theoretical work and to provide evidence that the multiquadric bases are indeed superior when they are used in conjunction with the ltdrm

radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions the development of numerical methods for the solution of partial differential equations pdes with nonlocal boundary conditions is important since such type of problems arise as mathematical models of various real world processes we use radial basis function rbf collocation technique for the solution of a multidimensional linear elliptic equation with classical dirichlet boundary condition and nonlocal integral conditions rbf based meshless methods are easily implemented and efficient especially for multidimensional problems formulated on complexly shaped domains in this paper properties of the method are investigated by studying two and three dimensional test examples with manufactured solutions we analyze the influence of the rbf shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix

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

on the structure and interpolation properties of quasi shift invariant spaces abstract the structure of certain types of quasi shift invariant spaces which take the form v x span l 2 x j j z for an infinite discrete set x x j r is investigated additionally the relation is explored between pairs x and y such that interpolation of functions in v x via interpolants in v y solely from the samples of the original function is possible and stable some conditions are given for which the sampling problem is stable and for which recovery of functions from their interpolants from a family of spaces v y is possible

multiquadric method for the numerical solution of a biphasic mixture model a computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model the model includes a set of constitutive equations for the fluid flows through the solid phase a set of momentum equations for stress strain equilibrium and a continuity equation for the solid phase and the fluid phase the numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading the numerical results are compared with spilker s penalty finite element method

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

reproducing kernel enhanced local radial basis collocation method standard radial basis functions rbfs offer exponential convergence however the method is suffered from the large condition numbers due to their nonlocal approximation the nonlocality of rbfs also limits their applications to small scale problems the reproducing kernel functions on the other hand provide polynomial reproducibility in a local approximation and the corresponding discrete systems exhibit relatively small condition numbers nonetheless reproducing kernel functions produce only algebraic convergence this work intends to combine the advantages of rbfs and reproducing kernel functions to yield a local approximation that is better conditioned than that of the rbfs while at the same time offers a higher rate of convergence than that of reproducing kernel functions further the locality in the proposed approximation allows its application to large scale problems error analysis of the proposed method is also provided numerical examples are given to demonstrate the improved conditioning and accuracy of the proposed method copyright 2007 john wiley sons ltd

the numerical simulation of the phase field crystal pfc and modified phase field crystal mpfc models via global and local meshless methods abstract in the current work we consider the two dimensional time dependent phase field crystal pfc and the modified phase field crystal models mpfc to obtain their numerical solutions for this purpose we apply two numerical meshless methods based on radial basis functions rbfs and also two meshless methods which are based on moving least squares method mls four techniques developed in this paper are globally radial basis functions grbfs radial basis functions pseudo spectral rbfs ps moving least squares mls and generalized moving least squares gmls approximations two methods based on rbfs are global and the other methods based on moving least squares are local procedures as is well known the meshless methods are suitable techniques for the numerical solution of partial differential equations on regular and non regular domains with different choices of grids in high dimensions applying the new methods on spatial domain and also using the semi implicit scheme for time variable yield a linear system of algebraic equations to solve this linear system the lu decomposition algorithm and command backslash in matlab software for gmls method are applied the implementation of boundary conditions in the rbfs ps is applied directly because the boundary conditions of the mentioned problems are periodic some numerical results show that the obtained simulations via four proposed methods are acceptable for approximating the solution of models investigated in the current paper moreover in the appendix of the paper a matlab code for gmls method is written

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

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

reduction of the semisimple 1 1 resonance the method of averaging is often used in hamiltonian systems of two degrees of freedom to find periodic orbits such periodic orbits can be reconstructed from the critical points of an associated reduced hamiltonian on a reduced space this paper details the construction of the reduced space and the reduced hamiltonian for the semisimple 1 1 resonance case the reduced space will be a 2 sphere in r3 and the reduced differential equations will be euler s equations restricted to this sphere the orbit projection from the energy surface in phase space to this sphere will be the hopf map the results of the paper are related to problems in physics on degeneracies due to symmetries of classical two dimensional harmonic oscillators and their quantum analogues for the hydrogen atom

