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

a meshfree approach to non newtonian free surface ice flow numerical models of glacier and ice sheet dynamics traditionally employ finite difference or finite element methods although these are highly developed and mature methods they suffer from some drawbacks such as inability to handle complex geometries finite differences or a costly assembly procedure for nonlinear problems finite elements additionally they are mesh based and therefore moving domains become a challenge in this paper we introduce a novel meshfree approach based on a radial basis function rbf method the meshfree nature of rbf methods enables efficient handling of moving margins and free ice surface rbf methods are also accurate easy to implement and allow for reduction the computational cost associated with the linear system assembly since stated in strong form to demonstrate the global rbf method we model the velocity field of ice flow in the haut glacier d arolla which is governed by the nonlinear stokes equations we test the method for different basal conditions and for a free moving surface we also compare the global rbf method with its localized counterpart the rbf partition of unity method rbf pum that allows for a significant gain in the computational efficiency both rbf methods are compared with the classical finite element method in terms of accuracy and efficiency we find that the rbf methods are more efficient than the finite element method and well suited for ice dynamics modeling especially the partition of unity approach

rbf methods in a stochastic volatility framework for greeks computation abstract in this paper we aim to apply a new proposed meshless approach for heston pde resolution in mathematical finance most classical models may be reformulated as partial differential equations pde which are commonly solved by finite difference and monte carlo methods alternatively bio inspired methods may be considered to select parameters and to solve said pdes in particular we focus on heston model pde and we solve it via radial basis functions rbf methods moreover some considerations on the computation of model parameters through learning approaches have been discussed we aim to price a vanilla call option focusing on the derivation of the formulae for its greeks profiles eventually we conclude rbfs are an accurate alternative method to price derivatives in a stochastic volatility framework especially since they allow for a fast and precise computation of greeks

a quasi radial basis functions method for american options pricing abstract based on the idea of quasi interpolation and radial basis functions approximation a fast and accurate numerical method is developed for solving the black scholes equation for valuation of american options prices since the method does not require solving a resultant full matrix the ill conditioning problem resulting from using the radial basis functions as a global interpolant can be avoided the method has been shown to be effective in solving problems with free boundary condition as indicated in the numerical computation for the american option pricing an excellent approximation of the solution as well as the free optimal exercise boundary can be obtained

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

semiclassical and spectral analysis of oceanic waves in this work we prove that the shallow water flow subject to strong wind forcing and linearized around an adequate stationary profile develops for large times closed trajectories due to the propagation of rossby waves while poincare waves are shown to disperse the methods used in this paper involve semi classical analysis and dynamical systems for the study of rossby waves while some refined spectral analysis is required for the study of poincare waves due to the large time scale involved which is of diffractive type

existence and uniqueness of weak solutions for precipitation fronts a novel hyperbolic free boundary problem in several space variables the determination of the large scale boundaries between moist and dry regions is an important problem in contemporary meteorology these phenomena have been addressed recently in a simplified tropical climate model through a novel hyperbolic free boundary formulation yielding three families drying slow moistening and fast moistening of precipitation fronts the last two wave types violate lax s shock inequalities yet are robustly realized this formal hyperbolic free boundary problem is given here a rigorous mathematical basis by establishing the existence and uniqueness of suitable weak solutions arising in the zero relaxation limit a new l2 contraction estimate is also established at positive relaxation values 2010 wiley periodicals inc

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

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

multilevel sparse grids collocation for linear partial differential equations with tensor product smooth basis functions abstract radial basis functions have become a popular tool for approximation and solution of partial differential equations pdes the recently proposed multilevel sparse interpolation with kernels musik algorithm proposed in georgoulis et al 2013 shows good convergence in this paper we use a sparse kernel basis for the solution of pdes by collocation we will use the form of approximation proposed and developed by kansa 1986 we will give numerical examples using a tensor product basis with the multiquadric mq and gaussian basis functions this paper is novel in that we consider space time pdes in four dimensions using an easy to implement algorithm with smooth approximations the accuracy observed numerically is as good with respect to the number of data points used as other methods in the literature see langer 2016 and wang et al 2016

