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Radial Basis Function Methods for Solving Partial Differential Equations-A Review

Gurpreet Singh Bhatia, Geeta Arora
2016 Indian Journal of Science and Technology  
We also discussed the preference of using meshfree methods like RBF over the mesh based methods. Findings: This paper presents a state-of-the-art review of the RBF methods.  ...  Some recent development of RBF approximation in solving PDEs is also discussed. The mathematical formulation of different RBF methods are discussed for better understanding.  ...  He evaluates methods based on different parameters like accuracy, storage and time taken by the method and finds multiquadric (one of the RBF) among the best ones 2 .  ... 
doi:10.17485/ijst/2016/v9i45/105079 fatcat:3thlei3p3fdfpagoizsjgdgi2q

Radial basis function partition of unity methods for pricing vanilla basket options

Victor Shcherbakov, Elisabeth Larsson
2016 Computers and Mathematics with Applications  
Abstract Mesh-free methods based on radial basis function (RBF) approximation are becoming widely used for solving PDE problems.  ...  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.  ...  The penalty method combined with RBFs is a good approach for pricing American options. It allows for removing the free boundary and transforming the problem to a fixed boundary problem.  ... 
doi:10.1016/j.camwa.2015.11.007 fatcat:65hblpgmt5h5xaqgy6g6uyq67q

An RBF–FD method for pricing American options under jump–diffusion models

Majid Haghi, Reza Mollapourasl, Michèle Vanmaele
2018 Computers and Mathematics with Applications  
The radial basis function (RBF) finite difference method is a local method resulting in a sparse linear system in contrast to global RBF-methods which lead to ill-conditioned dense matrix systems.  ...  For the corresponding European option we prove consistency, stability and second-order convergence in a discrete L2-norm.  ...  RMS error Conclusions We presented an efficient numerical method that is second order in time and space to evaluate European and American option prices based on a combination of the RBF-FD method and  ... 
doi:10.1016/j.camwa.2018.08.040 fatcat:abpcfblzpnabxjpoz7tqvnxvpq

A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels

Pankaj K. Mishra, Gregory E. Fasshauer, Mrinal K. Sen, Leevan Ling
2019 Computers and Mathematics with Applications  
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 practice, the use of small parameters in RBF-FD results in ill-conditioning, so that a stable algorithm is required for precise evaluation.  ... 
doi:10.1016/j.camwa.2018.12.027 fatcat:xccl7y2alnb2rieqcojfeecgie

Stable computations with flat radial basis functions using vector-valued rational approximations

Grady B. Wright, Bengt Fornberg
2017 Journal of Computational Physics  
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  ...  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,  ...  Acknowledgments The authors thank Nick Trefethen for many fruitful discussions regarding rational approximations when they were first putting the ideas of this paper together.  ... 
doi:10.1016/j.jcp.2016.11.030 fatcat:jbj6mawi75gfvikymxnukfylie

A Local Radial Basis Function Method for High-Dimensional American Option Pricing Problems

Rafael Company, Vera N. Egorova, Lucas Jodar, Fazlollah Soleymani
2018 Mathematical Modelling and Analysis  
In this work, we apply the local Wendland radial basis function (RBF) for solving the time-dependent multi dimensional option pricing nonlinear PDEs.  ...  Then, it is discussed that the valuation of a multi-asset option up to 4D can be computed using a modified shape parameter algorithm.  ...  Acknowledgement We wish to express our gratitude to anonymous reviewers for their helpful comments which improved the quality of the paper.  ... 
doi:10.3846/mma.2018.008 fatcat:st3zy2p7lbdbdanplsai5uf7tu

Radial basis function partition of unity operator splitting method for pricing multi-asset American options