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations a novel approach to the numerical solution of weakly singular volterra integral equa tions is presented using the cm multiquadric mq radial basis function rbf expansion rather than the more traditional finite difference finite element or polynomial spline schemes to avoid the collocation procedure that is usually ill conditioned we used a global minimization procedure combined with the method of successive approximations that utilized a small finite set of mq basis functions accurate solutions of weakly singular volterra integral equations are obtained with the minimal number of mq basis functions the expansion and optimization procedure was terminated whenever the global errors were less than 5 lo 2002 elsevier science ltd all rights reserved

geometry of the bifurcations of the normalized reduced henon heiles family this paper deals with the global geometry of the bifurcations of a family of hamiltonian functions that arises from normalizing the henon heiles family to fourth degree terms and then performing a reduction this gives a geometric explanation of the bifurcation diagram for the main resonance in the model of axisymmetric galaxies of braun and verhulst

dimension splitting data points redistribution for meshless approximation to better approximate nearly singular functions with meshless methods we propose a data points redistribution method extended from the well known one dimensional equidistribution principle with properly distributed data points nearly singular functions can be well approximated by linear combinations of global radial basis functions the proposed method is coupled with an adaptive trial subspace selection algorithm in order to reduce computational cost in our numerical examples clear exponential convergence with respect to the numbers of data points can be observed

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

osculatory interpolation in the method of fundamental solution for nonlinear poisson problems abstract the method of fundamental solution also known as the f trefftz method or the singularity method is an efficient numerical method for the solution of laplace equation for both two and three dimensional problems in recent years the method has also been applied for the solution of poisson equations by finding the particular solution to the nonhomogeneous terms in general approximate particular solutions are constructed using the interpolation of the nonhomogeneous terms by the radial basis functions the method has been validated in recent papers this paper presents an improvement of the solution procedure for such problems the improvement is achieved by using radial basis functions called osculatory radial basis functions such functions make use of the normal gradient at boundary to obtain improved interpolation the efficacy of the method is demonstrated for some prototypical nonlinear poisson problems and for multiple poisson equations

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

a meshfree stabilized collocation method scm based on reproducing kernel approximation abstract meshfree direct collocation method dcm based on strong form suffers low accuracy and instability compared with galerkin based meshfree methods meshfree weighted least squares collocation method wlscm can improve the accuracy and stability by minimizing the least squares functional corresponding to the governing equation and boundary conditions however usage of more discrete points in the least squares solution increases the computational cost which notably reduces the efficiency in this paper we propose a new meshfree stabilized collocation method scm and introduce the reproducing kernel function as the approximation auxiliary collocation points located in the local subdomains are employed for the stabilization their proper positions and corresponding weighting factors for integration using constant and cubic b spline weighting functions are derived to keep the consistency conditions therefore in this method the approximation can satisfy the consistency conditions not only on the points but also in the subdomains the suggested method has the comparable accuracy and stability compared with wlscm and the computation efficiency can match up to dcm numerical simulations demonstrate that scm can surpass dcm in accuracy and stability and outperform wlscm in efficiency and conditioning of the matrix according to the discrete equations where the condition number of matrix is related to the stability

proper orthogonal decomposition based model order reduction via radial basis functions for molecular dynamics systems summary model order reduction for molecular dynamics md systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi atomic interactions in high dimensions in the present work we introduce a proper orthogonal decomposition based method in conjunction with the radial basis function rbf approximation of the nonlinear and nonlocal potential energies and inter atomic forces for md systems this approach avoids coordinate transformation between the physical and reduced order coordinates and allows the potentials and inter atomic forces to be calculated directly in the reduced order space the rbf approximated potential energies and inter atomic forces in the reduced order space are discretized on the basis of the smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method the good approximation properties of rbfs in interpolating scattered data make them ideal candidates for the reduced order approximation of md inter atomic force calculations the proposed approach is validated by performing the reduced order simulations of dna molecules under various external loadings copyright 2013 john wiley sons ltd

treatment of material discontinuity in the element free galerkin method abstract a method is presented for application of the element free galerkin method efg to solid mechanics problems containing material discontinuities by using the efg method the trial and test functions for the weak form are constructed with moving least square interpolants in each material domain additional constraints are imposed at the variational level to ensure satisfaction of interface conditions as a result only nodal data no elemental connectivity is needed given c 1 continuity of the interpolant weighting function the dependent variable and its gradient are continuous in each material domain two linear elastic bi material examples are presented to illustrate the effectiveness of the method

the meshless numerical simulation of kelvin helmholtz instability during the wave growth of liquid liquid slug flow abstract kelvin helmholtz instability khi occurs at the interface of two fluids in which the heavier fluid flows at the bottom in the present numerical work the khi was analyzed by solving the modification of two dimensional 2 d incompressible navier stokes equations to capture the interface the cahn hilliard equation was implemented into the navier stokes equations the phenomena around the khi of two and three component fluids were investigated numerically by using radial basis function rbf combined with the domain decomposition method ddm in a primitive variable formulation here ddm is able to solve the large scale problem on the other hand the calculation accuracy decreases with the increase of the number of the subdomains for the above reason in the present works the domain was partitioned into 15 x 15 subdomains in order to reduce the decrease in accuracy due to the domain division next fractional step method was used to solve the modification of the navier stokes equations the numerical results indicate that the procedure used in the present work can easily handle the khi problem under the variations of the interface thickness density ratios and initial velocity differences moreover the interface evolutions obtained from the present method agree well with those of the finite difference method the effects of the interface thickness density ratio and magnitude of velocity difference on the khi were also investigated the decrease of the interface thickness produces a non smooth concentration profile and the increase of the interface thickness produces too much surface diffusion it was found also that the increase of the density ratio reduces the growth of khi the interface rolls up are strongly affected by the initial horizontal velocity difference finally the present study also shows that the rbf method is a reliable method to solve the khi on the complex domains

some extensions of radial basis functions and their applications in artificial intelligence abstract radial basis functions have recently found interesting applications in artificial intelligence and in particular in the problem of learning to perform a particular task from a set of examples however in many practical cases the radial basis functions method cannot be applied in a straight forward manner because it does not take into account some features that are typical of the problem of learning from examples in this paper we show some extensions of the standard theory introduced in order to deal with a large number of examples and with problems in which different variables play very different roles we present some examples and also point out some open problems

radial basis collocation methods for elliptic boundary value problems this paper introduces radial basis functions rbf into the collocation methods and the combined methods for elliptic boundary value problems first the ritz galerkin method rgm is chosen using the rbf and the integration approximation leads to the collocation method of rbf for poisson s equation next the combinations of rbf with finite element method fem finite difference method fdm etc can be easily formulated by following li 1 and hu and li 2 3 but more analysis of inverse estimates is explored in this paper since the rbfs have the exponential convergence rates and since the collocation nodes may be scattered in rather arbitrary fashions in various applications the rbf may be competitive to orthogonal polynomials for smooth solutions moreover for singular solutions we may use some singular functions and rbfs together numerical examples for smooth and singularity problems are provided to display effectiveness of the methods proposed and to support the analysis made

improved radial basis function methods for multi dimensional option pricing in this paper we have derived a radial basis function rbf based method for the pricing of financial contracts by solving the black scholes partial differential equation as an example of a financial contract that can be priced with this method we have chosen the multi dimensional european basket call option we have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter for other non optimal choices of shape parameter values the resulting convergence rate is algebraic we propose an adapted node point placement that improves the accuracy compared with a uniform distribution compared with an adaptive finite difference method the rbf method is 20 40 times faster in one and two space dimensions and has approximately the same memory requirements

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

a meshfree method for inverse wave propagation using collocation and radial basis functions abstract a stable and highly accurate meshfree scheme based on strong form collocation associated with radial basis functions and explicit time integration is proposed to solve ill posed inverse wave propagation problems tikhonov regularization technique and the l curve criterion are employed to deal with the noise measurement data before solving the governing equation to get a stable solution appropriate weights are first derived for imposing boundary conditions and known conditions from the measurement data to yield the optimal convergence for inverse problems compared to the required iterative procedures in common methods no iteration is needed in the proposed method which leads to the time saving computation and avoids the instability resulting from the iterative methods dispersion analysis describes that this method has much less dispersion error than the classical finite element method fem and popular weak form meshfree methods and stability analysis is performed to predict the critical time step for the explicit time integration another outstanding advantage of this method is that it is effortless to extend the solution scheme of one dimensional inverse problem to high dimensional inverse problems furthermore the influences of the high noise into measurement data are evaluated which demonstrate that even handling with very high noise in the input data the proposed algorithm can achieve stable and high accuracy solutions

