fast calculation of laurent expansions for matrix inverses previously described algorithms for calculating the laurent expansion of the inverse of a matrix valued analytic function become impractical already for singularity orders as low as around p 6 since they require over o 28p matrix multiplications and correspondingly large amounts of memory in place of using mathematically exact recursions we show that for floating point calculations a rational approximation approach can avoid this cost barrier without any significant loss in accuracy

stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions

holomorphic discrete series for hyperboloids of hermitian type abstract in the present paper we study the holomorphic discrete series h d s for hyperboloids of hermitian type they are the spacesg hwhereg so0 p 2 h so0 p 1 we find some complex hullsy ofg h they correspond to minimalg invariant cones in the lie algebra ofg consider the hardy spacesh2 y and give explicit expressions for the corresponding cauchy szego kernels earlier these expressions were known forp 2 gelfand gindikin andp 1 molchanov we compute the projectionse of thel2 g h onto the images of the hardy spaces the differenceq e e is most interesting it is an analogue of the hilbert transform we study the relations between the h d s forgand the h d s forg h it turns out that forp 2 there is a finite number of representations of the h d s forg hwhich are not representations of the h d s ofg we give explicit expressions for the kernels of the projections on the subspaces of the hardy spaces where the h d s forgacts

lie semialgebras are real phenomena in recent year there has been considerable interest in local semigroups in lie groups partly from the view point of geometric control theory partly from the view point of differential geometry for the former aspect see br hi js l for the latter lo o11 and 2 pa ro vi lie theory relates local subgroups of a lie group with subalgebras of its lie algebra in just the same spirit it relates certain local subsemigroups with lie semialgebras in its lie algebra here a lie semialgebra in a finite dimensional real lie algebra is a wedge i e a topologically closed subset which is stable under addition and multiplication by non negative scalars for which there is a neighborhood b of 0 in l such that for all x y b the campbellhausdorff product x y x y 89 y l is well defined and in addition stays in w for x ye w if w is a lie semialgebra in l then w w is a lie subalgebra hl 2 l if w is any wedge in l and if w contains the lie algebra w w generated by all commutators w w with w w w then w is a lie semialgebra lie semialgebras arising in this fashion we will call trivial we shall prove the following

a method for calculating meshless finite difference weights an exact method for calculating the finite difference weights for arbitrary distributed points in multiple dimensions is presented the method avoids the numerical ill conditioning associated with a small scale factor by delaying the limiting of 0 the gaussian radial basis function is approximated by a taylor s series expansion in the variable and the resultant matrix equation is solved using the fraction free lu decomposition method

analysis in space time bundles iii higher spin bundles the universal cosmos m is the unique four dimensional globally causal space time manifold to which the dirac and maxwell equations among others maximally and covariantly extend a systematic treatment is presented of general fields over m of arbitrary spin considered are fields induced from all irreducible representation of the isotropy group scale extended poincare group to g the connected causal group of m restricted to any species of such fields the k invariant canonical dirac operator k maximal essentially compact subgroup of g is shown g covariant for a unique conformal weight a normalized k finite basis for such fields is constructed the basis actions thereon of the dirac operator infinitesimal generators of g discrete symmetries second order casimir and the essentially unique third order noncentral quantum number enveloping algebra element invariant under k are derived composition series under g of a class of these field spaces namely the extension to m of the relativistic fields considered by bargman and wigner or arbitrary spin and conformal weight are determined distinguishing by invariance and causality features alone the essentially conventional positive energy mass 0 subspaces and massive invariant sub quotient spaces whose unitarity under g is given a new proof the completely positive subclass cf below of representations is determined a more detailed treatment of spin one bundles vector and two form of arbitrary conformal weight is included the exterior derivative transformations are diagonalized and the conformally invariant massive spin one scalar product is identified with a mathematical version of the conventional electromagnetic field lagrangian

