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

a mesh free numerical method for solution of the family of kuramoto sivashinsky equations in this paper radial basis function rbfs based mesh free method is implemented to find numerical solution of the kuramoto sivashinsky equations this approach has an edge over traditional methods such as finite difference and finite element methods because it does not require a mesh to discretize the problem domain and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method the accuracy of the method is assessed in terms of the error norms l 2 l number of nodes in the domain of influence free parameter dependent parameter rbfs and time step length numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations

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

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

a sufficient condition for the convergence of the inexact uzawa algorithm for saddle point problems in this paper the convergence property of the inexact uzawa algorithm for solving symmetric indefinite linear systems is studied a simple sufficient condition for the convergence of the inexact uzawa algorithm is obtained two examples and numerical experiments illustrating the conclusion are provided

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

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

multilevel sparse kernel based interpolation a multilevel kernel based interpolation method suitable for moderately high dimensional function interpolation problems is proposed the method termed multilevel sparse kernel based interpolation mlski for short uses both levelwise and directionwise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel based interpolants with different scaling in each direction the multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level on each level anisotropic radial basis functions are used for solving a number of small interpolation problems which are subsequently linearly combined to produce the interpolant mlski can be viewed as an extension of d boolean interpolation which is closely related to ideas in sparse grid and hyperbolic crosses literature to kernel ba

numerical solution of poisson s equation using radial basis function networks on the polar coordinate this paper introduces a variant of direct and indirect radial basis function networks drbfns and irbfns for the numerical solution of poisson s equation we use transformation from cartesian coordinates to polar ones and use drbfn and irbfn methods on the basis of a multiquadric approximation scheme we have experienced that the result shows better accuracy than previously known ones also our new way of solution does not influence the condition number

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

explicit and implicit meshless methods for linear advection diffusion type partial differential equations simple mesh grid free explicit and implicit numerical schemes for the solution of linear advection diffusion problems is developed and validated herein unlike the mesh or grid based methods these schemes use well distributed quasi random points and approximate the solution using global radial basis functions the schemes can be seen as generalized finite differences with random points instead of a regular grid system this allows the computation of problems with complex shaped boundaries in higher dimensions with no need for complex mesh grid structure and with no extra implementation difficulties

interpolation and approximation of 3 d and 4 d scattered data abstract given a set of 3 d or 4 d scattered data methods are presented that yield a bivariate or trivariate function that interpolate or approximate the given data the subroutine package includes hardy s multiquadric interpolant and multistage methods with many options available to the user thus several different interpolation and approximation methods can be generated the options available in the multistage methods can be effectively used if the data set is noisy rapidly varying or nonuniformly distributed the computational and storage requirements for the multistage methods are linear in the number of data points thus they are efficient on large data sets

the map and blend scattered data interpolant on a sphere abstract the map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere this technique consists of two different mappings of the sphere to planar domains solving two corresponding scattered data interpolation problems on the planar domains and then the interpolant on the sphere is formed by blending these two planar interpolants if the user has software for solving the scattered data interpolation problem on a planar domain only a small programming effort is needed to implement the map and blend interpolant on the sphere although any interpolant to scattered data on a planar domain can be used in our general technique we use the multiquadric radial basis method

a new approach for shape preserving interpolating curves in this paper we present an new approach to construct the so called shape preserving interpolation curves the basic idea is first to approximate the set of interpolated points with a class of mq quasi interpolation operators and then pass through the set with the use of multivariate interpolation by using compactly supported radial basis functions this approach possesses the advantages of certain shape preserving and good approximation behaviors the proposed algorithm is easy to implement

