A radial basis function partition of unity collocation method for convection-diffusion equations ⋆

Abstract Numerical solution of multi-dimensional PDEs is a challenging problem with respect to computational cost and memory requirements, as well as regarding representation of realistic geometries and adaption to solution features. Meshfree methods such as global radial basis function approximation have been successfully applied to several types of problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we instead propose to use a locally supported RBF collocation method based on a partition of unity approach to numerically solve time-dependent PDEs. We investigate the stability and accuracy of the method for convection-diffusion problems in two space dimensions as well as for an American option pricing problem. The numerical experiments show that we can achieve both spectral and high-order algebraic convergence for convection-diffusion problems, and that we can reduce the computational cost for the option pricing problem by adapting the node layout to the problem characteristics

A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure

Radial basis functions with partition of unity method for American options with stochastic volatility

In this article, we price American options under Heston's stochastic volatility model using a radial basis function (RBF) with partition of unity method (PUM) applied to a linear complementary formulation of the free boundary partial differential equation problem. RBF-PUMs are local meshfree methods that are accurate and flexible with respect to the problem geometry and that produce algebraic problems with sparse matrices which have a moderate condition number. Next, a Crank-Nicolson time discretisation is combined with the operator splitting method to get a fully discrete problem. To better control the computational cost and the accuracy, adaptivity is used in the spatial discretisation. Numerical experiments illustrate the accuracy and efficiency of the proposed algorithm

RBF approximation by partition of unity for valuation of options under exponential Lévy processes

International audienceThe prices of some European and American-style contracts on assets driven by a class of Markov processes containing, in particular, L\'{e}vy processes of pure jump type with infinite jump activity, are obtained numerically, as solutions of the partial integro-differential equations (PIDEs) they satisfy. This paper overcomes the ill-conditioning inherent in global meshfree methods by using localized RBF approximations known as the RBF partition of unity (RBF-PU) method for (PIDEs) arising in option pricing problems in L\'{e}vy driven assets. Then, Crank-Nicolson, LeapFrog (CNLF) is applied for time discretization. We treat the local term using an implicit step, and the nonlocal term using an explicit step, to avoid the inversion of the nonsparse matrix. For dealing with early exercise feature of American option and solving free boundary problem we use the implicit-explicit method combined with a penalty method. Efficiency and practical performance are demonstrated by numerical experiments for pricing European and American contracts. Suggested reviewers Mohan Kadalbajoo, Mehdi Dehghan, simon hubbert Submission Files Included in this PDF File Name [File Type] cover letter.pdf [Cover Letter] Research Highlights.pdf [Highlights] Levy_Fereshtian.pdf [Manuscript File] Submission Files Not Included in this PDF File Name [File Type] Paper_Levy.zip [LaTeX Source File] To view all the submission files, including those not included in the PDF, click on the manuscript title on your EVISE Homepage, then click 'Download zip file'

Pricing Financial Derivatives using Radial Basis Function generated Finite Differences with Polyharmonic Splines on Smoothly Varying Node Layouts

In this paper, we study the benefits of using polyharmonic splines and node layouts with smoothly varying density for developing robust and efficient radial basis function generated finite difference (RBF-FD) methods for pricing of financial derivatives. We present a significantly improved RBF-FD scheme and successfully apply it to two types of multidimensional partial differential equations in finance: a two-asset European call basket option under the Black--Scholes--Merton model, and a European call option under the Heston model. We also show that the performance of the improved method is equally high when it comes to pricing American options. By studying convergence, computational performance, and conditioning of the discrete systems, we show the superiority of the introduced approaches over previously used versions of the RBF-FD method in financial applications

RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure