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

implementation of a generalized exponential basis functions method for linear and non linear problems this is the accepted version of the following article mossaiby f ghaderian m rossi r implementation of a generalized exponential basis functions method for linear and non linear problems international journal for numerical methods in engineering on line jul 2015 which has been published in final form at http dx doi org 10 1002 nme 4985

a green s discrete transformation meshfree method for simulating transient diffusion problems this manuscript presents the formulation and application of the green s discrete transformation method gdtm for the meshfree simulation of transient diffusion problems including those with moving boundaries the gdtm implements a linear combination of time dependent green s basis functions defined on a set of source points to approximate the field in the form of a solution series a discrete transformation is implemented to evaluate unknown coefficients of this series which eliminates the need to use time integration schemes we will study the optimal number and location of the gdtm source points that yield the highest level of accuracy while maintaining a manageable condition number for the resulting linear system of equations the optimal values of these parameters which are inherently independent of the domain geometry are determined such that the basis functions have appropriate features for approximating the field a comprehensive convergence study is presented to show the precision and convergence rate of the gdtm for modeling various diffusion problems we also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies heat transfer in a turbine blade thermal response of a porous material and localized pitting corrosion in stainless steel copyright c 2015 john wiley sons ltd

local radial basis function based gridfree scheme for unsteady incompressible viscous flows a local radial basic function rbf based gridfree scheme has been developed to solve unsteady incompressible navier stokes equations in primitive variables the velocity pressure decoupling is obtained by making use of a fractional step algorithm the scheme is validated over a variety of benchmark problems and found a very good agreement with the existing results comparisons with the benchmark solutions show that the developed local rbf gridfree scheme is stable and produces accurate results on domains discretized even with non uniform distribution of nodal points

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

multi level meshless methods based on direct multi elliptic interpolation a short overview on the direct multi elliptic interpolation and the related meshless methods for solving partial differential equations is given a new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain an error estimation is also derived to implement the method quadtree based multi level methods are used the approach avoids the use of large dense and ill conditioned matrices and significantly reduces the computational cost

rbf fd formulas and convergence properties the local rbf is becoming increasingly popular as an alternative to the global version that suffers from ill conditioning in this paper we study analytically the convergence behavior of the local rbf method as a function of the number of nodes employed in the scheme the nodal distance and the shape parameter we derive exact formulas for the first and second derivatives in one dimension and for the laplacian in two dimensions using these formulas we compute taylor expansions for the error from this analysis we find that there is an optimal value of the shape parameter for which the error is minimum this optimal parameter is independent of the nodal distance our theoretical results are corroborated by numerical experiments

comparison of finite element techniques for solidification problems the accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed grid finite element techniques using the analytical solution for this problem as a reference in the finite element formulation of the problem different time stepping schemes are compared the implicit euler backward algorithm combined with an iterative scheme and two three time level methods the lees algorithm and a dupont algorithm which are both applied as non iterative schemes furthermore different methods for handling the evolution of latent heat are examined an approximation method suggested by lemmon and one suggested by del giudice both using the enthalpy formulation as well as a fictitious heat flow method presented by rolph and bathe results of calculations performed with the consistent heat capacity matrix are compared with those performed with a lumped heat capacity matrix

an alternative local collocation strategy for high convergence meshless pde solutions using radial basis functions this work proposes an alternative decomposition for local scalable meshless rbf collocation the proposed method operates on a dataset of scattered nodes that are placed within the solution domain and on the solution boundary forming a small rbf collocation system around each internal node unlike other meshless local rbf formulations that are based on a generalised finite difference rbf fd principle in the proposed finite collocation method the solution of the pde is driven entirely by collocation of pde governing and boundary operators within the local systems a sparse global collocation system is obtained not by enforcing the pde governing operator but by assembling the value of the field variable in terms of the field value at neighbouring nodes in analogy to full domain rbf collocation systems communication between stencils occurs only over the stencil periphery allowing the pde governing operator to be collocated in an uninterrupted manner within the stencil interior the local collocation of the pde governing operator allows the method to operate on centred stencils in the presence of strong convective fields the reconstruction weights assigned to nodes in the stencils being automatically adjusted to represent the flow of information as dictated by the problem physics this implicit upwinding effect mitigates the need for ad hoc upwinding stencils in convective dominant problems boundary conditions are also enforced within the local collocation systems allowing arbitrary boundary operators to be imposed naturally within the solution construction the performance of the method is assessed using a large number of numerical examples with two steady pdes the convection diffusion equation and the lame navier equations for linear elasticity the method exhibits high order convergence in each case tested greater than sixth order and the use of centred stencils is demonstrated for convective dominant problems in the case of linear elasticity the stress fields are reproduced to the same degree of accuracy as the displacement field and exhibit the same order of convergence the method is also highly stable towards variations in basis function flatness demonstrating significantly improved stability in comparison to finite difference type rbf collocation methods