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

an interval for the shape parameter in radial basis function approximation the radial basis functions rbfs depend on an auxiliary parameter called the shape parameter great theoretical and numerical efforts have been made to find the relationship between the accuracy of the rbf approximations and the value of the shape parameter in many cases the numerical approaches are based on minimization of an estimation of the error function such as rippa s approach and its modifications these approaches determine a value for the shape parameter in this paper we propose a practical approach to determine an interval instead of a value without any minimization and estimation of error function the idea is based on adding a loop on the shape parameter but not to minimize an error norm suitable values of the shape parameter are determined by take into account the practical convergence behavior of the problem the proposed method is applied to some illustrative examples including one and two dimensional interpolation and partial differential equations the results of the method are compared with those of some other approaches to confirm the reliability of the method furthermore numerical stability of the method of line respect to the shape parameter is considered for time dependent pdes the results show that the proposed method can be used as an independent method to find a shape parameter and also as an alternative for investigating the validity of the values of the other methods

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

geometrical theory of diffraction and spectral statistics we investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit by applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit 0 this result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space

a comparative study of numerical solutions of a class of kdv equation abstract in this work we present a comparative study of meshless method modified bernstein polynomials bp and b spline finite element method bs fem for the numerical solution of two different models of korteweg de vries kdv equation the multiquadric mq and gaussian ga radial basis functions rbfs are used due to exponential convergence rate excellent agreement is found between exact and rbfs solutions and same accuracy is obtained as bernstein polynomials and b spline finite element method

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

smooth interpolation of scattered data by local thin plate splines abstract an algorithm and the corresponding computer program for solution of the scattered data interpolation problem is described given points x k y k f k k 1 n a locally defined function f x y which has the property f x k y k f k k 1 n is constructed the algorithm is based on a weighted sum of locally defined thin plate splines and yields an interpolation function which is differentiable the program is available from the author

on the use of boundary conditions for variational formulations arising in financial mathematics the general intractability of derivative security valuation models to present techniques both analytic and numerical arguably remains one of the preeminant problem of mathematical finance it is the focus of this paper to examine the applicability of a promising recent development namely radial basis functions rbf to the problem of option valuation a black scholes framework is considered for american and european options written on a one and two risky assets the performance of rbf and finite differencing algorithms are examined with respect to artificial boundary conditions computational domain domain decomposition and mesh scaling

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

evolving space filling curves to distribute radial basis functions over an input space an evolutionary neural network training algorithm is proposed for radial basis function rbf networks the locations of basis function centers are not directly encoded in a genetic string but are governed by space filling curves whose parameters evolve genetically this encoding causes each group of codetermined basis functions to evolve to fit a region of the input space a network produced from this encoding is evaluated by training its output connections only networks produced by this evolutionary algorithm appear to have better generalization performance on the mackey glass time series than corresponding networks whose centers are determined by k means clustering

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

a quasi interpolation method for solving stiff ordinary differential equations based on the idea of quasi interpolation and radial basis functions approximation a numerical method is developed to quasi interpolate the forcing term of differential equations by using radial basis functions a highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions this overcomes the ill conditioning problem resulting from using the radial basis functions as a global interpolant error estimation is given for a particular second order stiff differential equation with boundary layer the result of computations indicates that the method can be applied to solve very stiff problems with the use of multiquadric a special class of radial basis functions it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation copyright 2000 john wiley sons ltd

a method for interpolating scattered data based upon a minimum norm network a method for interpolating scattered data is described ctiven xi yi zi i 1 n a bivariate function s with continuous first order partial derivatives is defined which has the property that s xi yi zi i 1 n the method is based upon a triangulation of the domain and a curve network which has certain minimum pseudonorm properties algorithms and examples are included

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

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

iterative solution of systems originating from integral equations and surface interpolation a simple procedure for generating diagonal dominance in certain linear systems is presented and the iterative solution of the transformed systems is discussed the method is presented for systems encountered in the numerical solution of fredholm equations of the first kind with singular kernels and for systems arising in the interpolation of surfaces by certain sets of radial functions in the second case the method provides new sets of basis functions generalizing in some sense the univariate b splines

on the iterative algorithm for large sparse saddle point problems in this paper we discuss some iterated method for solving the saddle point problem we propose some new schemes and prove its convergence the method has weaker convergence condition than the classic uzawa method the analysis is supported by numerical experiments