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

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

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

application of radial basis function with l stable pade time marching scheme for pricing exotic option in this work a high order l stable method for pricing exotic option has been discussed the spatial discretization is done by radial basis function based local grid free method to achieve first order ordinary differential equation followed by fourth order l stable method numerical study with one and two asset problems for digital option butterfly spread and barrier option is carried out with highly accurate results that are in good agreement with those obtained by other numerical methods in the literature

essential unitarization of symplectics and applications to field quantization abstract symplectic operators satisfying generic and group invariant spectral positivity conditions are studied the theory developed is applied and illustrated to determine the unique invariant frequency decomposition equivalently linear quantization with invariant vacuum state of the klein gordon equation in non static spacetimes let h be any linear topological symplectic space such that there exists a real linear and topological isomorphism of h with some complex hilbert space carrying into the imaginary part of the scalar product then any bounded invertible symplectic s sp h resp bounded infinitesimally symplectic a sp h which satisfies sv v 0 resp av v 0 for all nonzero v h where s i is invertible is realized uniquely and constructively as a unitary resp skewadjoint operator in a complex hilbert space which depends in general on the operator and typically only densely intersects h the essentially unique weakly and uniformly closed invariant convex cones in sp h are determined extending previously known results in the finite dimensional case a notion of skew adjoint extension of a closed semi bounded infinitesimally symplectic operator is defined strictly including the usual notion of positive self adjoint extension in a complex hilbert space all such skew adjoint extensions are parametrized as in the von neumann or birman krein vishik theories finally the unique complex hilbertian structure formulated on the space of solutions of the covariant klein gordon equation in generic conformal perturbations of flat space is uniquely determined by invariance under the scattering operator the invariant hilbert structure is explicitly calculated to first order for an infinite dimensional class of purely time dependent metric perturbations and higher order contributions are rigorously estimated

multidomain spectral solution of the euler gas dynamics equations abstract we present interface treatments for computing the euler gas dynamics equations by a multidomain chebyshev spectral collocation method these treatments are suitable for use at the corners of subdomains and for patched or overlapping subdomains they can be applied to interfaces placed in subsonic supersonic or transonic regions of a flow computed results indicate that the solutions are spectrally accurate can be more accurate and more efficient than a single domain calculation and that reflections of waves at interfaces are not significant

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

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

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

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

stable multi domain spectral penalty methods for fractional partial differential equations we propose stable multi domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order first a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials the approximation order is analyzed and verified through numerical examples based on the discrete fractional derivative we introduce stable multi domain spectral penalty methods for solving fractional advection and diffusion equations the equations are discretized in each sub domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes

stable computations with flat radial basis functions using vector valued rational approximations one commonly finds in applications of smooth radial basis functions rbfs that scaling the kernels so they are flat leads to smaller discretization errors however the direct numerical approach for computing with flat rbfs rbf direct is severely ill conditioned we present an algorithm for bypassing this ill conditioning that is based on a new method for rational approximation ra of vector valued analytic functions with the property that all components of the vector share the same singularities this new algorithm rbf ra is more accurate robust and easier to implement than the contour pade method which is similarly based on vector valued rational approximation in contrast to the stable rbf qr and rbf ga algorithms which are based on finding a better conditioned base in the same rbf space the new algorithm can be used with any type of smooth radial kernel and it is also applicable to a wider range of tasks including calculating hermite type implicit rbf fd stencils we present a series of numerical experiments demonstrating the effectiveness of this new method for computing rbf interpolants in the flat regime we also demonstrate the flexibility of the method by using it to compute implicit rbf fd formulas in the flat regime and then using these for solving poisson s equation in a 3 d spherical shell

unitarization of symplectics and stability for causal differential equations in hilbert space let sp h be the symplectic group for a complex hibert space h its lie algebra sp h contains an open invariant convex cone c0 each element of c0 commutes with a unique sympletic complex structure the cayley transform c x sp h i x 1 sp h is analyzed and compared with the exponential mapping as an application we consider equations of the form ddt s a t s where t a t c0 is strongly continuous and show that if a t dt 2 and t8 a t dt c0 the scattering operator s s limt t st t where st t is the solution such that st t i is in the range of b restricted to c0 it follows that s leaves invariant a unique complex structure in particular it is conjugate in sp h to a unitary operator