visualizing design spaces using two dimensional contextual self organizing maps visualization of design spaces is a complex problem that has the potential to provide many benefits design spaces can be easily visualized with two or three design variables using a range of methods however once a problem exceeds this limit direct visualization that captures all necessary behaviors becomes difficult to visualize these higher dimensions it is necessary to use visual cues such as color size and or symbols to show the added dimensions the disadvantage to using visual cues is the inability to expand much beyond three dimensions this research focuses on using contextual self organizing maps to provide a solution to visualizing high dimensional design spaces by using the dimensionality reduction capabilities of self organizing maps a visual representation is created by generating a self organizing map and applying objective function values as the contextual labels the map is then broken into three different maps containing separate contextual information namely the mean minimum a

minimal positive stencils in meshfree finite difference methods for the poisson equation meshfree finite difference methods for the poisson equation approximate the laplace operator on a point cloud desirable are positive stencils i e all neighbor entries are of the same sign classical least squares approaches yield large stencils that are in general not positive we present an approach that yields stencils of minimal size which are positive we provide conditions on the point cloud geometry so that positive stencils always exist the new discretization method is compared to least squares approaches in terms of accuracy and computational performance

an iterative multistep kernel based method for nonlinear volterra integral and integro differential equations of fractional order abstract an iterative multistep kernel based method is proposed for the nonlinear volterra integral equations and nonlinear volterra integro differential equations of fractional order which can produce reliable globally smooth numerical solutions an error estimate of the positive definite kernel interpolation and the convergence and error analysis of the proposed iterative scheme are investigated here we focused on positive definite radial basis kernels and further a new and applicable shape parameter selection strategy is proposed the proposed multi step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long time simulations in order to show the efficiency and versatility of the proposed method some numerical experiments are reported the comparison of the numerical results with the analytical solutions and the best reported results in the literature confirm the good accuracy and applicability of the proposed method

the sample solution approach for determination of the optimal shape parameter in the multiquadric function of the kansa method abstract the kansa method with the multiquadric radial basis function mq rbf is inherently meshfree and can achieve an exponential convergence rate if the optimal shape parameter is available however it is not an easy task to obtain the optimal shape parameter for complex problems whose analytical solution is often a priori unknown this has long been a bottleneck for the mq kansa method application to practical problems in this paper we present a novel sample solution approach ssa for achieving a reasonably good shape parameter of the mq rbf in the kansa method for the solution of problems whose analytical solution is unknown the basic assumption behind the ssa is that the optimal shape parameter is considered to be largely depended on the shape of computational domain the type of the boundary conditions the number and distribution of nodes and the governing equation in the procedure of the ssa we set up a pseudo problem as the sample solution whose solution is known it is not difficult to obtain the optimal parameter of the mq rbf in the numerical solution of the pseudo problem the ssa suggests that the optimal shape parameter of the pseudo problem can also achieve an approximately optimal accuracy in the solution of the original problem numerical examples and comparisons are provided to verify the proposed ssa in terms of accuracy and stability in solving homogeneous problems and non homogeneous modified helmholtz problems in several complex domains even using chaotic distribution of collocation points

stability and convergence of radial basis function finite difference method for the numerical solution of the reaction diffusion equations stability convergence and application of radial basis function finite difference rbf fd scheme is studied for solving the reaction diffusion equations rdes we show that the explicit rbf fd method is stable and stability condition depends on the shape parameter of related radial basis function the generalized multiquadric gmq is applied as radial basis function and weight coefficients are explicitly presented for equispaced node distribution also two methods are presented to compute the optimal shape parameter the combination of these methods with the gmq fd method will produce two efficient algorithms for numerical solution of rdes the variable gmq fd vgmq fd and the constant gmq fd cgmq fd we test the scheme on traveling wave and compare its accuracy with the conventional finite difference method fdm

development and validation of a 3d rbf spectral model for coastal wave simulation abstract with the objective of simulating wave propagation in the nearshore zone for engineering scale applications a two dimensional 2dv model based on the euler zakharov equations 73 54 is extended to three dimensions 3d to maintain the flexibility of the approach with the goal of applying the model to irregularly shaped domains the horizontal plane is discretized with scattered nodes the horizontal derivatives are then estimated using the radial basis function finite difference rbf fd method while a spectral approach is used in the vertical dimension a sensitivity analysis examined the robustness of the rbf fd approach as a function of rbf parameters when estimating the derivatives of a representative function for a targeted stencil size between 20 and 30 nodes piecewise smooth ps polyharmonic spline phs functions are recommended avoiding the use of infinitely smooth is rbfs which are less appropriate for the desired applications because of their dependence on a shape parameter comparisons of simulation results to observations from two wave basin experiments show that nonlinear effects induced by complex bottom bathymetries are reproduced well by the model with the recommended rbf approach validating the use of this method for 3d simulations of wave propagation

