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

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

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

the effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking stretching sheet the effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking stretching sheet are analyzed the similarity transformation reduces the time independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations the resulting five parameter problem is solved by the homotopy perturbation method the results are presented graphically to interpret various physical parameters appearing in the problem

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

numerical experiments with one dimensional adaptive cubic algorithm abstract a code and some numerical experiments with a one dimensional cubic algorithm are presented it is demonstrated that the algorithm is applicable for full global optimization of a large class of functions including discontinuous and unbounded functions experiments with such functions show that successive runs yield monotonically improving results which descend onto the set of all global optimizers if the sequence of experimental runs is properly organized

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

moments in infinite channel queues we consider a new approach to the problem of evaluating the moments of the steady state number of customers in system b for an infinite channel queue having general input our new approach uses the rate conservation law as a consequence we show that for a variety of g gi queues var b as long as service times have finite first moment in addition we give some ordering relations for the variance of b when comparing two alternative systems with convex ordered inter arrival time distributions

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

combinatorial resampling particle filter an effective and efficient method for articulated object tracking particle filter pf is a method dedicated to posterior density estimations using weighted samples whose elements are called particles in particular this approach can be applied to object tracking in video sequences in complex situations and in this paper we focus on articulated object tracking i e objects that can be decomposed as a set of subparts one of pf s crucial step is a resampling step in which particles are resampled to avoid degeneracy problems in this paper we propose to exploit mathematical properties of articulated objects to swap conditionally independent subparts of the particles in order to generate new particle sets we then introduce a new resampling method called combinatorial resampling that resamples over the particle set resulting from all the admissible swappings the so called combinatorial set in essence combinatorial resampling cr is quite similar to the combination of a crossover operator and a usual resampling but there exists a fundamental difference between cr and the use of crossover operators we prove that cr is sound i e in a bayesian framework it is guaranteed to represent without any bias the posterior densities of the states over time by construction the particle sets produced by cr better represent the density to estimate over the whole state space than the original set and therefore cr produces higher quality samples unfortunately the combinatorial set is generally of an exponential size and therefore to be scalable we show how it can be implicitly constructed and resampled from thus resulting in both an efficient and effective resampling scheme finally through experimentations both on challenging synthetic and real video sequences we also show that our resampling method outperforms all classical resampling methods both in terms of the quality of its results and in terms of computation times

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

approximate bayesian multibody tracking visual tracking of multiple targets is a challenging problem especially when efficiency is an issue occlusions if not properly handled are a major source of failure solutions supporting principled occlusion reasoning have been proposed but are yet unpractical for online applications this paper presents a new solution which effectively manages the trade off between reliable modeling and computational efficiency the hybrid joint separable hjs filter is derived from a joint bayesian formulation of the problem and shown to be efficient while optimal in terms of compact belief representation computational efficiency is achieved by employing a markov random field approximation to joint dynamics and an incremental algorithm for posterior update with an appearance likelihood that implements a physically based model of the occlusion process a particle filter implementation is proposed which achieves accurate tracking during partial occlusions while in cases of complete occlusion tracking hypotheses are bound to estimated occlusion volumes experiments show that the proposed algorithm is efficient robust and able to resolve long term occlusions between targets with identical appearance

optimization of implicitly defined functions gradient methods for local optimization of implictly defined functions are presented the methods are structurally unstable under data perturbations that create null points of fy f y where f x y 0 defines y x then a set filtration method for global optimization of implictly defined functions under constraints is proposed which is structurally stable under data perturbations example are considered illustrating theory and numerical algorithms

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

c ondensation conditional density propagation forvisual tracking the problem of tracking curves in dense visual clutter is challenging kalman filtering is inadequate because it is based on gaussian densities which being unimo dal cannot represent simultaneous alternative hypotheses the condensation algorithm uses factored sampling previously applied to the interpretation of static images in which the probability distribution of possible interpretations is represented by a randomly generated set condensation uses learned dynamical models together with visual observations to propagate the random set over time the result is highly robust tracking of agile motion notwithstanding the use of stochastic methods the algorithm runs in near real time

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

numerical solution of nonlinear volterra fredholm hammerstein integral equations via tau collocation method with convergence analysis in this paper we consider the nonlinear volterra fredholm hammerstein integral equations the approximate solution for the nonlinear volterra fredholm hammerstein integral equations is obtained by using the tau collocation method to do this the nonlinear volterra fredholm hammerstein integral equations is transformed into a system of nonlinear algebraic equations in matrix form thus by solving this system unknown coefficients are obtained the spectral rate of convergence for the proposed method is established in the l 2 norm the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the tau collocation method is of high accuracy more convenient and efficient for solving nonlinear volterra fredholm hammerstein integral equations