a note on the iterative algorithm for large sparse saddle point problems abstract in this paper we discuss some more generalized iterative schemes for solving the large sparse saddle point problems proposed by ling and hu x f ling x z hu on the iterative algorithm for large sparse saddle point problems appl math comput 178 2006 372 379 and apply the preconditioned gmres to accelerate the convergence numerical experiments are reported that show the effectiveness of the proposed methods

a numerical solution of the nonlinear controlled duffing oscillator by radial basis functions in this research a new numerical method is applied to investigate the nonlinear controlled duffing oscillator this method is based on the radial basis functions rbfs to approximate the solution of the optimal control problem by using the collocation method we apply legendre gauss lobatto points for rbfs center nodes in order to use the numerical integration method more easily then the method of lagrange multipliers is used to obtain the optimum of the problems for this purpose different applications of rbfs are used the differential and integral expressions which arise in the dynamic systems the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients illustrative examples are included to demonstrate the validity and applicability of the technique

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

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

seismic modeling with radial basis function generated finite differences rbf fd a simplified treatment of interfaces abstract in a previous study of seismic modeling with radial basis function generated finite differences rbf fd we outlined a numerical method for solving 2 d wave equations in domains with material interfaces between different regions the method was applicable on a mesh free set of data nodes it included all information about interfaces within the weights of the stencils allowing the use of traditional time integrators and was shown to solve problems of the 2 d elastic wave equation to 3rd order accuracy in the present paper we discuss a refinement of that method that makes it simpler to implement it can also improve accuracy for the case of smoothly variable model parameter values near interfaces we give several test cases that demonstrate the method solving 2 d elastic wave equation problems to 4th order accuracy even in the presence of smoothly curved interfaces with jump discontinuities in the model parameters

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

cauchy characteristic evolution and waveforms we investigate a new methodology for the computation of waves generated by isolated sources this approach consists of a global spacetime evolution algorithm based on a cauchy initial value formulation in a bounded interior region and based on characteristic hypersurfaces in the exterior we match the two schemes at their common interface the characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate interval so that our numerical solution extends to infinity and accurately models the free space problem the matching interface need not be situated far from the sources the wavefronts may have arbitrary nonspherical geometry and strong nonlinearity may be present in both the interior and the exterior regions stability and second order convergence of the algorithms to the exact solution of the infinite domain problem are established numerically in three space dimensions the matching algorithm is compared with examples of both local and nonlocal radiation boundary conditions proposed in the literature for linear problems matching outperformed the local radiation conditions chosen for testing and was about as accurate for the same grid resolution as the exact nonlocal conditions however since the computational cost of the nonlocal conditions is many times that of matching this algorithm may be used with higher grid resolutions yielding a significantly higher final accuracy for strongly nonlinear problems matching was significantly more accurate than all other methods tested this seems to be due to the fact that currently available local and nonlocal conditions are based on linearizing the governing equations in the far field while matching consistently takes nonlinearity into account in both interior and exterior regions

numerical solution of two dimensional elliptic pdes with nonlocal boundary conditions in the present paper two numerical methods are analyzed for the solution of two dimensional poisson equation with two different types of nonlocal boundary conditions the first numerical method is a collocation method based on haar wavelet whereas the second numerical method is a meshless method based on different types of radial basis functions rbfs a two point boundary condition and an integral boundary condition are the two types of nonlocal boundary conditions considered in the present work for the collocation method based on haar wavelet a new approach is formulated which involves the approximation of a fourth order mixed derivative by a haar expansion which is integrated subsequently to get wavelet approximation of the solution for the meshless method based on rbfs the algorithm is implemented using two different splitting schemes with and without shape parameter splitting for numerical solution of the model the comparative analysis of the meshless methods with and without shape parameter splitting scheme is performed between themselves as well as with the haar wavelet accuracy and efficiency wise performance is confirmed through application of the algorithms on the benchmark tests