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

a rbf partition of unity collocation method based on finite difference for initial boundary value problems abstract meshfree radial basis function rbf methods are popular tools used to numerically solve partial differential equations pdes they take advantage of being flexible with respect to geometry easy to implement in higher dimensions and can also provide high order convergence since one of the main disadvantages of global rbf based methods is generally the computational cost associated with the solution of large linear systems in this paper we focus on a localizing rbf partition of unity method rbf pum based on a finite difference fd scheme specifically we propose a new rbf pum fd collocation method which can successfully be applied to solve time dependent pdes this approach allows to significantly decrease ill conditioning of traditional rbf based methods moreover the rbf pum fd scheme results in a sparse matrix system reducing the computational effort but maintaining at the same time a high level of accuracy numerical experiments show performances of our collocation scheme on two benchmark problems involving unsteady convection diffusion and pseudo parabolic equations

efficient computation of partition of unity interpolants through a block based searching technique in this paper we propose a new efficient interpolation tool extremely suitable for large scattered data sets the partition of unity method is used and performed by blending radial basis functions rbfs as local approximants and using locally supported weight functions in particular we present a new space partitioning data structure based on a partition of the underlying generic domain in blocks this approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points leading to an optimized searching routine complexity analysis and numerical experiments in two and three dimensional interpolation support our findings some applications to geometric modelling are also considered moreover the associated software package written in matlab is here discussed and made available to the scientific community

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

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

a lock free material finite element for non linear oscillations of laminated plates abstract the objective of the present paper is to propose an efficient accurate and robust four node shear flexible composite plate element with six degrees of freedom per node to investigate the non linear oscillatory behavior of unsymmetrical laminated plates the degrees of freedom considered are three displacement u v w along x y and z axis two rotations x y abouty and x axis and twist xy the elementc employs coupled displacement field which is derived using moment shear equilibrium and in plane equilibrium of composite strips along the x andy axis the displacement field so derived not only depend on the element co ordinates but are a function of extensional bending extensional bending and transverse shear stiffness coefficients as well a bi cubic polynomial distribution with 16 generalized undetermined coefficients for the transverse displacement is assumed the element stiffness and mass matrices are computed numerically by employing 3 3 gauss legendre product rules the element is found to be free of shear locking and does not exhibit any spurious modes the element is found to be free of shear locking and does not exhibit any spurious modes in order to compute the non linear frequencies linear mode shape corresponding to fundamental frequency is assumed as the spatial distribution and non linear finite element equations are reduced to a single non linear second order ordinary differential equation this equation is solved by employing direct numerical integration method a series of numerical examples is solved to demonstrate the efficacy of the proposed material finite element

greeks computation in the option pricing problem by means of rbf pu methods abstract in this article we focus on option greeks computation by means of radial basis functions rbf with partition of unity methods we start by presenting rbf applications to the financial world we price single underlying european and american barrier options and an american basket option on two correlated underlyings furthermore we derive the expression for greeks calculation via rbf and we compare the results of the corresponding solution with the finite difference method we conclude rbfs are an accurate and precise alternative method to price contingent claims offering an appealing solution to evaluate greeks with better results than finite difference methods

a simple justification of the singular limit for equatorial shallow water dynamics the equatorial shallow water equations at low froude number form a symmetric hyperbolic system with large variable coefficient terms although such systems are not covered by the classical klainerman majda theory of singular limits the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long wave equations as the height and froude number tend to 0 their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions a simpler proof of a slight generalization is presented here in the spirit of the classical theory 2008 wiley periodicals inc

improved radial basis function methods for multi dimensional option pricing in this paper we have derived a radial basis function rbf based method for the pricing of financial contracts by solving the black scholes partial differential equation as an example of a financial contract that can be priced with this method we have chosen the multi dimensional european basket call option we have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter for other non optimal choices of shape parameter values the resulting convergence rate is algebraic we propose an adapted node point placement that improves the accuracy compared with a uniform distribution compared with an adaptive finite difference method the rbf method is 20 40 times faster in one and two space dimensions and has approximately the same memory requirements

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

pseudo differential energy estimates of singular perturbations in this paper we study singular limits of hyperbolic systems which exhibit large time oscillations and in particular pseudo differential energy estimates that enable us to obtain uniform existence time and a priori bounds on the solutions 1997 john wiley sons inc

a kirchhoff mode method for c0 bilinear and serendipity plate elements abstract a new formulation for the c0 quadrilateral and serendipity plate elements is presented it is an extension of the mode decomposition approach that has been successfully applied in evaluating the c0 triangular linear plate element in contrast to the triangular element the concept of an equivalent discrete kirchhoff configuration is utilized the elements are applied to several examples and the performance of these elements is shown to be quite good