compact finite difference scheme and rbf meshless approach for solving 2d rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives abstract this article is devoted to the comparison between a high order difference scheme and radial basis functions rbfs meshless approach for the numerical solution of two dimensional fractional rayleigh stokes problem the fractional derivative of problem is described in the riemann liouville sense in the high order difference scheme we discretize the spaces derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme also in the rbf meshless method we discretize the time fractional derivatives of the mentioned equation by integrating both sides of it then we will use the kansa s approach to approximate the spatial derivatives we prove the stability and convergence for high order difference scheme using fourier analysis and for time discrete scheme in rbf method via energy approach we compare the results of compact finite difference method and rbf meshless approach in terms of accuracy and cpu time also we compare the numerical results of compact difference scheme with other methods in the literature to show the high accuracy and efficiency of the proposed method

the method of variably scaled radial kernels for solving two dimensional magnetohydrodynamic mhd equations using two discretizations the crank nicolson scheme and the method of lines mol abstract mhd equations have many applications in physics and engineering the model is coupled equations in velocity and magnetic field and has a parameter namely hartmann the value of hartmann number plays an important role in the equations when this parameter increases using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem in the present paper a numerical meshless method based on radial basis functions rbfs is provided to solve mhd equations for approximating the spatial variable a new approach which is introduced by bozzini et al 2015 is applied the method will be used here is based on the interpolation with variably scaled kernels the methodology of the new technique is defining the scale function c on the domain r d then the interpolation problem from the data locations x j r d transforms to the new interpolation problem in the data locations x j c x j r d 1 bozzini et al 2015 the radial kernels used in the current work are multiquadrics mq inverse quadric iq and wendland s function of course the latter one is based on compactly supported functions to discretize the time variable two techniques are applied one of them is the crank nicolson scheme and another one is based on mol the numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem also in appendix we provide a computational algorithm for implementing the new technique in matlab software

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

compactly embedded cartan algebras and invariant cones in lie algebras on considere une theorie generale des cones invariants et la signification d une classification potentielle de toutes les paires l w d une algebre de lie l et d un cone invariant w on montre le role fondamental joue par les sous algebres de cartan

on choosing a radial basis function and a shape parameter when solving a convective pde on a sphere radial basis function rbf approximations have been used for some time to interpolate data on a sphere as well as on many other types of domains their ability to solve to spectral accuracy convection type pdes over a sphere has been demonstrated only very recently in such applications there are two main choices that have to be made i which type of radial function to use and ii what value to choose for their shape parameter denoted by e and with flat basis functions stretched out in the radial direction corresponding to e 0 the recent rbf qr algorithm has made it practical to compute stably also for small values of e results from solving a convective type pde on a sphere are compared here for many choices of radial functions over the complete range of e values from very large down to the limit of e 0 the results are analyzed with a methodology that has similarities to the customary fourier analysis in equispaced 1 d periodic settings in particular we find that high accuracy can be maintained also over very long time integrations we furthermore gain insights into why rbfs sometimes offer higher accuracy than spherical harmonics since the latter arise as an often non optimal special case of the former anticipated future application areas for rbf based methods in spherical geometries include weather and climate modeling

on sophus lie s fundamental theorem sophus lie s third theorem asserts that for every finite dimensional lie algebra there is a local lie group with the given algebra as tangent algebra at the identity in particular each subalgebra of the lie algebra of a local lie group determines a local subgroup whose lie algebra is the given one recent interest in subsemigroups and local subsemigroups of lie groups motivate the investigation to which extent local semigroups in lie groups are determined by infinitesimal generators it is known that the set l s of tangent vectors at the identity of a local semigroup s with identity in a local lie group g is an additively closed convex subset of the lie algebra l g which is topologically closed and contains 0 its largest vector subspace h is a subalgebra and all automorphisms of l g of the form eadx with x h leave l s invariant as a whole with adx y x y we call an additively and topologically closed convex subset w containing 0 in a lie algebra l a lie wedge if it satisfies eadxw w for all x w t w in this paper we show that lie wedges are precisely the infinitesimal generators of local semigroups in lie groups more precisely we show that in a banach lie algebra l for every lie wedge w whose edge w w has a closed vector space complement in l there is an open ball b around 0 in l such that the cambell hausdorff series x y x y 12 x y converges absolutely for x y b and there is a subset s b with i 0 s ii s s b s and l s w

invariant convex cones and causality in semisimple lie algebras and groups the convex cones in a simple lie algebra g invariant under the adjoint group g of g are studied using a earlier abstract classification of such cones we find explicit algebraic presentations of such cones in all the classical hermitian symmetric lie algebras nontrivial such cones exist only in these cases the g orbits in such cones are listed the notion of a temporal action of a lie group with an invariant causal orientation upon a causally oriented manifold is defined the canonical actions of such classical groups g as above on the shilov boundaries of the associated tube type hermitian symmetric spaces are shown to be temporal actions corollaries are 1 the existence of nontrivial lie semigroups s in the infinite sheeted coverings g of g which are invariant under conjugation by g and satisfy s s 1 e and 2 the global causality i e no closed time like curves of such covering groups g

on pseudo spectral time discretizations in summation by parts form abstract fully implicit discrete formulations in summation by parts form for initial boundary value problems must be invertible in order to provide well functioning procedures we prove that under mild assumptions pseudo spectral collocation methods for the time derivative lead to invertible discrete systems when energy stable spatial discretizations are used

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

simple space time symmetries generalizing conformal field theory we study simple space time symmetry groups g which act on a space time manifold m g h which admits a g invariant global causal structure we classify pairs g m which share the following additional properties of conformal field theory 1 the stability subgroup h of o m is the identity component of a parabolic subgroup of g implying factorization h man where m generalizes lorentz transformations a dilatations and n special conformal transformations 2 special conformal transformations n act trivially on tangent vectors v tom the allowed simple lie groups g are the universal coverings of su m m so 2 d sp l r so 4n and e7 25 and h are particular maximal parabolic subgroups they coincide with the groups of fractional linear transformations of euclidean jordan algebras whose use as generalizations of minkowski space time was advocated by gunaydin mod phys lett a 8 1407 1993 all these groups g admit positive energy representations it will also be shown that the classical confor

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

a new class of oscillatory radial basis functions radial basis functions rbfs form a primary tool for multivariate interpolation and they are also receiving increased attention for solving pdes on irregular domains traditionally only nonoscillatory radial functions have been considered we find here that a certain class of oscillatory radial functions including gaussians as a special case leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat this flat limit is important in that it generalizes traditional spectral methods to completely general node layouts interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions such as multiquadrics and inverse multiquadratics

lebesgue function for multivariate interpolation by radial basis functions multivariate interpolation and approximation is a powerful tool for intuition of real world in this paper we study lebesgue function and lebesgue constant for multivariate interpolation by conditionally positive definite rbfs also we give a conjecture about similar bernstein and erdos problem to multivariate case

boundary optimized diagonal norm sbp operators abstract by using non equispaced grid points near boundaries we derive boundary optimized first derivative finite difference operators of orders up to twelve the boundary closures are based on a diagonal norm summation by parts sbp framework thereby guaranteeing linear stability on piecewise curvilinear multi block grids the new operators lead to significantly more efficient numerical approximations compared with traditional sbp operators on equidistant grids we also show that the non uniform grids make it possible to derive operators with fewer one sided boundary stencils than their traditional counterparts numerical experiments with the 2d compressible euler equations on a curvilinear multi block grid demonstrate the accuracy and stability properties of the new operators

quasi interpolation scheme for arbitrary dimensional scattered data approximation based on natural neighbors and rbf interpolation abstract a local quasi interpolation scheme is presented in the paper which can be applied to arbitrary dimensional scattered data approximation with high accuracy high efficiency and high stability under our new quasi interpolation scheme we treat the natural neighbor set of the point to be estimated in a given scattered node set as a local interpolation node set based on the local node set a local radial basis function interpolation with algebraic precision of any degree is constructed through a modified taylor expansion of sampled function at each natural neighbor to compute approximate values given the exclusion of the point to be estimated from its natural neighbor set i e interpolation node set the interpolation scheme we construct is a local quasi interpolation scheme and its approximation error is given the numerical experimental results show that when the derivative information is unavailable our method outperforms the famous natural neighbor interpolation method in both approximation accuracy and efficiency as algebraic precision increases the approximation accuracy can be further improved