a nonlinear empirical prescription for simultaneously interpolating and smoothing contours over an irregular grid a simple empirical interpolation procedure is developed based on the approximation that the magnitudes in the contouring space vary according to the inverse square of the distance this prescription is no more complicated than using linear interpolation but has the advantage that an interpolated point is now a function of its n nearest neighbours in all directions using such an interpolation approximation a method has been devised for tracing the contours through the reference grid points plus extra grid points whereby each triangular element now has 3d 3 extra grid points where d is the number of calculated points on each side of the triangle this procedure does not increase the computational burden but merely serves the purpose of guiding the path of the contour by taking into account the nonlinearity of the space between the original reference grid points within an inverse square approximation this technique has the added benefit that the resulting contours are simultaneously smoothed out

on a new family of radial basis functions mathematical analysis and applications to option pricing abstract in this paper we introduce a new family of infinitely smooth and nearly locally supported radial basis functions rbfs derived from the general solution of a heat equation arising from the american option pricing problem these basis functions are expressed in terms of the repeated integrals of the complementary error function and provide highly efficient tools to solve the free boundary partial differential equation resulting from the related option pricing model we introduce an integral operator with a function dependent lower limit which is employed as a basic tool to prove the radial positive definiteness of the proposed basis functions and could be of independent interest in the rbf theory we then show that using the introduced functions as expansion bases in the context of an rbf based meshless collocation scheme we could exactly impose the transparent boundary condition accompanying the heat equation we prove that the condition numbers of the resulting collocation matrices are orders of magnitude less than those arising from other popular rbf families used in current literature some other properties of these bases such as their fourier transforms as well as some useful representations in terms of positive borel measures will also be discussed

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

polynomials and potential theory for gaussian radial basis function interpolation we explore a connection between gaussian radial basis functions and polynomials using standard tools of potential theory we find that these radial functions are susceptible to the runge phenomenon not only in the limit of increasingly flat functions but also in the finite shape parameter case we show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices while not imposing restrictions on the shape parameter or number of points

weighted radial basis collocation method for boundary value problems this work introduces the weighted radial basis collocation method for boundary value problems we first show that the employment of least squares functional with quadrature rules constitutes an approximation of the direct collocation method standard radial basis collocation method however yields a larger solution error near boundaries the residuals in the least squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted the error analysis shows unbalanced errors between domain neumann boundary and dirichlet boundary least squares terms a weighted least squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors it is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates

multidimensional interpolation using osculatory radial basis functions abstract a procedure has been developed for the interpolation of functions defined in two dimensions using the values of the function and its normal derivatives at the boundary the interpolating functions used are combinations of two classes of radial basis functions this permits an interpolation scheme with osculatory features i e the interpolant has a specified slope at the boundary interpolation points the improved accuracy of the method is demonstrated over interpolation schemes using the traditional radial basis functions with no osculation especially for the calculation of the derivatives of the function at various internal points a brief discussion on the utility of the new interpolation technique in solving nonlinear poisson problems using the dual reciprocity boundary element method is provided

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

recent developments in the numerical evaluation of particular solutions in the boundary element method abstract we survey recent work on the numerical evaluation of particular solutions in the boundary elements method bem particularly rbf approximations and related mathematical convergence theory current research suggests at least in r 2 that the best choice for bases is the generalized multiquadrics numerical results are presented to support this observation

a weighted meshfree collocation method for incompressible flows using radial basis functions abstract a weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible navier stokes equations the velocities and pressure are solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability no artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation radial basis collocation method in a lagrangian form is quite easy to capture the moving boundary or free surface in flow problems moreover solid boundary conditions can be enforced directly and no special treatments are required further critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated numerical studies validate the high accuracy as well as good stability of the presented method

