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

a rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pdes we obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to lipschitz continuous viscosity solutions of uniformly elliptic partial differential equations 2007 wiley periodicals inc

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

multiquadric method for the numerical solution of a biphasic mixture model a computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model the model includes a set of constitutive equations for the fluid flows through the solid phase a set of momentum equations for stress strain equilibrium and a continuity equation for the solid phase and the fluid phase the numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading the numerical results are compared with spilker s penalty finite element method

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

solving time dependent differential equations by multiquadric trigonometric quasi interpolation multiquadric mq quasi interpolation is a popular method for the numerical solution of differential equations however mq quasi interpolation is not well suited for the equations with periodic solutions this is mainly due to the fact that its kernel the mq function is not a periodic function a reasonable way of overcoming the difficulty is to use a quasi interpolant whose kernel itself is also periodic in these cases the paper constructs such a quasi interpolant error estimates of the quasi interpolant are also provided the quasi interpolant possesses many fair properties of the mq quasi interpolant i e simplicity efficiency stability etc moreover it is more suitable than the mq quasi interpolant for periodic problems since the quasi interpolant as well as its derivatives are periodic examples of solving both linear and nonlinear partial differential equations whose solutions are periodic by the quasi interpolant and the mq quasi interpolant are compared at the end of the paper numerical results show that the quasi interpolant outperforms the mq quasi interpolant for periodic problems

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

on time inhomogeneous controlled diffusion processes in domains time inhomogeneous controlled diffusion processes in both cylindrical and noncylindrical domains are considered bellman s principle and its applications to proving the continuity of value functions are investigated

a univariate quasi multiquadric interpolationwith better smoothness in this paper we propose a multilevel univariate quasi interpolation scheme usingmultiquadric basis it is practical as it does not require derivative values of the function being interpolated it has a higher degree of smoothness than the original level 0 formula as it allows a shape parameter c o h our level 1 quasi interpolation costs o email protected n flops to set up it preserves strict convexity and monotonicity when c o h we prove the proposed scheme converges with a rate of o h 2 email protected h furthermore if both a and are relatively small compared with email protected the convergence rate will increase we verify numerically that c h is a good shape parameter to use for our method hence we need not find the optimal parameter for all test functions both convergence speed and error are optimized for c between 0 5h and 1 5h our method can be generalized to a multilevel scheme we include the numerical results for the level 2 scheme the shape parameter of the level 2 scheme can be chosen between 2h to 3h

a comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations abstract in this article we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations a series of test examples was solved with these two schemes different problems with different type of governing equations and boundary conditions particular emphasis was paid to the ability of these schemes to solve the steady state convection diffusion equation at high values of the peclet number from the examples tested in this work it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better however the unsymmetric method has the advantage of being simpler to implement two main features about the results obtained in this work are worthy of special attention first with the symmetric method it was possible to solve convection diffusion problems at a very high peclet number without the need of any artificial damping term and second with these two approaches symmetric and unsymmetric it is possible to impose free boundary conditions for problems in unbounded domains

convergent semi lagrangian methods for the monge ampere equation on unstructured grids this paper is concerned with developing and analyzing convergent semi lagrangian methods for the fully nonlinear elliptic monge ampere equation on general triangular grids this is done by establishing an equivalent in the viscosity sense hamilton jacobi bellman formulation of the monge ampere equation a significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built in property of the new formulation moreover this new approach allows one to tap the wealthy numerical methods such as semi lagrangian schemes for hamilton jacobi bellman equations to solve monge ampere type equations it is proved that the considered numerical methods are monotone pointwise consistent and uniformly stable consequently its solutions converge uniformly to the unique convex viscosity solution of the monge ampere dirichlet problem a superlinearly convergent howard s algorithm which is a newton type method is utilized as the nonlinear

numerical solution of burgers fisher equation by cubic b spline quasi interpolation abstract in this paper numerical solution of the burgers equation is presented based on the cubic b spline quasi interpolation at first the cubic b spline quasi interpolation is introduced moreover the numerical scheme is presented 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 the accuracy of the proposed method is demonstrated by some test problems the numerical results are found in good agreement with exact solutions the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

geometric properties of random disk packings random packings ofn 2000 rigid disks in the plane subject to periodic boundary conditions on a square primitive cell have been generated by a concurrent construction which treats all disks on an equal footing as opposed to previously investigated sequential constructions the particles start with random positions and velocities and as they move about they grow uniformly in size from points to jammed disks the collection of packings displays several striking geometric features these include for largen typically polycrystalline textures with irregular grain boundaries and linear shear fractures the packings occasionally contain monovacancies and trapped but unjammed rattler disks the latter appear to be confined to the grain boundaries the linear shear fractures preserve bond orientational order but disrupt translational order within the crystalline grains a new efficient event driven simulation algorithm is employed to generate the histories of colliding and jamming disks on a computer which can process one million floating point instructions per second the algorithm processes more than one million pairwise collisions per hour