exact polynomial reproduction for oscillatory radial basis functions on infinite lattices until now only nonoscillatory radial basis functions rbfs have been considered in the literature it has recently been shown that a certain family of oscillatory rbfs based on j bessel functions gives rise to nonsingular interpolation problems and seems to be the only class of functions not to diverge in the limit of flat basis functions for any node layout this paper proves another interesting feature of these functions exact polynomial reproduction of arbitrary order on an infinite lattice in n first a closed form expression is derived for calculating the expansion coefficients for any order polynomial in any dimension then a proof is given showing that the resulting interpolant using this class of oscillatory rbfs will give exact polynomial reproduction examples in one and two dimensions are presented it is specifically noted that such closed form expressions cannot be derived for other classes of rbfs due to the fact that j bessel rbfs reproduce polynomials via a different mechanism

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

on the scaling ranges of detrended fluctuation analysis for long term memory correlated short series of data abstract we examine the scaling regime for the detrended fluctuation analysis dfa the most popular method used to detect the presence of long term memory in data and the fractal structure of time series first the scaling range for dfa is studied for uncorrelated data as a function of time series length l and the correlation coefficient of the linear regression r 2 at various confidence levels next a similar analysis for artificial short series of data with long term memory is performed in both cases the scaling range is found to change linearly both with l and r 2 we show how this dependence can be generalized to a simple unified model describing the relation l r 2 h where h 1 2 h 1 stands for the hurst exponent of the long range autocorrelated signal our findings should be useful in all applications of dfa technique particularly for instantaneous local dfa where a huge number of short time series has to be analyzed at the same time without possibility of checking the scaling range in each of them separately

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

parametrization of causal actions of universal covering groups and global hyperbolicity abstract the transformation group of the universal covering of the real projective line obtained by lifting ordinary projective transformations is given explicitly in terms of canonical coordinates a similar formulation is given of the action of the universal covering of su 2 2 upon the universal covering of the shilov boundary of its associated bounded hermitian symmetric domain structured as r1 su 2 the former group g isomorphic to s u 1 1 has a unique and continuous bi invariant global partial ordering similar to that expressing space time causality relations corresponding to its bivariant lorentzian metric the partial ordering is the same as that induced by the ordering of the real line which the transformation group preserves as an application the compactness of the intervals g 1 g 2 g g g 1 g g 2 for g 1 g 2 g necessary for global hyperbolicity of the metric is studied it is shown that g1 g2 is compact if and only if g g1 for all g in a neighborhood of g2 where is the generator of the center of g satisfying e in particular the interior of e is a maximal open global hyperbolic submanifold s u 1 1 is not globally hyperbolic

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

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

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

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

vector field approximation using radial basis functions in this article we investigate the performance of rbf pde methods for approximating solenoidal fields it is well known that global rbf collocation methods present a trade off principle which means that smoothness implies high convergence order plus ill conditioning on the other hand local methods for solving this problem have recently appeared in the literature in this paper we perform a numerical investigation of the differences between rbf global and local methods in order to investigate the possible advantage of using local methods for the approximation of vector fields more precisely we compare the local hermite interpolation technique using inverse multiquadrics against the non symmetric collocation method of kansa

on the role of polynomials in rbf fd approximations radial basis function generated finite difference rbf fd approximations generalize classical grid based finite differences fd from lattice based to scattered node layouts this greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas many different types of radial functions have been considered in this rbf fd context in this study we find that i polyharmonic splines phs in conjunction with supplementary polynomials provide a very simple way to defeat stagnation also known as saturation error and ii give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter in follow up studies we will focus on how to best use these hybrid rbf polynomial bases for fd approximations in the contexts of solving elliptic and hyperbolic type pdes

a criterion for the determination of optimal scaling ranges in dfa and mf dfa abstract we develop a criterion based on a brute force algorithm to systematically determine optimal fitting regions for fluctuation functions in detrended fluctuation analysis dfa and multifractal detrended fluctuation analysis mf dfa we analyze and compare results with several artificially generated time series with known parameters as illustrative examples of this technique we show that crossover time scales can also be objectively and efficiently estimated using the introduced algorithm finally we employ the proposed methodology to study the scaling behavior of a natural time series monthly sunspot data previously filtered with empirical mode decomposition emd in this case we find results comparable with those found in the literature

generalized clenshaw curtis quadrature rule with application to a collocation least squares method abstract this paper deals with an extension of one dimensional clenshaw curtis quadrature rule to r d d 2 on a convex domain as one of its applications we apply this quadrature rule to a collocation least squares method using arbitrary abscissas for a first order system of an elliptic boundary value problem so that the convergence analysis on a numerical solution can be shown in a standard sobolev norm

auto correlated behavior of wti crude oil volatilities a multiscale perspective abstract in this paper we investigate the long range auto correlated behavior of wti crude oil volatility series employing multifractal detrended fluctuation analysis our findings show that the for small time scales the auto correlations of volatilities were multifractal while for large time scales the auto correlations were nearly monofractal based on multiscale analysis we also investigate the dynamics of auto correlations for different intervals of time scales and find that several shocks could make significant effects on the auto correlated behaviors for small time scales analyzing the dynamics of multifractality degrees of auto correlations for small time scales we find that the stronger auto correlations were always related to the lower degrees of multifractality at last we have discussions on the determination factors of price behavior the predictive implications of scaling behavior in volatilities for oil markets and the reasons why long range auto correlations of volatility were always strong for both small time scales and large time scales our results are very important theoretically and practically

the orthogonal gradients method a radial basis functions method for solving partial differential equations on arbitrary surfaces much work has been done on reconstructing arbitrary surfaces using the radial basis function rbf method but one can hardly find any work done on the use of rbfs to solve partial differential equations pdes on arbitrary surfaces in this paper we investigate methods to solve pdes on arbitrary stationary surfaces embedded in r 3 using the rbf method we present three rbf based methods that easily discretize surface differential operators we take advantage of the meshfree character of rbfs which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension two out of the three methods which we call the orthogonal gradients ogr methods are the result of our work and are hereby presented for the first time

radial basis function interpolation in the limit of increasingly flat basis functions abstract we propose a new approach to study radial basis function rbf interpolation in the limit of increasingly flat functions the new approach is based on the semi analytical computation of the laurent series of the inverse of the rbf interpolation matrix described in a previous paper 3 once the laurent series is obtained it can be used to compute the limiting polynomial interpolant the optimal shape parameter of the rbfs used for interpolation and the weights of rbf finite difference formulas among other things

summation by parts operators with minimal dispersion error for coarse grid flow calculations we present a procedure for constructing summation by parts operators with minimal dispersion error both near and far from numerical interfaces examples of such operators are constructed and compar

high order implicit time marching methods based on generalized summation by parts operators this article extends the theory of classical finite difference summation by parts fd sbp time marching methods to the generalized summation by parts gsbp framework dual consistent gsbp time marching methods are shown to retain a and l stability as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization this also implies that the solution approximated at the end of each time step is superconvergent in addition gsbp time marching methods constructed with a diagonal norm are bn stable this article also formalizes the connection between fd sbp gsbp time marching methods and implicit runge kutta methods through this connection the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems it is also exploited to derive conditions under which nonlinearly stable gsbp time marching methods can be constructed the gsbp approach to time marching can simplify the construction of high order f

a preconditioned implementation of pseudospectral methods on arbitrary grids in this article a preconditioner for pseudospectral methods in terms of expansion coefficients on arbitrary grids is presented the preconditioner is first developed for application in ordinary differential equations and then is generalized for partial differential equations a fully discrete pseudospectral approximation is then used to solve bivariate problems this preconditioner in effect makes the pseudospectral methods insensitive to specific grid points the numerical results show the efficiency of the method

convergence analysis for a second order elliptic equation by a collocation method using scattered points a collocation method using scattered points applied to a second order elliptic differential equation is analyzed by establishing a new quadrature formula for the space of the polynomials we show that a polynomial solution possesses stability and preserves a similar convergence property occurred in the classical high order collocation method

fractional geometric mean reversion processes abstract we use a method developed in carmona et al 2003 2 to study the fractional geometric mean reversion processes our obtained results hold for any h 1 4 1