the shape parameter in the gaussian function this is the fifth of our series of works about the shape parameter we now explore the parameter b contained in the famous gaussian function e b x 2 email protected r n in the theory of radial basis functions rbfs the gaussian is frequently used in virtue of its good error bound and numerical tractability however the optimal choice of b has been unknown people conversant with rbfs know that b is very influential but do not have a reliable criterion of its choice the purpose of this paper is to uncover its mystery in particular we have greatly improved the result of madych 1992 in 15 and we present a concrete function of b which shows the influence of b in the error estimate of gaussian interpolation and with which the optimal b can always be found

proof of non integrability for the henon heiles hamiltonian near an exceptional integrable case abstract in this paper we describe an analytical method for establishing non integrability in hamiltonian systems which are small perturbations of integrable ones we illustrate the method with an application to a system of henon heiles type

meshfree point collocation method for elasticity and crack problems a generalized diffuse derivative approximation is combined with a point collocation scheme for solid mechanics problems the derivatives are obtained from a local approximation so their evaluation is computationally very efficient this meshfree point collocation method has other advantages it does not require special treatment for essential boundary condition nor the time consuming integration of a weak form neither the connectivity of the mesh nor differentiability of the weight function is necessary the accuracy of the solutions is exceptional and generally exceeds that of element free galerkin method with linear basis the performance and robustness are demonstrated by several numerical examples including crack problems copyright 2004 john wiley sons ltd

homoclinic and heteroclinic orbits of reversible vectorfields under perturbation on demontre l existence de liens homocline heterocline entre les orbites periodiques hyperboliques d une classe de systemes hamiltoniens admettant un nombre suffisant de symetries sous une reduction isoenergetique le probleme se reduit a un probleme comprenant des champs vectoriels reversibles avec perturbations dependantes du temps admettant les memes symetries de renversement

geometrical aspects of ziglin s non integrability theorem for complex hamiltonian systems on place la methode de yoshida dans un cadre geometrique conceptuel et on etend son applicabilite

on certain block toeplitz matrices related to radial functions abstract interpolation of smooth functions and the discretization of elliptic pdes by means of radial functions lead to structured linear systems which for equidistant grid points have almost the block toeplitz structure we prove upper bounds for the condition numbers of the n n toeplitz matrices which discretize the model problem u x f x x 0 1 u 0 a u 1 b over an equally spaced grid of n 2 points in 0 1 by means of the collocation method based on radial functions of the multiquadric inverse multiquadric and gaussian type these bounds are asymptotically sharp

subdomain radial basis collocation method for heterogeneous media summary strong form collocation in conjunction with radial basis approximation functions offer implementation simplicity and exponential convergence in solving partial differential equations however the smoothness and nonlocality of radial basis functions pose considerable difficulties in solving problems with local features and heterogeneity in this work we propose a simple subdomain strong form collocation method in which the approximation in each subdomain is constructed separately proper interface conditions are then imposed on the interface under the subdomain strong form collocation construction it is shown that both neumann and dirichlet boundary conditions should be imposed on the interface to achieve the optimum convergence error analysis and numerical tests consistently confirm the need to impose the optimal interface conditions the performance of the proposed methods in dealing with heterogeneous media is also validated copyright q 2009 john wiley sons ltd

a cartesian grid collocation technique with integrated radial basis functions for mixed boundary value problems in this paper high order systems are reformulated as first order systems which are then numerically solved by a collocation method the collocation method is based on cartesian discretisation with 1d integrated radial basis function networks 1d irbfn maiduy trancong 2007 the present method is enhanced by a new boundary interpolation technique based on 1d irbfn which is introduced to obtain variable approximation at irregular points in irregular domains the proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains the main results obtained are a the boundary conditions for the reformulated problem are of dirichlet type only b the integrated rbfn approximation avoids the well known reduction of convergence rate associated with differential formulations c the primary variable e g displacement temperature and the dual variable e g stress temperature gradient have similar convergence order d the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated numerical experiments show that the proposed method achieves very good accuracy and high convergence rates

symbolic dynamics for the henon heiles hamiltonian on the critical level abstract we present a computer assisted proof of the existence of a rich symbolic dynamic structure for the henon heiles hamiltonian system at the critical energy e 1 6