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

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

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

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

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

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

a probabilistic exclusion principle for tracking multiple objects tracking multiple targets is a challenging problem especially when the targets are identical in the sense that the same model is used to describe each target in this case simply instantiating several independent 1 body trackers is not an adequate solution because the independent trackers tend to coalesce onto the best fitting target this paper presents an observation density for tracking which solves this problem by exhibiting a probabilistic exclusion principle exclusion arises naturally from a systematic derivation of the observation density without relying on heuristics another important contribution of the paper is the presentation of partitioned sampling a new sampling method for multiple object tracking partitioned sampling avoids the high computational load associated with fully coupled trackers while retaining the desirable properties of coupling

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

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

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

kernel based object tracking a new approach toward target representation and localization the central component in visual tracking of nonrigid objects is proposed the feature histogram based target representations are regularized by spatial masking with an isotropic kernel the masking induces spatially smooth similarity functions suitable for gradient based optimization hence the target localization problem can be formulated using the basin of attraction of the local maxima we employ a metric derived from the bhattacharyya coefficient as similarity measure and use the mean shift procedure to perform the optimization in the presented tracking examples the new method successfully coped with camera motion partial occlusions clutter and target scale variations integration with motion filters and data association techniques is also discussed we describe only a few of the potential applications exploitation of background information kalman tracking using motion models and face tracking

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

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

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

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

view invariant action recognition based on artificial neural networks in this paper a novel view invariant action recognition method based on neural network representation and recognition is proposed the novel representation of action videos is based on learning spatially related human body posture prototypes using self organizing maps fuzzy distances from human body posture prototypes are used to produce a time invariant action representation multilayer perceptrons are used for action classification the algorithm is trained using data from a multi camera setup an arbitrary number of cameras can be used in order to recognize actions using a bayesian framework the proposed method can also be applied to videos depicting interactions between humans without any modification the use of information captured from different viewing angles leads to high classification performance the proposed method is the first one that has been tested in challenging experimental setups a fact that denotes its effectiveness to deal with most of the open issues in action recognition

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

stochastic global optimization using tangent minorants for lipschitz functions abstract this paper deals with global stochastic optimization where the decision variable belongs to a compact subset x of r the objective function is the mathematical expectation of a partial bivariate lipschitz function f x depending on a decision variable x and a random variable whose probability distribution depends on x in the first part of the present paper we propose a branch and bound algorithm based on tangent minorants that provides a global minimum in the second part we consider the case where the function f is discontinuous we show the manner we correct that discontinuity without modifying the global minimum of e f x we also illustrate how to extend this framework to multidimensional stochastic optimization problems by using the alienor method then we validate the proposed method by applying it to some test functions and compare it to known algorithms

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

the beta algorithm abstract a set monotonic non gradient algorithm is proposed for finding global minima of general non convex mathematical programming problems the algorithm is based on the cubic algorithm e a galperin j math anal appl m 112 1985 635 640 equipped with a semi certain distinction operator and the marginal comparison constant generator

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

the combined laplace transform adomian decomposition method for handling nonlinear volterra integro differential equations in this work a combined form of the laplace transform method with the adomian decomposition method is developed for analytic treatment of the nonlinear volterra integro differential equations the combined method is capable of handling both equations of the first and second kind illustrative examples will be examined to support the proposed analysis

study of the effects of the reynolds number on circular porous slider via variational iteration algorithm ii abstract in this paper the problem of the porous slider where the fluid is injected through the porous bottom is considered the similarity transformation reduces the governing equations into coupled nonlinear ordinary differential equations the resulting equations are solved by using he s variational iteration algorithm ii the resulting series solution contains the well known reynolds number the influence of the reynolds number on the velocity field has been discussed graphically