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

a radial basis functions based finite differences method for wave equation with an integral condition the hyperbolic partial differential equation which contains integral condition in place of classical boundary condition arises in many application of modern physics and technologies in this article we propose a numerical method to solve the hyperbolic equation with nonlocal boundary condition using radial basis function based finite difference method several numerical experiments are presented and compared with some existing method to demonstrate the efficiency of the proposed method

the numerical solution of nonlinear high dimensional generalized benjamin bona mahony burgers equation via the meshless method of radial basis functions in this paper a numerical technique is proposed for solving the nonlinear generalized benjamin bona mahony burgers equation firstly we obtain a time discrete scheme by approximating the first order time derivative via forward finite difference formula then we use kansa s approach to approximate the spatial derivatives we prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method also we show that convergence order of the time discrete scheme is o t we solve the two dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular triangular and circular domains and also the three dimensional case is solved on the cubical and spherical domains the aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations also several test problems including the three dimensional case are given numerical examples confirm the efficiency of the proposed scheme

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 stabilized radial basis finite difference rbf fd method with hybrid kernels abstract recent developments have made it possible to overcome grid based limitations of finite difference fd methods by adopting the kernel based meshless framework using radial basis functions rbfs such an approach provides a meshless implementation and is referred to as the radial basis generated finite difference rbf fd method in this paper we propose a stabilized rbf fd approach with a hybrid kernel generated through a hybridization of the gaussian and cubic rbf this hybrid kernel was found to improve the condition of the system matrix consequently the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard rbf fd methods coupled with present stable algorithms unlike other rbf fd approaches the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity and the size of the stencils as an application we solve the frequency domain acoustic wave equation in a 2d half space in order to suppress spurious reflections from truncated computational boundaries absorbing boundary conditions have been effectively implemented

limit problems for interpolation by analytic radial basis functions interpolation problems for analytic radial basis functions like the gaussian and inverse multiquadrics can degenerate in two ways the radial basis functions can be scaled to become increasingly flat or the data points coalesce in the limit while the radial basis functions stay fixed both cases call for a careful regularization which if carried out explicitly yields a preconditioning technique for the degenerating linear systems behind these interpolation problems this paper deals with both cases for the increasingly flat limit we recover results by larsson and fornberg together with lee yoon and yoon concerning convergence of interpolants towards polynomials with slight modifications the same technique can also handle scenarios with coalescing data points for fixed radial basis functions the results show that the degenerating local lagrange interpolation problems converge towards certain hermite birkhoff problems this is an important prerequisite for dealing with approximation by radial basis functions adaptively using freely varying data sites

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

linear combination of power law functions for detecting multiscaling using detrended fluctuation analysis abstract in many instances the fluctuation function obtained from detrended fluctuation analysis dfa cannot be described by a uniform power law function along scales in fact the manifestation of crossover scales may reflect the simultaneous action of different stochastic mechanisms displayed predominantly within certain scale ranges this note proposes the use of a linear combination of power law functions for adjusting dfa data the idea is that each power law function recast the dominance of certain stochastic mechanisms e g the mean reversion and long term trends at specific scale domains different values of the scaling exponents are numerically estimated by means of a nonlinear least squares fitting of power law functions examples of crude oil market and heart rate variability are discussed with some detail for illustrating the advantages of taking a linear combination of power law functions for describing scaling behavior from dfa

integration matrix based on arbitrary grids with a preconditioner for pseudospectral method a matrix representation of integration for arbitrary grids is introduced suitable results are then obtained to be used along with differentiation matrix preconditioner to implement pseudospectral method on integro differential equations using arbitrary grids numerical examples are given to clarify the efficiency of the new method