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

on the iterative algorithm for saddle point problems abstract in this paper we point out that the generalized iteration method presented in a recent paper by ling and hu x f ling x z hu on the iterative algorithm for large sparse saddle point problems appl math comput 178 2006 372 379 belongs to the inexact uzawa algorithm and an improper statement in a theorem in that paper is corrected

trace formula for a dielectric microdisk with a point scatterer two dimensional dielectric microcavities are of widespread use in microoptics applications recently a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the cavity in this paper we extend this trace formula to a dielectric disk with a small scatterer this system has been introduced for microlaser applications because it has long lived resonances with strongly directional far field we show that its resonance spectrum contains signatures not only of periodic rays but also of diffractive rays that occur in keller s geometrical theory of diffraction we compare our results with those for a closed cavity with dirichlet boundary conditions

a collocation meshless method based on local optimal point interpolation this paper deals with the use of the local optimal point interpolating lopi formula in solving partial differential equations pdes with a collocation method lopi is an interpolating formula constructed by localization of optimal point interpolation formulas that reproduces polynomials and verifies the delta kronecker property this scheme results in a truly meshless method that produces high quality output and accurate solutions

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

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

cooperative competitive genetic evolution of radial basis function centers and widths for time series prediction in a radial basis function rbf network the rbf centers and widths can be evolved by a cooperative competitive genetic algorithm the set of genetic strings in one generation of the algorithm represents one rep network not a population of competing networks this leads to moderate computation times for the algorithm as a whole selection operates on individual rbfs rather than on whole networks selection therefore requires a genetic fitness function that promotes competition among rbfs which are doing nearly the same job while at the same time promoting cooperation among rbfs which cover different parts of the domain of the function to be approximated niche creation resulting from a fitness function of the form w sub i sup spl beta e w sub i sup spl beta 1 spl beta 2 can facilitate the desired cooperative competitive behavior the feasibility of the resulting algorithm to evolve networks of gaussian inverse multiquadric and thin plate spline rbfs is demonstrated by predicting the mackey glass time series for each type of rbf and for networks of 25 50 75 100 125 and 150 rbf units prediction errors for the evolved gaussian rbf networks are 50 70 lower than rbf networks obtained by k means clustering

trace formula for dielectric cavities general properties the construction of the trace formula for open dielectric cavities is examined in detail using the krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity the contribution of each periodic orbit is the product of the two factors the first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived the coefficient of the perimeter term differs from the one for closed cavities due to unusual high energy asymptotics of the s matrix for the scattering on the cavity corrections to the leading semi classical formula are briefly discussed obtained formulas agree well with numerical calculations for circular dielectric cavities

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

pricing european and american options by radial basis point interpolation a rbpi method for pricing european and american options is proposed rbpi has not yet been used for option pricing the overall efficiency is improved by coupling rbpi with other approaches three different algorithms for american options are tested and compared numerical results demonstrate the effectiveness of the techniques employed we propose the use of the meshfree radial basis point interpolation rbpi to solve the black scholes model for european and american options the rbpi meshfree method offers several advantages over the more conventional radial basis function approximation nevertheless it has never been applied to option pricing at least to the very best of our knowledge in this paper the rbpi is combined with several numerical techniques namely an exponential change of variables which allows us to approximate the option prices on their whole spatial domain a mesh refinement algorithm which turns out to be very suitable for dealing with the non smooth options payoff and an implicit euler richardson extrapolated scheme which provides a satisfactory level of time accuracy finally in order to solve the free boundary problem that arises in the case of american options three different approaches are used and compared the projected successive overrelaxation method psor the bermudan approximation and the penalty approach numerical experiments are presented which demonstrate the computational efficiency of the rbpi and the effectiveness of the various techniques employed

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

a meshless integration free and boundary only rbf technique based on the radial basis function rbf nonsingular general solution and dual reciprocity method drm this paper presents an inherently meshless integration free boundary only rbf collocation technique for numerical solution of various partial differential equation systems the basic ideas behind this methodology are very mathematically simple in this study the rbfs are employed to approximate the inhomogeneous terms via the drm while nonsingular general solution leads to a boundary only rbf formulation for homogenous solution the present scheme is named as the boundary knot method bkm to differentiate it from the other numerical techniques in particular due to the use of nonsingular general solutions rather than singular fundamental solutions the bkm is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions the efficiency and utility of this new technique are validated through a number of typical numerical examples completeness concern of the bkm due to the sole use of the nonsingular part of complete fundamental solution is also discussed