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

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

numerical analysis of graded mesh methods for a class of second kind integral equations on the real line abstract in this paper we are concerned with the numerical analysis of the collocation method based on graded meshes of second kind integral equations on the real line of the form s s r s t z t t dt s r where l 1 r z l r and bc r the space of bounded continuous complex valued functions on r are assumed known and the function bc r is to be determined we introduce some new graded meshes for the collocation method of the integral equation which are different from those used previously for the wiener hopf integral equation in the case when the solution decays exponentially at infinity and establish optimal local and global l norm error estimates under the condition that the solution decays only polynomially at infinity

a meshfree generalized finite difference method for surface pdes abstract in this paper we propose a novel meshfree generalized finite difference method gfdm approach to discretize pdes defined on manifolds derivative approximations for the same are done directly on the tangent space in a manner that mimics the procedure followed in volume based meshfree gfdms as a result the proposed method not only does not require a mesh it also does not require an explicit reconstruction of the manifold in contrast to some existing methods it avoids the complexities of dealing with a manifold metric while also avoiding the need to solve a pde in the embedding space a major advantage of this method is that all developments in usual volume based numerical methods can be directly ported over to surfaces using this framework we propose discretizations of the surface gradient operator the surface laplacian and surface diffusion operators possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced and a few practical applications are presented

error estimates of local multiquadric based differential quadrature lmqdq method through numerical experiments summary in this article we present an error estimate of the derivative approximation by the local multiquadricbased differential quadrature lmqdq method radial basis function is different from the polynomial approximation in which taylor series expansion is not applicable so the present analysis is performed through the numerical solution of poisson equation it is known that the approximation error of lmqdq method depends on three factors i e local density of knots h free shape parameter c and number of supporting knots ns by numerical experiments their contribution to the approximation error and correlation were studied and analysed in this paper an error estimate o h c n is thereafter proposed in which n is a positive constant and determined by the number of supporting knots ns copyright 2005 john wiley sons ltd

local rbf fd solutions for steady convection diffusion problems this paper describes the application of radial basis function rbf based finite difference type scheme rbf fd for solving steady convection diffusion equations numerical studies are made using multiquadric mq rbf by varying the shape parameter in mq the accuracy of the solution is seen to be highly improved for large values of reynolds numbers the developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non oscillatory copyright 2007 john wiley sons ltd

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

local method of approximate particular solutions for two dimensional unsteady burgers equations the local method of approximate particular solutions lmaps is first proposed to solve burgers equations in avoiding the ill conditioning problem the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low order linear systems within the local supporting domain then the global solutions can be obtained by reformulating the local matrix in the global and sparse matrix the obtained large sparse linear systems are directly solved instead of using more complicated iterative method the numerical experiments have shown that the proposed method is suitable for solving the two dimensional unsteady burgers equations with high accuracy and efficiency

improved treatment of wall boundary conditions for a particle method with consistent spatial discretization abstract in recent years consistent spatial discretization schemes for meshfree particle methods to numerically simulate incompressible flow have been studied by many researchers this study is focused on the treatment of solid wall boundary conditions for one of those schemes namely the least squares mps lsmps scheme and proposes a new technique to deal with no slip and free slip wall boundary conditions with the proposed treatment wall geometries are expressed by surface meshes i e polygons in 3d and line segments in 2d thus complicated geometries can be handled easily based on a taylor series expansion wall boundary conditions are incorporated into the differential operators acting on fluid particles located in the vicinity of a wall through a least squares approach as a consequence neumann boundary conditions can be treated quite efficiently to verify consistency of the proposed discretization scheme a convergence study was carried out as numerical examples couette flow plane poiseuille flow gravity driven flow in a 3d square duct a rigid rotation problem taylor green vortices and lid driven cavity flow have been calculated using the proposed boundary treatment with both no slip and free slip conditions applied as a result the present method agreed well with the reference solutions which verified its computational accuracy

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

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

optimal variable shape parameter for multiquadric based rbf fd method in this follow up paper to our previous study in bayona et al 2011 2 we present a new technique to compute the solution of pdes with the multiquadric based rbf finite difference method rbf fd using an optimal node dependent variable value of the shape parameter this optimal value is chosen so that to leading order the local approximation error of the rbf fd formulas is zero in our previous paper bayona et al 2011 2 we considered the case of an optimal constant value of the shape parameter for all the nodes our new results show that if one allows the shape parameter to be different at each grid point of the domain one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique we analyze the same examples studied in bayona et al 2011 2 both with structured and unstructured grids and compare our new results with those obtained previously we also find that if there are a significant number of nodes for which no optimal value of the shape parameter exists then the improvement in accuracy deteriorates significantly in those cases we use generalized multiquadrics as rbfs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter

numerical methods based on rational variable substitution for wiener hopf equations of the second kind this paper considers numerical methods for wiener hopf equations of the second kind y t 0 k t s y s ds g t 0 t by applying rational variable substitution to integrals on the semi infinite interval 0 and using the well known clenshaw curtis quadrature to the resulted integral we get a clenshaw curtis rational ccr quadrature rule we then apply the ccr quadrature to wiener hopf equations the reduction of singularities in the transformed equation is considered numerical examples are given to illustrate the efficiency of the numerical methods proposed in this paper

operational quadrature methods for wiener hopf integral equations we study the numerical solution of wiener hopf integral equations by a class of quadrature methods which lead to discrete wiener hopf equations with quadrature weights constructed from the fourier transform of the kernel or rather from the laplace transforms of the kernel halves as the analytical theory of wiener hopf equations is likewise based on the fourier transform of the kernel this approach enables us to obtain results on solvability and stability and error estimates for the discretization the discrete wiener hopf equations are solved by using an approximate wiener hopf factorization obtained with fft numerical experiments with the milne equation of radiative transfer are included

a comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations abstract in this article we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations a series of test examples was solved with these two schemes different problems with different type of governing equations and boundary conditions particular emphasis was paid to the ability of these schemes to solve the steady state convection diffusion equation at high values of the peclet number from the examples tested in this work it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better however the unsymmetric method has the advantage of being simpler to implement two main features about the results obtained in this work are worthy of special attention first with the symmetric method it was possible to solve convection diffusion problems at a very high peclet number without the need of any artificial damping term and second with these two approaches symmetric and unsymmetric it is possible to impose free boundary conditions for problems in unbounded domains

product integration collocation methods for noncompact integral operator equations we discuss the numerical solution of a class of second kind integral equations in which the integral operator is not compact such equations arise for example when boundary integral methods are applied to potential problems in a two dimensional domain with corners in the boundary we are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes provided some simple modifications are made to the underlying basis functions these are sufficient to ensure stability but do not damage the rate of convergence numerical experiments show that such modifications are necessary in certain circumstances

frequency optimized rbf fd for wave equations abstract we present a method to obtain optimal rbf fd formulas which maximize their frequency range of validity the optimization is based on the idea of keeping an error of interest dispersion phase or group velocity errors below a given threshold for a wavenumber interval as large as possible to find the weights of these optimal finite difference formulas we solve an optimization problem in a previous work we developed a method to optimize the frequency range of validity for finite difference weights that method required to solve a system of nonlinear equations with as many unknowns as half of the number of weights which is a very hard task when the number of nodes gets large the current method requires solving an optimization problem with only one parameter which makes finding a global minimum easier and thus can be used for bigger stencils we also study which of the standard rbf are more appropriate for this problem and introduce a new rbf that depends on two parameters this new rbf improves the resulting frequency response of the rbf fd methods while keeping the cost of the optimization problem low

a combination of proper orthogonal decomposition discrete empirical interpolation method pod deim and meshless local rbf dq approach for prevention of groundwater contamination abstract the main presented idea is to reduce the used cpu time for employing the local radial basis functions differential quadrature lrbf dq method to this end the proper orthogonal decomposition discrete empirical interpolation method pod deim has been combined with the lrbf dq technique for checking the ability of the new procedure the groundwater equation is solved this equation has been classified in category of system of advection diffusion equations the solutions of advection equations have some shock thus special numerical methods should be applied for example discontinuous galerkin and finite volume methods moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes

a fast two grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half plane we consider the numerical treatment of second kind integral equations on the real line of the form phi s psi s int infty infty k s t z t phi t dt s in mathbb r abbreviated k z in which l 1 z l and bc the space of bounded continuous functions on are assumed known and bc is to be determined we first derive sharp error estimates for the finite section approximation reducing the range of integration to a a via bounds on i k z 1 as an operator on spaces of weighted continuous functions numerical solution by a simple discrete collocation method on a uniform grid on is then analysed in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix vector multiply via the fft to utilise this possibility we propose a modified two grid iteration a feature of which is that the coarse grid matrix is approximated by a banded matrix and analyse convergence and computational cost in cases wherez is not compactly supported a combined finite section and two grid algorithm can be applied and we extend the analysis to this case as an application we consider acoustic scattering in the half plane with a robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied our final result is that ifz related to the boundary impedance in the application takes values in an appropriate compact subsetq of the complex plane then the difference between s and its finite section approximation computed numerically using the iterative scheme proposed is c 1 khlog 1 kh 1 1 2 ka 1 2 in the interval a a 1 forkh sufficiently small wherek is the wavenumber andh the grid spacing moreover this numerical approximation can be computed in c 2 n logn operations wheren 2a h is the number of degrees of freedom the values of the constantsc 1 andc 2 depend only on the setq and not on the wavenumberk or the support ofz

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

meshless methods for one dimensional oscillatory fredholm integral equations abstract in this paper efficient and simple algorithms based on levin s quadrature theory and our earlier work involving local radial basis function rbf and chebyshev differentiation matrices are adopted for numerical solution of one dimensional highly oscillatory fredholm integral equations this work is focused on the comparative performance of local rbf meshless and pseudospectral procedures we have tested the proposed methods on phase functions with and without stationary phase point s both on uniform and chebyshev grid points the proposed procedures are shown accurate and efficient and therefore provide a reliable platform for the numerical solution of integral equations from the numerical results we draw some conclusions about accuracy efficiency and robustness of the proposed approaches

overlapping blocks by growing a partition with applications to preconditioning starting from a partitioning of an edge weighted graph into subgraphs we develop a method which enlarges the respective sets of vertices to produce a decomposition with overlapping subgraphs the vertices to be added when growing a subset are chosen according to a criterion which measures the strength of connectivity with this subset by using our method on the directed graph associated with a matrix we obtain an overlapping decomposition of the set of variables which can be used for algebraic additive and multiplicative schwarz preconditioners we present a complexity analysis of this block growing method thus proving its computational efficiency numerical results for problems stemming from various application areas show that with these overlapping schwarz preconditioners we usually substantially improve gmres convergence as compared to preconditioners based on a nonoverlapping decomposition or an overlapping decomposition based on level sets without other criteria as well as incomplete lu