effect of nonstationarities on detrended fluctuation analysis detrended fluctuation analysis dfa is a scaling analysis method used to quantify long range power law correlations in signals many physical and biological signals are noisy heterogeneous and exhibit different types of nonstationarities which can affect the correlation properties of these signals we systematically study the effects of three types of nonstationarities often encountered in real data specifically we consider nonstationary sequences formed in three ways i stitching together segments of data obtained from discontinuous experimental recordings or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts a cutting procedure commonly used in preparing data prior to signal analysis ii adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes and iii generating a signal comprised of segments with different properties e g different standard deviations or different correlation exponents we compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities we find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on a the fraction and size of the parts cut out from the signal b the concentration of spikes and their amplitudes c the proportion between segments with different standard deviations or different correlations and d the correlation properties of the stationary signal we show how to develop strategies for preprocessing raw data prior to analysis which will minimize the effects of nonstationarities on the scaling properties of the data and how to interpret the results of dfa for complex signals with different local characteristics

corner corrected diagonal norm summation by parts operators for the first derivative with increased order of accuracy combined with simultaneous approximation terms summation by parts sbp operators offer a versatile and efficient methodology that leads to consistent conservative and provably stable discretizations however diagonal norm operators with a repeating interior point operator that have thus far been constructed suffer from a loss of accuracy while on the interior these operators are of degree 2p at a number of nodes near the boundaries they are of degree p and therefore of global degree p meaning the highest degree monomial for which the operators are exact at all nodes this implies that for hyperbolic problems and operators of degree greater than unity they lead to solutions with a global order of accuracy lower than the degree of the interior point operator in this paper we develop a procedure to construct diagonal norm first derivative sbp operators that are of degree 2p at all nodes and therefore can lead to solutions of hyperbolic problems of order 2 p 1 this is accomplished by adding nonzero entries in the upper right and lower left corners of sbp operator matrices with a repeating interior point operator this modification necessitates treating these new operators as elements where mesh refinement is accomplished by increasing the number of elements in the mesh rather than increasing the number of nodes the significant improvements in accuracy of this new family for the same repeating interior point operator are demonstrated in the context of the linear convection equation

enhancing finite differences with radial basis functions polynomials are used together with polyharmonic spline phs radial basis functions rbfs to create local rbf finite difference rbf fd weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences fd the presented method replicates the convergence properties of fd but for arbitrary node layouts it is tested on the 2d compressible navier stokes equations at low mach number relevant to atmospheric flows test cases are taken from the numerical weather prediction community and solved on bounded domains thus attention is given on how to handle boundaries with the rbf fd method as well as a novel implementation for hyperviscosity comparisons are done on cartesian hexagonal and quasi uniform node layouts consideration and guidelines are given on phs order polynomial degree and stencil size the main advantages of the present method are 1 capturing the basic physics of the problem surprisingly well even at very coarse resolutions 2 high order accuracy without the need of tuning a shape parameter and 3 the inclusion of polynomials eliminates stagnation saturation errors a matlab code is given to calculate the differentiation weights for this novel approach

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

the spatial operator in the incompressible navier stokes oseen and stokes equations abstract we investigate the spatial operator in the incompressible navier stokes oseen and stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions the theoretical results are derived for a general form of energy stable boundary conditions and applied to a few commonly used ones the analysis is done on a system that simultaneously covers the nonlinear incompressible navier stokes the oseen and the stokes equations when the spectrum of the spatial operator is investigated we restrict the analysis to the oseen and stokes equations the continuous analysis carries over to the discrete setting by using the summation by parts framework

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

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

fast high dimensional node generation with variable density abstract we present an algorithm for producing discrete distributions with a prescribed nearest neighbor distance function our approach is a combination of quasi monte carlo q mc methods and weighted riesz energy minimization the initial distribution is a stratified q mc sequence with some modifications a suitable energy functional on the configuration space is then minimized to ensure local regularity the resulting node sets are good candidates for building meshless solvers and interpolants as well as for other purposes where a point cloud with a controlled separation covering ratio is required applications of a three dimensional implementation of the algorithm in particular to atmospheric modeling are also given