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

bayesian object localisation in images a bayesian approach to intensity based object localisation is presented that employs a learned probabilistic model of image filter bank output applied via monte carlo methods to escape the inefficiency of exhaustive search an adequate probabilistic account of image data requires intensities both in the foreground i e over the object and in the background to be modelled some previous approaches to object localisation by monte carlo methods have used models which we claim do not fully address the issue of the statistical independence of image intensities it is addressed here by applying to each image a bank of filters whose outputs are approximately statistically independent distributions of the responses of individual filters over foreground and background are learned from training data these distributions are then used to define a joint distribution for the output of the filter bank conditioned on object configuration and this serves as an observation likelihood for use in probabilistic inference about localisation the effectiveness of probabilistic object localisation in image clutter using bayesian localisation is illustrated because it is a monte carlo method it produces not simply a single estimate of object configuration but an entire sample from the posterior distribution for the configuration this makes sequential inference of configuration possible two examples are illustrated here coarse to fine scale inference and propagation of configuration estimates over time in image sequences

stereo matching using belief propagation in this paper we formulate the stereo matching problem as a markov network and solve it using bayesian belief propagation the stereo markov network consists of three coupled markov random fields that model the following a smooth field for depth disparity a line process for depth discontinuity and a binary process for occlusion after eliminating the line process and the binary process by introducing two robust functions we apply the belief propagation algorithm to obtain the maximum a posteriori map estimation in the markov network other low level visual cues e g image segmentation can also be easily incorporated in our stereo model to obtain better stereo results experiments demonstrate that our methods are comparable to the state of the art stereo algorithms for many test cases

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

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

solution and control of pde via global optimization methods abstract based on the concept of equivalent solutions not to be confused with approximations to the exact solution a new consideration is given to ill posed 1 2 and overdetermined pde problems and to problems with nonexistent solutions 3 then a new method based on full global optimization techniques is developed for solution and control of processes described by partial differential equations the ideas are illustrated by examples and a case study is presented in comparison with the quasi reversibility method 4

two methods for solving optimization problems arising in electronic measurements and electrical engineering in this paper we introduce a common problem in electronic measurements and electrical engineering finding the first root from the left of an equation in the presence of some initial conditions we present examples of electrotechnical devices analog signal filtering where it is necessary to solve it two new methods for solving this problem based on global optimization ideas are introduced the first uses the exact a priori given global lipschitz constant for the first derivative the second method adaptively estimates local lipschitz constants during the search both algorithms either find the first root from the left or determine the global minimizers in the case when the objective function has no roots sufficient conditions for convergence of the new methods to the desired solution are established in both cases the results of numerical experiments for real problems and a set of test functions are also presented

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

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

global one dimensional optimization using smooth auxiliary functions in this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has lipschitzean first derivatives to solve this problem smooth auxiliary functions which are adaptively improved during the course of the search are constructed three new algorithms are introduced the first used the exact a priori known lipschitz constant for derivatives the second when this constant is unknown estimates it during the course of the search and finally the last method uses neither the exact global lipschitz constant nor its estimate but instead adaptively estimates the local lipschitz constants in different sectors of the search region during the course of optimization convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given 1998 the mathematical programming society inc published by elsevier science b v

multiquadric interpolation improved abstract our concern in this paper is the improvement of localization properties of multiquadric interpolation on a multivariate integer grid

the integer cubic algorithm abstract the integer and mixed cubic algorithms are proposed for full global solution of integer and mixed optimization problems the paper presents further development of the ideas in 1 4

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

the alpha algorithm and the application of the cubic algorithm in case of unknown lipschitz constant abstract a new algorithm for full global optimization of a lipschitzian function over an arbitrary bounded set in r n is presented a procedure is proposed to handle functions with unknown lip constants application to the cubic algorithm is given and illustrative examples are presented

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

pseudorandom lattices for global optimization abstract rationalization processes are proposed to improve uniformity in small samples for pseudorandom lattices in 0 1 n constructed from sequences produced by random number generators on this basis the space filtration and space contraction algorithms are developed for the solution of multimodal global optimization problems strong convergence to the global minimum value and convergence in measure onto the set of all global minimizers are proved numerical experiments are presented to illustrate a better uniformity provided by a rationalization process and the use of the space filtration algorithm for global optimization

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

global optimization in problems with uncertainties the gamma algorithm abstract problems with uncertainties can be viewed and formalized making use of multifunctions or general set valued functions a new concept of global optimality is proposed which allows us to solve global optimization problems with uncertainties in natural setting without imposing artificial constraints on uncertainties nor introducing a kind of partial ordering in order to apply conventional optimality concepts and optimization techniques nor considering solution in probability with the new concept deterministic optimization requires two optimization procedures a study of the subject is presented with many illustrative examples then a monotonic iterative algorithm is developed which renders approximate solutions with precision specified in advance a notion of piecewise continuous function of several variables is proposed and the method is then generalized for uncertain functions defined by a closed set valued function with piecewise continuous upper and lower boundaries the max2 f reduction precision and decomposition lemmas are proved to facilitate practical applications deferred deletions of sets with discontinuities are introduced and convergence theorem is proved for the modified algorithm

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

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