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

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

tractability of parameterized completion problems on chordal strongly chordal and proper interval graphs we study the parameterized complexity of three np hard graph completion problems the minimum fill in problem asks if a graph can be triangulated by adding at most k edges we develop o ck m and o k2mn f k algorithms for this problem on a graph with n vertices and m edges here f k is exponential in k and the constants hidden by the big o notation are small and do not depend on k in particular this implies that the problem is fixed parameter tractable fpt the proper interval graph completion problem motivated by molecular biology asks if a graph can be made proper interval by adding no more than k edges we show that the problem is fpt by providing a simple search tree based algorithm that solves it in o ck m time similarly we show that the parameterized version of the strongly chordal graph completion problem is fpt by giving an o ck m log n time algorithm for it all of our algorithms can actually enumerate all possible k completions within the same time bounds

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

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

minimal interval completion through graph exploration given an arbitrary graph g v e and an interval graph h v f with email protected f we say that h is an interval completion of g the graph h is called a minimal interval completion of g if for any sandwich graph h v f with email protected f f h is not an interval graph in this paper we give a o nm time algorithm computing a minimal interval completion of an arbitrary graph the output is an interval model of the completion

lex bfs and partition refinement with applications to transitive orientation interval graph recognition and consecutive ones testing by making use of lexicographic breadth rst search lex bfs and partition renement with pivots we obtain very simple algorithms for some well known problems in graph theory we give a o n mlogn algorithm for transitive orientation of a comparability graph and simple linear algorithms to recognize interval graphs convex graphs y semichordal graphs and matrices that have the consecutive ones property previous approaches to these problems used dicult preprocessing steps such as computing pq trees or modular decomposition the algorithms we give are easy to understand and straightforward to prove they do not make use of sophisticated data structures and the complexity analysis is straightforward c 2000 elsevier science b v all rights reserved

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

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

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

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

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

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

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

pathwidth bandwidth and completion problems to proper interval graphs with small cliques we study two related problems motivated by molecular biology given a graph g and a constant k does there exist a supergraph g of g that is a unit interval graph and has clique size at most k given a graph g and a proper k coloring c of g does there exist a supergraph g of g that is properly colored by c and is a unit interval graph we show that those problems are polynomial for fixed k on the other hand we prove that the first problem is equivalent to deciding if the bandwidth of g is at most k 1 hence it is np hard and w t hard for all t we also show that the second problem is w 1 hard this implies that for fixed k both of the problems are unlikely to have an o n alpha algorithm where alpha is a constant independent of k a central tool in our study is a new graph theoretic parameter closely related to pathwidth an unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph

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

an o n2 time algorithm for the minimal interval completion problem an interval completion of an arbitrary graph g is an interval graph h on the same vertex set obtained from g by adding new edges if the set of newly added edges is inclusion minimal among all possibilities we say that h is a minimal interval completion of g we give an o n 2 time algorithm to obtain a minimal interval completion of an arbitrary graph this improves the previous o nm time bound for the problem and lowers this bound for the first time below the best known bound for minimal chordal completion

calderon zygmund theory for product domains hp spaces this article deals with the boundedness properties of singular integrals that generalize the double hilbert transform on product domains the action of these operators on hardy spaces and on l log l is discussed

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