local rbf fd technique for solving the two dimensional modified anomalous sub diffusion equation abstract the main aim of this paper is to propose an efficient and suitable numerical procedure based on the local meshless collocation method for solving the two dimensional modified anomalous sub diffusion equation the fractional derivative is based on the riemann liouville fractional integral firstly a finite difference scheme with o has been employed to discrete the time variable and also the local radial basis finite difference lrbf fd method is used to discrete the spatial direction for the presented numerical technique we prove the unconditional stability and also obtain an error bound we employ a test problem to show the accuracy of the proposed technique also we solve the mentioned model on irregular domain to show the efficincy of the developed technique

error bounds for kernel based numerical differentiation the literature on meshless methods shows that kernel based numerical differentiation formulae are robust and provide high accuracy at low cost this paper analyzes the error of such formulas using the new technique of growth functions it allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives since differentiation formulas based on polynomials also have error bounds in terms of growth functions we have a convenient way to compare kernel based and polynomial based formulas it follows that kernel based formulas are comparable in accuracy to the best possible polynomial based formulas a variety of examples is provided

generalized finite differences using fundamental solutions summary it is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions the fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a dirac delta distribution fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary only global approximations in the domain of interest in this work the same characteristic of the fundamental solutions that of solving the homogeneous equation exactly is used but not to build a global approximation on the contrary local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid copyright q 2009 john wiley sons ltd

multilevel compact radial functions based computational schemes for some elliptic problems abstract compactly supported radial basis functions cs rbfs have been recently introduced in the context of the dual reciprocity method as a possible cure of dense matrices and ill conditioning problems when using the classical radial basis functions however the support scaling factor and slow convergence rate of the cs rbfs have also raised issues on the effectiveness of the cs rbfs in this paper two multilevel schemes have been proposed to alleviate these problems

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

projection and multi projection methods for nonlinear integral equations on the half line abstract in this article galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear hammerstein type integral equation on the half line for both convolution and non convolution kernels using the space of piecewise polynomial subspace we prove that the approximate and iterated approximate solution in galerkin and collocation methods converges with the order o n r and o n 2 r respectively in the infinity norm where n 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation we improve these results further by considering multi projection methods in fact we show that iterated approximate solution in multi galerkin multi collocation converges with the order o n 4 r numerical results are presented to confirm the theoretical results

the use of proper orthogonal decomposition pod meshless rbf fd technique to simulate the shallow water equations abstract the main aim of this paper is to develop a fast and efficient local meshless method for solving shallow water equations in one and two dimensional cases the mentioned equation has been classified in category of advection equations the solutions of advection equations have some shock thus especial numerical methods should be employed for example discontinuous galerkin and finite volume methods here based on the proper orthogonal decomposition approach we want to construct a fast meshless method to this end we consider shallow water models and obtain a suitable time discrete scheme based on the predictor corrector technique then by applying the proper orthogonal decomposition technique a new set of basis functions can be built for the solution space in which the size of new solution space is less than the original problem thus by employing the new bases the cpu time will be reduced some examples have been studied to show the efficiency of the present numerical technique

on some convergence results for fdm with irregular mesh abstract the proof of convergence of the finite difference method with arbitrary irregular meshes for some class of elliptic problems is presented by the use of the truncation error technique and stability analysis it was showed that max i u i u h i ch i e the solution u h converges linearly with the size of the star correctness of this theorem was also confirmed by numerical tests

compact moving least squares an optimization framework for generating high order compact meshless discretizations abstract a generalization of the optimization framework typically used in moving least squares is presented that provides high order approximation while maintaining compact stencils and a consistent treatment of boundaries the approach which we refer to as compact moving least squares resembles the capabilities of compact finite differences but requires no structure in the underlying set of nodes an efficient collocation scheme is used to demonstrate the capabilities of the method to solve elliptic boundary value problems in strong form stably without the need for an expensive weak form the flexibility of the approach is demonstrated by using the same framework to both solve a variety of elliptic problems and to generate implicit approximations to derivatives finally an efficient preconditioner is presented for the steady stokes equations and the approach s efficiency and high order of accuracy is demonstrated for domains with curvi linear boundaries

two algorithms for fast 2d node generation application to rbf meshless discretization of diffusion problems and image halftoning abstract mesh generation techniques for traditional mesh based numerical approaches such as fem and fvm have now reached a good degree of maturity there is no such an acknowledged background when dealing with node generation techniques for meshless numerical approaches despite their theoretical simplicity and efficiency furthermore node generation can be put in connection with some well known image approximation techniques two node generation algorithms are here proposed and employed in the numerical solution of 2d steady state diffusion problems by means of a local radial basis function rbf meshless method finally such algorithms are also tested for grayscale image approximation through stippling

a boundary meshless method using chebyshev interpolation and trigonometric basis function for solving heat conduction problems a boundary meshless method has been developed to solve the heat conduction equations through the use of a newly established two stage approximation scheme and a trigonometric series expansion scheme to approximate the particular solution and fundamental solution respectively as a result no fundamental solution is required and the closed form of approximate particular solution is easy to obtain the effectiveness of the proposed computational scheme is demonstrated by several examples in 2d and 3d we also compare our proposed method with the finite difference method and the other meshless method showed in sarler and vertnik comput math appl 2006 51 1269 1282 excellent numerical results have been observed copyright 2007 john wiley sons ltd

adaptive meshless centres and rbf stencils for poisson equation we consider adaptive meshless discretisation of the dirichlet problem for poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions new meshless stencil selection and adaptive refinement algorithms are proposed in 2d numerical experiments show that the accuracy of the solution is comparable with and often better than that achieved by the mesh based adaptive finite element method

