fractional derivative for interpolation in rn and so n applications in functionally graded materials and rigid body transformations abstract an integration of fractional calculus data interpolation and lie group leads to new treatments of interpolation problems a local and centered fractional derivative is introduced which enables the fractional interpolant to create a c smooth r curve by extending integer order cost functions to fractional ones the fractional interpolant fills the gaps between the traditional linear interpolating segments cubic splines and higher order splines interpolating rigid body transformations has enormous applications in mechanics computer graphics and robotics previous approaches have provided the re parametrization methods the recursive methods e g de casteljau s algorithm and the gradient methods to approximate the correct interpolating curve on so 3 a closed form model that merges the tangent spaces of knots is proposed the model first constructs a family of candidate curves and then combines them into an improved approximation applications in modeling functionally graded materials and rigid body transformations are discussed

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

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

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

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

computationally efficient spatial interpolators based on spartan spatial random fields this paper addresses the spatial interpolation of scattered data in d dimensions the problem is approached using the theory of spartan spatial random fields ssrfs focusing on a specific gaussian ssrf i e the fluctuation gradient curvature fgc model a family of spatial interpolators predictors is formulated by maximizing the fgc ssrf probability density function at each prediction point conditioned by the data an analytical expression for the general uniform bandwidth spartan gubs predictor is derived the linear weights of this predictor involve weighted summations of kernel functions over the sample and prediction points approximations for the sums are obtained at the asymptotic limit of a dense sampling network leading to simplified explicit expressions of the weights an asymptotic locally adaptive spartan alas predictor is defined by means of a kernel family that involves a tunable local parameter the relevant equations are fully developed in d 2 using simulated data in two dimensions we show that the alas prediction accuracy is comparable to that of ordinary kriging ok which is an optimal spatial linear predictor solp the numerical complexity of the alas predictor increases linearly with the sample size in contrast with the cubic dependence of ok for large data sets the alas predictor is shown to be orders of magnitude faster than ok at the cost of a slightly higher prediction dispersion the performance of the alas predictor and ok are compared on a data set of rainfall measurements using cross validation measures

general variational approach to the interpolation problem abstract the talmi and gilat variational approach to the interpolation problem in arbitrary dimension is presented together with the corresponding physical model the connection of this approach to some known spline methods is demonstrated and new interpolation functions are derived for one two and three dimensional cases they are designed to be flexible through the use of meaningful parameters and to give good approximations of both the function itself and its derivatives as well

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

on the mathematical theory of the aharonov bohm effect we consider the schr dinger operator h i a 2 in the space l2 2 with a magnetic potential a x a x2 x1 x 2 where a is an arbitrary function on the unit circle our goal is to study spectral properties of the corresponding scattering matrix s 0 we obtain its stationary representation and show that its singular part up to compact terms is a pseudodifferential operator of zero order whose symbol is an explicit function of a we deduce from this result that the essential spectrum of s does not depend on and consists of two complex conjugated and perhaps overlapping closed intervals of the unit circle finally we calculate the diagonal singularity of the scattering amplitude kernel of s considered as an integral operator in particular we show that for all these properties only the behaviour of a potential at infinity is essential the preceding papers on this subject treated the case a const and used the separation of variables in the schr dinger equation in the polar coordinates this technique does not of course work for arbitrary a from an analytical point of view our paper relies on some modern tools of scattering theory and well known properties of pseudodifferential operators

schwarz method for earthquake source dynamics dynamic faulting under slip dependent friction in a linear elastic domain in plane and 3d configurations is considered the use of an implicit time stepping scheme newmark method allows much larger values of the time step than the critical cfl time step and higher accuracy to handle the non smoothness of the interface constitutive law slip weakening friction the finite element form of the quasi variational inequality is solved by a schwarz domain decomposition method by separating the inner nodes of the domain from the nodes on the fault in this way the quasi variational inequality splits into two subproblems the first one is a large linear system of equations and its unknowns are related to the mesh nodes of the first subdomain i e lying inside the domain the unknowns of the second subproblem are the degrees of freedom of the mesh nodes of the second subdomain i e lying on the domain boundary where the conditions of contact and friction are imposed this nonlinear subproblem is solved by the same schwarz algorithm leading to some local nonlinear subproblems of a very small size numerical experiments are performed to illustrate convergence in time and space instability capturing energy dissipation and the influence of normal stress variations we have used the proposed numerical method to compute source dynamics phenomena on complex and realistic 2d fault models branched fault systems

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

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

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

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

critical friction for wedged configurations abstract a wedged configuration with coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads we analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3 d context the critical friction coefficient w is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses for friction coefficients with w the wedged problem has at least a solution and for the wedged problem is stated in a discrete framework using a mixed finite element approach and the discrete critical friction coefficient h w is introduced as the solution of a global minimization problem involving a non differentiable and non convex functional the existence of h the displacement field of a critical wedged state is proved and a specific numerical method based on a genetic algorithm was developed to compute the critical wedged configurations some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations

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

besov and triebel lizorkin regularity for the hodge decomposition and applications to magnetic potentials abstract the hodge decomposition is a fundamental result in the theory of differential forms it has been extensively studied from different perspectives and it is the source of several applications in physics and mathematics for example it is important in the analysis of navier stokes and maxwell s equations it can be used to prove results on inverse scattering problems in quantum mechanics and it is useful for the study of magnetic potentials associated to a magnetic field the regularity properties of the hodge decomposition have been studied in several works in the last decades most prominently it has been proved that sobolev regularity holds true in this paper we prove regularity for the hodge decomposition in a much larger scale of function spaces besov and triebel lizorkin spaces on compact riemannian manifolds with boundary we prove regularity in such spaces for the hodge morrey decomposition and the hodge morrey friedrichs decomposition we furthermore state and prove a refinement for the hodge morrey friedrichs decomposition with regularity that allows expressing every differential form as the sum of the differential of a form plus the co differential of another form plus a harmonic field belonging to a finite dimensional space this holds true even if the original form is barely differentiable with sth order of differentiability for every real number s 0 we finally provide an application to the study of magnetic potentials associated to magnetic fields more precisely we prove the existence of a magnetic potential associated to a magnetic field the former being one time more differentiable than the latter in terms of besov and triebel lizorkin regularity this is an important motivation for us since we have the intention to apply our results to further investigations in relativistic inverse time dependent scattering problems in quantum mechanics where besov and triebel lizorkin regularity for magnetic potentials is relevant

unconditionally stable higher order newmark methods by sub stepping procedure abstract in this paper a procedure to construct unconditionally stable higher order step by step time integration algorithms based on the newmark method is presented the second order accurate newmark method is extended to third fourth and even higher order by linearly combining the numerical results evaluated at some specific sub step locations the weighting factors and the sub step locations are algorithmic parameters they are chosen to eliminate the leading truncation error terms and to control the stability characteristics the ranges of parameters for unconditionally stable third and fourth order algorithms are given besides by using complex time steps a third order l stable and a fourth order a stable non dissipative algorithms are derived furthermore the accuracy of the excitation responses is shown to be improved by the present procedure as well to maintain high order accuracy the excitation may need to be modified the computational procedure is discussed it is found that there are no changes in the implementation of the underlying newmark method and no extra computations of high order gradients as the numerical results at the sub step locations are obtained independent of one another the evaluations can be computed in parallel comparisons with other higher order algorithms are presented

warping digital images using thin plate splines this note is to direct attention brack to thin plate spline because evaluation algorithms which became generaly available early in 1992 have dramatically reduced their computational complexity thin plate splines make very effective warping functions but special attention must be paid when calculating their coefficients and when evaluating them many times

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

a continuum based finite element formulation for the implicit solution of multibody large deformation frictional contact problems in this paper a formulation is presented for the finite element treatment of multibody large deformation frictional contact problems the term multibody is used to mean that when two bodies mechanically contact both may be deformable a novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first before deriving the corresponding finite element equations this feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general in particular the approach yields a characterization of the frictional constraint assuming a coulomb law suitable for arbitrary discretizations in either two or three dimensions a geometric framework is constructed within which both frictionless and frictional response are naturally described making subsequent finite element discretization a straightforward substitution of finite dimensional solution spaces for their continuum counterparts to our knowledge this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature the development includes exact linearization of the statement of virtual work which enables optimal convergence properties for newton raphson solution strategies and which appears to be highly desirable if not essential for the general robustness of implicit finite element techniques since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non linear behaviour this assertion is substantiated through presentation of a variety of examples in both two and three dimensions

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

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

scattered and track data interpolation using an efficient strip searching procedure abstract a new local algorithm for bivariate interpolation of large sets of scattered and track data is presented the method which changes partially depending on the kind of data is based on the partition of the interpolation domain in a suitable number of parallel strips and starting from these on the construction for any data point of a square neighbourhood containing a convenient number of data points then the well known modified shepard s formula for surface interpolation is applied with some effective improvements the proposed algorithm is very fast owing to the optimal nearest neighbour searching and achieves good accuracy computational cost and storage requirements are analyzed moreover the efficiency and reliability of the algorithm are shown by several numerical tests also performed by renka s algorithm for a comparison

image surface approximation with irregular samples a method to approximate image surfaces using irregular samples is proposed experimental results show that a high compression ratio can be achieved for simple or complicated images a method accounting for the three dimensional effects is proposed as a way of selecting the irregular samples from the image approximation spline knots are chosen along the contours and the knot replacement is sensitive to the curvature along contours and the density of contours in the near neighborhood in general the proposed method was shown to reduce the reconstruction error both qualitatively and quantitatively and can be used to model the image surface for the purpose of data compression and representation this approach retains key features in images and throws away redundant data

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

domain decomposition method for dynamic faulting under slip dependent friction the anti plane shearing problem on a system of finite faults under a slip dependent friction in a linear elastic domain is considered using a newmark method for the time discretization of the problem we have obtained an elliptic variational inequality at each time step an upper bound for the time step size which is not a cfl condition is deduced from the solution uniqueness criterion using the first eigenvalue of the tangent problem finite element form of the variational inequality is solved by a schwarz method assuming that the inner nodes of the domain lie in one subdomain and the nodes on the fault lie in other subdomains two decompositions of the domain are analyzed one made up of two subdomains and another one with three subdomains numerical experiments are performed to illustrate convergence for a single time step convergence of the schwarz algorithm influence of the mesh size influence of the time step convergence in time instability capturing energy dissipation optimal time step and an application to a relevant physical problem interacting parallel fault segments

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

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

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

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

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