on averaging reduction and symmetry in hamiltonian systems abstract the existence of periodic orbits for hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged hamiltonian on an associated reduced space alternatively in classical kinetic plus potential energy hamiltonians the existence of such orbits can often be established by elementary geometrical arguments the present paper unifies the two approaches by exploiting discrete symmetries including reversing diffeomorphisms that occur in a given system the symmetries are used to locate the periodic orbits in the averaged hamiltonian and thence in the original hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries in applications to the henon heiles hamiltonian it is illustrated how higher order averaging can sometimes be used to overcome degeneracies encountered at first order

integrated multiquadric radial basis function approximation methods promising numerical results using once and twice integrated radial basis functions have been recently presented in this work we investigate the integrated radial basis function irbf concept in greater detail connect to the existing rbf theory and make conjectures about the properties of irbf approximation methods the irbf methods are used to solve pdes

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

better bases for radial basis function interpolation problems radial basis function interpolation involves two stages the first is fitting solving a linear system corresponding to the interpolation conditions the second is evaluation the systems occurring in fitting problems are often very ill conditioned changing the basis in which the radial basis function space is expressed can greatly improve the conditioning of these systems resulting in improved accuracy and in the case of iterative methods improved speed of solution the change of basis can also improve the accuracy of evaluation by reducing loss of significance errors in this paper new bases for the relevant space of approximants and associated preconditioning schemes are developed which are based on floater s mean value coordinates positivity results and scale independence results are shown for schemes of a general type numerical results show that the given preconditioning scheme usually improves conditioning of polyharmonic spline and multiquadric interpolation problems in r 2 and r 3 by several orders of magnitude the theory indicates that using the new basis elements evaluated indirectly for both fitting and evaluation will reduce loss of significance errors on evaluation numerical experiments confirm this showing that such an approach can improve overall accuracy by several significant figures

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

local radial basis function based differential quadrature method and its application to solve two dimensional incompressible navier stokes equations local radial basis function based differential quadrature method is presented in detail in this paper the method is a natural mesh free approach like the conventional differential quadrature dq method it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots which may be distributed randomly however different from the conventional dq method the weighting coefficients in present method are determined by taking the radial basis functions rbfs instead of high order polynomials as the test functions the method works in a similar fashion as conventional finite difference schemes but with truly mesh free property in this paper we mainly concentrate on the multiquadric rbfs since they have exponential convergence the effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied and how the value of optimal c varies with the number of local support knots is also numerically demonstrated the proposed method is validated by its application to the simulation of natural convection in a square cavity excellent numerical results are obtained on an irregular knot distribution

interpolation of track data with radial basis methods abstract the multiquadric and thin plate spline radial basis methods together with the triangle based minimum norm network algorithm and the modified quadratic shepards method are applied to various sets of data that are sampled densely along tracks in the plane the effectiveness of these methods on track data has been questioned in the past we observe that both radial basis methods and the minimum norm network method performed well on smoothly varying track data sets while the multiquadric method with a small value for the parameter r 2 was the only method that was effective on rapidly varying track data

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

subdomain radial basis collocation method for fracture mechanics the direct approximation of strong form using radial basis functions rbfs commonly called the radial basis collocation method rbcm has been recognized as an effective means for solving boundary value problems nevertheless the non compactness of the rbfs precludes its application to problems with local features such as fracture problems among others this work attempts to apply rbcm to fracture mechanics by introducing a domain decomposition technique with proper interface conditions the proposed method allows 1 natural representation of discontinuity across the crack surfaces and 2 enrichment of crack tip solution in a local subdomain with the proper domain decomposition and interface conditions exponential convergence rate can be achieved while keeping the discrete system well conditioned the analytical prediction and numerical results demonstrate that an optimal dimension of the near tip subdomain exists the effectiveness of the proposed method is justified by the mathematical analysis and demonstrated by the numerical examples copyright 2010 john wiley sons ltd