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

an improved localized method of approximate particular solutions for solving elliptic pdes in this paper we improve the localized method of approximate particular solutions lmaps in yao et al 2011 by utilizing the polyharmonic splines ps radial basis function rbf for solving elliptic partial differential equations pdes lmaps has been widely circulated since it is published in 2010 the multiquadric mq has been considered as the most popular choice among all rbfs however adjusting the shape parameter is a critical issue when utilizing the original lmaps in this paper we modified lmaps by combining conditionally positive definite rbf ps and an additional low degree of polynomial basis in the localization process the accuracy of the proposed lmaps is significantly improved we can simply increase the order of ps to achieve even higher accuracy other than the unexpected high accuracy there is no need to deal with the difficult issues of choosing optimal shape parameter this is a huge advantage in the rbf simulations of pdes in the numerical experiments we will present the pros and cons of improved lmaps ilmaps using ps and some commonly used rbfs mq matern and gaussian versus the original lmaps olmaps

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

the generalized finite difference method for long time dynamic modeling of three dimensional coupled thermoelasticity problems abstract in this study a new framework for the efficient and accurate solutions of three dimensional 3d dynamic coupled thermoelasticity problems is presented in our computations the krylov deferred correction kdc method a pseudo spectral type collocation technique is introduced to perform the large scale and long time temporal simulations the generalized finite difference method gfdm a relatively new meshless method is then adopted to solve the resulting boundary value problems the gfdm uses the taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables the method thus is truly meshless that can be applied for solving problems merely defined over irregular clouds of points for problem with complicated geometries this paper also examines a new distance criterion for adaptive selection of nodes in the gfdm simulations preliminary numerical experiments show that the kdc accelerated gfdm methods are very promising for accurate and efficient long time and large scale dynamic simulations

elite ensemble of optimal input pruned neural networks using trust tech the ensemble of optimal input pruned neural networks using trust tech elite method for constructing high quality ensemble through an optimal linear combination of accurate and diverse neural networks is developed the optimization problems in the proposed methodology are solved by a global optimization a global optimization method called transformation under stability retraining equilibrium characterization trust tech whose main features include its capability in identifying multiple local optimal solutions in a deterministic systematic and tier by tier manner elite creates a diverse population via a feature selection procedure of different local optimal neural networks obtained using tier 1 trust tech search in addition the capability of each input pruned network is fully exploited through a trust tech based optimal training finally finding the optimal linear combination weights for an ensemble is modeled as a nonlinear programming problem and solved using trust tech and the interior point method where the issue of non convexity can be effectively handled extensive numerical experiments have been carried out for pattern classification on the synthetic and benchmark datasets numerical results show that elite consistently outperforms existing methods on the benchmark datasets the results show that elite can be very promising for constructing high quality neural network ensembles

self organizing nets for optimization given some optimization problem and a series of typically expensive trials of solution candidates sampled from a search space how can we efficiently select the next candidate we address this fundamental problem by embedding simple optimization strategies in learning algorithms inspired by kohonen s self organizing maps and neural gas networks our adaptive nets or grids are used to identify and exploit search space regions that maximize the probability of generating points closer to the optima net nodes are attracted by candidates that lead to improved evaluations thus quickly biasing the active data selection process toward promising regions without loss of ability to escape from local optima on standard benchmark functions our techniques perform more reliably than the widely used covariance matrix adaptation evolution strategy the proposed algorithm is also applied to the problem of drag reduction in a flow past an actively controlled circular cylinder leading to unprecedented drag reduction

an adaptive rbf finite collocation approach to track transport processes across moving fronts we develop a radial basis function finite collocation method rbf fc that utilises an adaptive quadtree dataset to cluster nodes around critical features in the domain for the solution of transport processes occurring in moving front problems an adaptive quadtree dataset is used to generate an improved distribution of solution centres in the domain around which local hermitian collocation systems are formed in these local systems the governing pde and boundary operators of the problem are enforced by collocation globally the systems are linked via reconstruction of the solution variable in terms of the solution value at neighbouring nodes producing a sparse global matrix with a solution cost that scales linearly with the number of nodes in the domain by generating an adaptive set of nodal points with the quadtree dataset we can reduce the global solution error in the rbf fc method whilst maintaining low solution times the proposed method is validated on a steady state boundary layer capture problem and a transient advection problem infinite peclet number in a 2d domain we then couple the method with an interface tracking technique to solve the transient heat transfer in a hele shaw cell viscous fingering problem we demonstrate the effectiveness of the proposed method for the solution of convection dominated transport in problems across moving fronts