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

semi lagrangian schemes for linear and fully non linear diffusion equations for linear and fully non linear diffusion equations of bellmanisaacs type we introduce a class of monotone approximation schemes relying on monotone interpolation as opposed to classical numerical methods these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices such schemes have to have a wide stencil in general besides providing a unifying framework for several known first order accurate schemes our class of schemes also includes more efficient versions and a new second order scheme that converges only for essentially monotone solutions the methods are easy to implement and analyze and they are more efficient than some other known schemes we prove stability and convergence of the schemes in the general case and provide error estimates in the convex case which are robust in the sense that they apply to degenerate equations and non smooth solutions the methods are extensively tested

generalized b splines as a tool in isogeometric analysis the concept of isogeometric analysis has been proposed in 13 where nurbs are considered as basis of the analysis thanks to their ability to construct an exact geometric model in several practical applications and to their popularity in commercial cad systems in this paper we propose an alternative to the rational model presenting an isogeometric analysis approach based on generalized b splines geometric models exactly represented by generalized b splines include those generated by nurbs moreover generalized b splines possess all fundamental properties of algebraic b splines and nurbs including classical refinement processes as h p k refinements finally since generalized b splines are not confined to rational functions they behave completely similar to algebraic b splines with respect to differentiation and integration this seems to be of interest in the treatment of some relevant problems

an efficient dense and stable particular elements generation method based on geometry summary in this paper a new discrete elements generation method based on geometry is proposed to fill geometric domains with particles disks or spheres by generating particles each one with a random radius or with a radius calculated from the iteration to ensure no overlaps exist between particles and identifying unstable particles and changing them to stable ones a dense and stable packing can be created a partitioning particle radius interval method and a particle stability inspection and improvement method are introduced to guarantee the algorithm s success and the stability of the particles some packings were created to evaluate the performances of the new method the results showed that the algorithm was very efficient and was able to create isotropic packings of low porosities and large coordinate numbers the partitioning particle radius interval method improved the generation efficiency significantly and increased the packing densities through the comparisons with several existing methods proposed recently the method proposed in this work is found to be more efficient and can fill geometric domains with the lowest porosities in addition the stability of the particles is guaranteed and no complex triangular or tetrahedral mesh is required in particle generation thereby making the new method simpler copyright 2016 john wiley sons ltd

some regularity and convergence results for parabolic hamilton jacobi bellman equations in bounded domains abstract we study parabolic hamilton jacobi bellman hjb equations in bounded domains with strong dirichlet boundary conditions we work under the assumption of the existence of a sufficiently regular barrier function for the problem to obtain well posedness and regularity of a related switching system and the convergence of its components to the hjb equation in particular we show existence of a viscosity solution to the switching system by a novel construction of sub and supersolutions and application of perron s method error bounds for monotone schemes for the hjb equation are then derived from estimates near the boundary where the standard regularisation procedure for viscosity solutions is not applicable and are found to be of the same order as known results for the whole space we deduce error bounds for some common finite difference and truncated semi lagrangian schemes

packing spherical discrete elements for large scale simulations abstract we introduce a new geometric method to generate sphere packings with restricted overlap values sample generation is an important but time consuming step that precedes a calculation performed with the discrete element method dem at present there does not exist any software dedicated to dem which would be similar to the mesh software that exists for finite element methods fem a practical objective of the method is to build very large sphere packings several hundreds of thousands in a few minutes instead of several days as the current dynamic methods do the developed algorithm uses a new geometric procedure to position very efficiently the polydisperse spheres in a tetrahedral mesh the algorithm implemented into yade open dem open source software consists in filling tetrahedral meshes with spheres in addition to the features of the tetrahedral mesh the input parameters are the minimum and maximum radii or their size ratio and the magnitude of authorized overlaps the filling procedure is stopped when a target solid fraction or number of spheres is reached based on this method an efficient tool can be designed for dems used by researchers and engineers the generated packings can be isotropic and the number of contacts per sphere is very high due to its geometric procedure in this paper different properties of the generated packings are characterized and examples from real industrial problems are presented to show how this method can be used the current c version of this packing algorithm is part of yade open dem 20 available on the web https yade dem org

quasi interpolation for linear functional data quasi interpolation has been studied extensively in the literature however most studies of quasi interpolation are usually only for discrete function values or a finite linear combination of discrete function values note that in practical applications more commonly we can sample the linear functional data the discrete values of the right hand side of some differential equations rather than the discrete function values e g remote sensing seismic data etc therefore it is more meaningful to study quasi interpolation for the linear functional data the main result of this paper is to propose such a quasi interpolation scheme error estimate of the scheme is also given in the paper based on the error estimate one can find a quasi interpolant that provides an optimal approximation order with respect to the smoothness of the right hand side of the differential equation the scheme can be applied in many situations such as the numerical solution of the differential equation construction of the lyapunov function and so on respective examples are presented in the end of this paper

continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations using the maximum principle for semicontinuous functions 3 4 we prove a general continuous dependence on the nonlinearities estimate for bounded holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations furthermore we provide existence uniqueness and holder continuity results for bounded viscosity solutions of such equations our results are general enough to encompass hamilton jacobi bellman isaacs s equations of zero sum two player stochastic dierential games an immediate conse quence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations

an efficient algorithm to generate random sphere packs in arbitrary domains particle based methods based on material models using spheres can provide good approximations for many physical phenomena at both the micro and macroscales the point of departure for the simulations in general is a dense arrangement of spherical particles sphere pack inside a given container for generic domains the generation of a sphere pack may be complex and time consuming especially if the pack must comply with a prescribed sphere size distribution and the stability requirements of the simulation the primary goal of this paper is to present an efficient algorithm that is capable of producing packs with millions of spheres following a statistical sphere size distribution inside complex arbitrary domains this algorithm uses a new strategy to ensure that the sphere size distribution is preserved even when large particles are rejected in the growing process the paper also presents numerical results that enable an evaluation of the proposed algorithm

solving partial differential equation by using multiquadric quasi interpolation abstract in this paper we use a kind of univariate multiquadric mq quasi interpolation to solve partial differential equation pde 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 temporal derivative of the dependent variable the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement our numerical experiment includes two examples one is solving viscid burgers equation for initial trapezoidal conditions another is simulating the interaction of two waves travelling in opposite direction from the numerical experiment we can see that the present scheme is valid