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

improved accuracy of multiquadric interpolation using variable shape parameters given n scattered data points we examined the problem of finding n variable multiquadric mq shape parameters or r2 values because the problem of finding the optimal r2 values is a nonlinear one we optimized these parameters numerically by minimizing the root mean square rms errors the resulting r2 values varied over many orders of magnitude we have tested this approach on a number of univariate and bivariate franke s problems and found that the rms error reduction was substantial

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

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

point perturbations of circle billiards the spectral statistics of the circular billiard with a point scatterer is investigated in the semiclassical limit the spectrum is demonstrated to be composed of two uncorrelated level sequences the first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point scatterer the nearest neighbour spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

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

bifurcations of periodic orbits and uniform approximations we derive uniform approximations for contributions to gutzwiller s periodic orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space there orbits lie close together and give collective contributions while the individual contributions of gutzwiller s type would diverge at the bifurcation new results for the tangent the period doubling and the period tripling bifurcation are given they are obtained by going beyond the local approximation and including higher order terms in the normal form of the action the uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results

on the optimal shape parameters of radial basis functions used for 2 d meshless methods abstract a radial point interpolation meshless or radial pim method was proposed by authors to overcome the possible singularity associated with only polynomial basis the radial pim used multiquadric mq or gaussian as basis functions these two radial basis functions all included shape parameters although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed how these shape parameters affect the accuracy of the radial pim has not been studied yet this paper studied the effect of shape parameters on the numerical accuracy of radial pim a range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix error of energy and irregularity of node distribution it is observed that the widely used shape parameters for mq and reciprocal mq basis are not even close to their optimums the optimal shape parameters are found in this paper to be simply q 1 03 and r 1 42 for mq basis and c 0 003 0 03 for gaussian basis

a numerical study of compact approximations based on flat integrated radial basis functions for second order differential equations abstract in this paper we propose a simple but effective preconditioning technique to improve the numerical stability of integrated radial basis function irbf methods the proposed preconditioner is simply the inverse of a well conditioned matrix that is constructed using non flat irbfs much larger values of the free shape parameter of irbfs can thus be employed and better accuracy for smooth solution problems can be achieved furthermore to improve the accuracy of local irbf methods we propose a new stencil namely combined compact irbf ccirbf in which i the starting point is the fourth order derivative and ii nodal values of first and second order derivatives at side nodes of the stencil are included in the computation of first and second order derivatives at the middle node in a natural way the proposed stencil can be employed in uniform nonuniform cartesian grids the preconditioning technique in combination with the ccirbf scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by poisson burgers convection diffusion and navier stokes equations highly accurate and stable solutions are obtained in some cases the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning

application of a trace formula to the spectra of flat three dimensional dielectric resonators the length spectra of flat three dimensional dielectric resonators of circular shape were determined from a microwave experiment they were compared to a semiclassical trace formula obtained within a two dimensional model based on the effective index of refraction approximation and a good agreement was found it was necessary to take into account the dispersion of the effective index of refraction for the two dimensional approximation furthermore small deviations between the experimental length spectrum and the trace formula prediction were attributed to the systematic error of the effective index of refraction approximation in summary the methods developed in this article enable the application of the trace formula for two dimensional dielectric resonators also to realistic flat three dimensional dielectric microcavities and lasers allowing for the interpretation of their spectra in terms of classical periodic orbits

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

a meshfree method for the numerical solution of the rlw equation in this paper we present a meshfree technique for the numerical solution of the regularized long wave rlw equation this approach is based on a global collocation method using the radial basis functions rbfs different kinds of rbfs are used for this purpose accuracy of the new method is tested in terms of l 2 and l error norms in case of non availability of the exact solution performance of the new method is compared with existing methods stability analysis of the method is established propagation of single and double solitary waves wave undulation and conservation properties of mass energy and momentum of the rlw equation are discussed