multicloud multigrid convergence with a meshless operator the primary objective of this work is to develop and test a new convergence acceleration technique we call multicloud multicloud is well founded in the mathematical basis of multigrid but relies on a meshless operator on coarse levels the meshless operator enables extremely simple and automatic coarsening procedures for arbitrary meshes using arbitrary fine level discretization schemes the performance of multicloud is compared with established multigrid techniques for structured and unstructured meshes for the euler equations on two dimensional test cases results indicate comparable convergence rates per unit work for multicloud and multigrid however because of its mesh and scheme transparency multicloud may be applied to a wide array of problems with no modification of fine level schemes as is often required with agglomeration techniques the implication is that multicloud can be implemented in a completely modular fashion allowing researchers to develop fine level algorithms independent of the convergence accelerator for complex three dimensional problems

doubly stochastic radial basis function methods abstract we propose a doubly stochastic radial basis function dsrbf method for function recoveries instead of a constant we treat the rbf shape parameters as stochastic variables whose distribution were determined by a stochastic leave one out cross validation loocv estimation a careful operation count is provided in order to determine the ranges of all the parameters in our methods the overhead cost for setting up the proposed dsrbf method is o n 2 for function recovery problems with n basis numerical experiments confirm that the proposed method not only outperforms constant shape parameter formulation in terms of accuracy with comparable computational cost but also the optimal loocv formulation in terms of both accuracy and computational cost

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

a cell centred finite volume method for the poisson problem on non graded quadtrees with second order accurate gradients this paper introduces a two dimensional cell centred finite volume discretization of the poisson problem on adaptive cartesian quadtree grids which exhibits second order accuracy in both the solution and its gradients and requires no grading condition between adjacent cells at t junction configurations which occur wherever resolution differs between neighboring cells use of the standard centred difference gradient stencil requires that ghost values be constructed by interpolation to properly recover second order accuracy in the resulting numerical gradients prior work addressing block structured grids and graded trees has shown that quadratic rather than linear interpolation is required the gradients otherwise exhibit only first order convergence which limits potential applications such as fluid flow however previous schemes fail or lose accuracy in the presence of the more complex t junction geometries arising in the case of general non graded quadtrees which place no restrictions on the resolution of neighboring cells we therefore propose novel quadratic interpolant constructions for this case that enable second order convergence by relying on stencils oriented diagonally and applied recursively as needed the method handles complex tree topologies and large resolution jumps between neighboring cells even along the domain boundary and both dirichlet and neumann boundary conditions are supported numerical experiments confirm the overall second order accuracy of the method in the l norm

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

optimal constant shape parameter for multiquadric based rbf fd method radial basis functions rbfs have become a popular method for interpolation and solution of partial differential equations pdes many types of rbfs used in these problems contain a shape parameter and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter in this paper we focus on pde problems solved with a multiquadric based rbf finite difference rbf fd method we propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error the algorithm is based on analytical approximations to the local rbf fd error derived in 1 we show through several examples in 1d and 2d both with structured and unstructured nodes that very accurate solutions compared to finite differences can be achieved using the optimal value of the constant shape parameter

consistency conditions for the influence graphs generalized finite difference method the modeling of physical phenomena is most often conveyed through partial differential equations governing continuous fields computational approaches are used afterwards in order to approximate the solutions to these equations in doing so a discretized version of the initial model must be set up using one of the numerous available methods this paper illustrates how influence graphs weighted digraphs with geometrical attributes may behave in the same way as differential operators in so far as a set of appropriate conditions of consistency are satisfied these conditions are written explicitly for the most often used cases including differential boundary operators on the one hand a consistent neighborhood may be regarded as a generalization of a classical finite difference scheme on the other hand a consistent influence graph is a discrete object modeling a physical behavior as well as its continuous counterpart numerical experimentations address firstly the second order diffusion model and lastly the fourth order kirchhoff theory of plate deflection

meshfree explicit local radial basis function collocation method for diffusion problems this paper formulates a simple explicit local version of the classical meshless radial basis function collocation kansa method the formulation copes with the diffusion equation applicable in the solution of a broad spectrum of scientific and engineering problems the method is structured on multiquadrics radial basis functions instead of global the collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node the computational effort thus grows roughly linearly with the number of the nodes the developed approach thus overcomes the principal large scale problem bottleneck of the original kansa method two test cases are elaborated the first is the boundary value problem nafems test associated with the steady temperature field with simultaneous involvement of the dirichlet neumann and robin boundary conditions on a rectangle the second is the initial value problem associated with the dirichlet jump problem on a square the accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes number of nodes in the domain of influence multiquadrics free parameter and timestep length on uniform and nonuniform node arrangements the developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the dirichlet jump where the approximation properties appear similar

the convergence of nystrom methods for wiener hopf equations we consider second kind integral equations on the half line where the integral operator is a compact perturbation of a convolution operator it is shown that these may be solved numerically by nystrom methods based on composite quadrature rules provided the underlying mesh is graded to correctly match the behaviour of the solution we prove the same rates of convergence that occur when the methods are applied to equations on finite intervals numerical examples are given

a localized approach for the method of approximate particular solutions the method of approximate particular solutions maps has been recently developed to solve various types of partial differential equations in the maps radial basis functions play an important role in approximating the forcing term coupled with the concept of particular solutions and radial basis functions a simple and effective numerical method for solving a large class of partial differential equations can be achieved one of the difficulties of globally applying maps is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points thereby affecting our ability to solve large scale science and engineering problems in this paper we develop a localized scheme for the method of approximate particular solutions lmaps the new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation in this paper this local numerical scheme is used for solving large scale problems up to one million interpolation points three numerical examples in two dimensions are used to validate the proposed numerical scheme