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

an rbf fd sparse scheme to simulate high dimensional black scholes partial differential equations abstract the objective of this research is to investigate the numerical solution of high dimensional black scholes partial differential equations pdes to do this the weights of the radial basis function finite difference rbf fd scheme using inverse multi quadric function are used with an emphasis on the hotzone the proposed rbf fd technique is capable to price multi asset options efficiently

penalty methods for the numerical solution of american multi asset option problems we derive and analyze a penalty method for solving american multi asset option problems a small non linear penalty term is added to the black scholes equation this approach gives a fixed solution domain removing the free and moving boundary imposed by the early exercise feature of the contract explicit implicit and semi implicit finite difference schemes are derived and in the case of independent assets we prove that the approximate option prices satisfy some basic properties of the american option problem several numerical experiments are carried out in order to investigate the performance of the schemes we give examples indicating that our results are sharp finally the experiments indicate that in the case of correlated underlying assets the same properties are valid as in the independent case

numerical simulation of reaction diffusion neural dynamics models and their synchronization desynchronization application to epileptic seizures abstract cognitive disorders especially epilepsy are closely linked with synchronization desynchronization of neurons in the brain in this paper the dynamical modeling and behavior analysis of a fitzhugh nagumo neuron and also synchronization control of a network of fitzhugh nagumo neurons which promise the understanding of cognitive processing are studied to numerically simulate these models and in order to overcome their difficulties such as computational complexity multi dimensionality non linearity having large spatial and temporal domains and also having a discrete initial data we propose the use of a strongly meshless technique based on wendland compactly supported radial basis function in conjunction with an operator splitting algorithm the main advantage of the proposed algorithm is its computational complexity for example o n x 1 3 n x 2 3 of the inversion of all matrices which is much lower than the complexity of pure radial basis functions method o n x 1 3 n x 2 3 of the inversion of matrices numerical experiments are presented showing that the proposed approaches are extremely accurate and fast

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

radial basis function generated finite differences for option pricing problems abstract in this paper we present a numerical method to price options based on radial basis function generated finite differences rbf fd in space and the backward differentiation formula of order 2 bdf 2 in time we use gaussian rbfs that depend on a shape parameter e the choice of this parameter is crucial for the performance of the method we chose e as const h 1 and we derive suitable values of the constant for different stencil sizes in 1d and 2d this constant is independent of the problem parameters such as the volatilities of the underlying assets and the interest rate in the market in the literature on option pricing with rbf fd a constant value of the shape parameter is used we show that this always leads to ill conditioning for decreasing h whereas our proposed method avoids such ill conditioning we present numerical results for problems in 1d 2d and 3d demonstrating the useful features of our method such as discretization sparsity flexibility in node placement and easy dimensional extendability which provide high computational efficiency and accuracy

fast averaging for long and short wave scaled equatorial shallow water equations with coriolis parameter deviating from linearity the equatorial shallow water equations at low froude number form a symmetric hyperbolic system with large terms containing a variable coefficient the coriolis parameter f which depends on the latitude the limiting behavior of the solutions as the froude number tends to zero was investigated rigorously a few years ago using the common approximation that the variations of f with latitude are linear in that case the large terms have a peculiar structure due to special properties of the harmonic oscillator hamiltonian which can be exploited to prove strong uniform a priori estimates in adapted functional spaces it is shown here that these estimates still hold when f deviates from linearity even though the special properties on which the proofs were based have no obvious generalization as in the linear case existence uniqueness and convergence properties of the solutions corresponding to general unbalanced data are deduced from the estimates

analysis of the nonlinear vibrations of unsymmetrically laminated composite beams large amplitude free vibrations of unsymmetrically laminated beams using von karman large deflection theory are investigated herein one dimensional finite elements based on classical lamination theory first order shear deformation theory and higher order shear deformation theory having 8 10 and 12 degrees of freedom per node respectively are developed

radial basis function partition of unity method for modelling water flow in porous media abstract water flow in variably saturated porous media is modelled by using the highly nonlinear parabolic richards equation the nonlinearity is due to the hydraulic conductivity and moisture content variables the latter were estimated by using experimental models including gardner burdine mualem and van genuchten models the aim of this work is to develop a new technique based on the radial basis function partition of unity method rbfpum and gardner model in order to solve richards equation in one and two dimensions we have used gardner model to handle the nonlinearity issue and the rbfpum is used to approximate the solution of the linearized richards equation our proposed algorithm is based on testing many configurations of the partitions number and selecting the optimal shape parameter for each case then we pick up the optimal configuration partitions number shape parameter that yields the best solution in terms of error and conditioning number by following this procedure an optimal solution is ensured for our given problem as numerical tests we consider the vertical infiltration of water in soils in order to validate our proposed method

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

a penalty method for american options with jump diffusion processes the fair price for an american option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem we develop an implicit discretization method for pricing such american options the jump diffusion correlation integral term is computed using an iterative method coupled with an fft while the american constraint is imposed by using a penalty method we derive sufficient conditions for global convergence of the discrete penalized equations at each timestep finally we present numerical tests which illustrate such convergence

multi dimensional option pricing using radial basis functions and the generalized fourier transform we show that the generalized fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi dimensional option pricing we derive a general algorithm including a transformation of the black scholes equation into the heat equation that can be used in any number of dimensions numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes furthermore the gain increases with the number of dimensions