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

comment on a nonlinear empirical prescription for simultaneously interpolating and smoothing contours over an irregular grid by f duggan abstract the article discussed introduces inverse distance weighting idw as an interpolation technique for scattered measurements idw has been extensively researched and tested and its limitations widely published in the present context however it is considered desirable to augment duggan s contribution by enumerating the drawbacks of the method other aspects of the article which require particular clarification are also noted

radial basis function interpolation in the quantum trajectory method optimization of the multi quadric shape parameter this paper investigates multi quadric radial basis function rbf interpolation and its application in the quantum trajectory method qtm for wave packet propagation in the multi quadric r r2 2 1 2 r is the radial distance from the observation point to the origin of the basis function and is known as the shape parameter due to its affect on the functional form of the basis function the quality of any rbf interpolation scheme is dictated by the choice of this parameter many recent studies have investigated a suitable means for obtaining an optimized time independent parameter the purpose of this study however is to not only to find this optimized shape parameter but also to analyze its time dependence in four different dynamical models the anisotropic harmonic oscillator the downhill ramp the uphill ramp and a harmonic oscillator coupled with a downhill ramp to obtain the optimized shape parameter at each time step an algorithm similar to the leave one out method of cross validation is utilized the results for each of the four models are presented and the feasibility and necessity of employing a shape parameter optimization algorithm for each of the models is discussed

a compact five point stencil based on integrated rbfs for 2d second order differential problems in this paper a compact 5 point stencil for the discretisation of second order partial differential equations pdes in two space dimensions is proposed we employ integrated radial basis functions in one dimension 1d irbfs to construct the approximations for the dependent variable and its derivatives over the three nodes in each direction of the stencil certain nodal values of the second order derivatives are incorporated into the approximations with the help of the integration constants in the case of elliptic pdes one algebraic equation is formed at each interior node and the obtained final system of which each row has 5 non zero entries is solved iteratively using a picard scheme in the case of parabolic pdes discretised with a crank nicolson procedure a set of three simultaneous algebraic equations is established at each interior node and the three equations are then combined to form two tridiagonal equations through the implicit elimination approach linear and non linear test problems including lid driven cavity flow and natural convection between the outer square and the inner cylinder are considered to verify the proposed stencil

multiquadric collocation method with integralformulation for boundary layer problems singularly perturbed boundary value problems often have solutions with very thinlayers in which the solution changes rapidly this paper concentrates on the case where theses layers occur near the boundary although our method can be applied to problems with interior layers one technique to deal with the increased resolution requirements in these layers is the use of domain transformations a coordinate stretching based transform allows to move collocation points into the layer a requirement to resolve the layer accurately previously such transformations have been studied in the context of finite difference and spectral collocation methods in this paper we use radial basis functions rbfs to solve the boundary value problem specifically we present a collocation method based on multiquadric mq functions with an integral formulation combined with a coordinate transformation we find that our scheme is ultimately more accurate than a recently proposed adaptive mq scheme the rbf scheme is also amenable to adaptivity

the role of the multiquadric shape parameters in solving elliptic partial differential equations this study examines the generalized multiquadrics mq f j x x x j 2 c j 2 b in the numerical solutions of elliptic two dimensional partial differential equations pdes with dirichlet boundary conditions the exponent b as well as c j 2 can be classified as shape parameters since these affect the shape of the mq basis function we examined variations of b as well as c j 2 where c j 2 can be different over the interior and on the boundary the results show that increasing b has the most important effect on convergence followed next by distinct sets of c j 2 w w c j 2 w additional convergence accelerations were obtained by permitting both c j 2 w w and c j 2 w to oscillate about its mean value with amplitude of approximately 1 2 for odd and even values of the indices our results show high orders of accuracy as the number of data centers increases with some simple heuristics

automatic interpolation methods for mapping piezometric surfaces abstract the purpose of analysis of spatial data should determine the approach to be adopted if estimation is concerned with detailed local interpretation rather than some form of global averaging then kriging would seem to be an inappropriate method this is especially so for the interpolation of precise observations which consequently are better estimated by deterministic methods five such efficient direct automatic interpolation methods are discussed and applied to the contour mapping of a piezometric surface

multi dimensional option pricing using radial basis functions and the generalized fourier transform we show that the generalized fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi dimensional option pricing we derive a general algorithm including a transformation of the black scholes equation into the heat equation that can be used in any number of dimensions numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes furthermore the gain increases with the number of dimensions