a new hybrid boundary node method based on taylor expansion and the shepard interpolation method summary a novel meshless method based on the shepard and taylor interpolation method stim and the hybrid boundary node method hbnm is proposed based on the shepard interpolation method and taylor expansion the stim is developed to construct the shape function of the hbnm in the stim the shepard shape function is used as the basic function which is the zero level shape function and the high power basic functions are constructed through taylor expansion four advantages of the stim are the interpolation property the arbitrarily high order consistency the absence of inversion for the whole process of shape function construction and the low computational expense these properties are desirable in the implementation of meshless methods by combining the stim and the hbnm a much more effective meshless method is proposed to solve the elasticity problems compared with the traditional hbnm the stim can improve accuracy because of the use of high power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process numerical examples are given to demonstrate the accuracy and efficiency of the proposed method copyright 2015 john wiley sons ltd

minimal numerical differentiation formulas we investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector error bounds are given in terms of a growth function that carries the information about the geometry of the centers specific forms of weighted ell 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments the results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods

circulant matrix and conformal mapping for solving partial differential equations in recent years a fast radial basis function rbf solver for surface interpolation has been developed by karageorghis et al 2007 in this paper we follow up their work by extending the idea of fast evaluation of the surface interpolation to efficiently solve various types of partial differential equations pdes we look into the possibility of solving pdes with conformal mapping to map the rbf circular points on a disk to the interior of an irregular domain we also compare the pros and cons for these two approaches in both approaches the circulant matrix formulation and the matrix decomposition method are the central ideas of fast computation in numerical experiments we test these two approaches for solving pdes using up to 160000 rbf interpolation points with greater accuracy and efficiency

a meshfree local rbf collocation method for anti plane transverse elastic wave propagation analysis in 2d phononic crystals in this paper a meshfree or meshless local radial basis function rbf collocation method is proposed to calculate the band structures of two dimensional 2d anti plane transverse elastic waves in phononic crystals three new techniques are developed for calculating the normal derivative of the field quantity required by the treatment of the boundary conditions which improve the stability of the local rbf collocation method significantly the general form of the local rbf collocation method for a unit cell with periodic boundary conditions is proposed where the continuity conditions on the interface between the matrix and the scatterer are taken into account the band structures or dispersion relations can be obtained by solving the eigenvalue problem and sweeping the boundary of the irreducible first brillouin zone the proposed local rbf collocation method is verified by using the corresponding results obtained with the finite element method for different acoustic impedance ratios various scatterer shapes scatterer arrangements lattice forms and material properties numerical examples are presented and discussed to show the performance and the efficiency of the developed local rbf collocation method compared to the fem for computing the band structures of 2d phononic crystals

meshfree numerical approach based on the finite pointset method for static linear elasticity problems abstract in this work a promising meshfree numerical approach is extended and implemented for the first time in the field of linear static elasticity a real meshfree method the so called finite pointset method fpm is applied and implemented in order to solve the strong classical form of the governing partial differential equations pdes the well known navier cauchy equations fpm numerical performance is demonstrated for several classical and well known studied benchmark problems and comparison with other approaches is also reported numerical results suggest that fpm method is stable with good convergence behavior for 2d linear static elasticity problems

decision manifolds a supervised learning algorithm based on self organization in this paper we present a neural classifier algorithm that locally approximates the decision surface of labeled data by a patchwork of separating hyperplanes which are arranged under certain topological constraints similar to those of self organizing maps soms we take advantage of the fact that these boundaries can often be represented by linear ones connected by a low dimensional nonlinear manifold thus influencing the placement of the separators the resulting classifier allows for a voting scheme that averages over the classification results of neighboring hyper planes our algorithm is computationally efficient both in terms of training and classification further we present a model selection method to estimate the topology of the classification boundary we demonstrate the algorithm s usefulness on several artificial and real world data sets and compare it to the state of the art supervised learning algorithms

an element free smoothed radial point interpolation method efs rpim for 2d and 3d solid mechanics problems abstract this paper presents a novel element free smoothed radial point interpolation method efs rpim for solving 2d and 3d solid mechanics problems the idea of the present technique is that field nodes and smoothing cells scs used for smoothing operations are created independently and without using a background grid which saves tedious mesh generation efforts and makes the pre process more flexible in the formulation we use the generalized smoothed galerkin gs galerkin weak form that requires only discrete values of shape functions that can be created using the rpim by varying the amount of nodes and scs as well as their ratio the accuracy can be improved and upper bound or lower bound solutions can be obtained by design the scs can be of regular or irregular polygons in this work we tested triangular quadrangle n sided polygon and tetrahedron as examples stability condition is examined and some criteria are found to avoid the presence of spurious zero energy modes this paper is the first time to create gs galerkin weak form models without using a background mesh that tied with nodes and hence the efs rpim is a true meshfree approach the proposed efs rpim is so far the only technique that can offer both upper and lower bound solutions numerical results show that the efs rpim gives accurate results and desirable convergence rate when comparing with the standard finite element method fem and the cell based smoothed fem cs fem