Victor Shcherbakov
2016 BIT Numerical Mathematics  
Radial basis function partition of unity operator splitting method for pricing multi-asset American options.  ...  Acknowledgements The author would like to thank Marjon Ruijter (Rabobank, Utrecht) for the provided reference solution for the two-dimensional problem, Elisabeth Larsson (Uppsala University) for fruitful  ...  discussions, and Slobodan Milovanović (Uppsala University) for help with visualising results.  ... 
doi:10.1007/s10543-016-0616-y fatcat:36hbov3ehzcxzcnpxbvprvvuau

Basket Option Pricing Under Jump Diffusion Models

Ali Safdari-Vaighani
2017 Zenodo  
The aim of this paper is to show how basket option prices in the jump diffusion models, mainly on the Merton model, can be computed using RBF based approximation methods.  ...  For a test problem, the RBF-PU method is applied for numerical solution of partial integral differential equation arising from the two-asset European vanilla put options.  ...  for RBF based method and FD method.  ... 
doi:10.5281/zenodo.1131169 fatcat:h6lhya4pqvdl7ohzpgfzddmsem

Solving PDEs with radial basis functions

Bengt Fornberg, Natasha Flyer
2015 Acta Numerica  
a major contender also for very large simulations on advanced distributed memory computer systems.  ...  During the last few years, radial basis functions, in particular in their 'local' RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE 'toy problems' to becoming  ...  Historically, one can recognize an evolutionary path FD ⇒ PS ⇒ RBFRBF-FD that starts by extending from FD methods (first applied to PDEs just over a century ago [66] ) to PS methods.  ... 
doi:10.1017/s0962492914000130 fatcat:p3zcgxnpmzawnpvfzmculu2c4q

A Radial Basis Function Scheme for Option Pricing in Exponential Lévy Models

Raymond Brummelhuis, Ron T. L. Chan
2013 Applied Mathematical Finance  
We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE.  ...  Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods.  ...  Acknowledgment This paper is based on the PhD thesis of the second-named author.  ... 
doi:10.1080/1350486x.2013.850902 fatcat:qde6tyi3pngpnmqgea6kwq6m74

On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere

Bengt Fornberg, Cécile Piret
2008 Journal of Computational Physics  
Anticipated future application areas for RBF-based methods in spherical geometries include weather and climate modeling.  ...  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).  ...  We will therefore need a much larger number of terms in the expansions to attain the desired accuracy in the RBF-QR method.  ... 
doi:10.1016/j.jcp.2007.11.016 fatcat:n7obeiyzjfajpdbumzma27m7ji

Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids

Bengt Fornberg, Natasha Flyer
2005 Advances in Computational Mathematics  
To better understand the accuracy that can be obtained, we survey here derivative approximations based on RBFs using a similar Fourier analysis approach that has become the standard way for assessing the  ...  Radial basis function (RBF) interpolation can be very effective for scattered data in any number of dimensions.  ...  More generally (on a periodic domain), RBFs based on φ(r) = r 2n−1 and splines of order 2n − 1 give identical interpolants.  ... 
doi:10.1007/s10444-004-1812-x fatcat:og52nxrkrzanbgv2isuvzuyiry

Comparisons between pseudospectral and radial basis function derivative approximations

B. Fornberg, N. Flyer, J. M. Russell
2009 IMA Journal of Numerical Analysis  
Radial basis functions (RBFs) were introduced about the same time for interpolation on scattered nodes in irregular geometries.  ...  Fourier-based pseudospectral (PS) methods have been used since the 1970s for obtaining spectrally accurate solutions to PDEs in periodic geometries.  ...  A simpler, and maybe more natural option is to use ψ 2 (x − x c , y − y c ) = φ (2 sin 2 ( x−x c 2 ) + sin 2 ( y−y c 2 )). (6.3) While (6.3) can be evaluated directly for all choices of RBFs and values  ... 
doi:10.1093/imanum/drn064 fatcat:d2tuv3agxbdf3nlazxja5u2awi

A Numerical Study of Radial Basis Function Based Methods for Options Pricing under the One Dimension Jump-diffusion Model [article]

Ron T. L. Chan, Simon Hubbert
2011 arXiv   pre-print
The aim of this chapter is to show how option prices in jump-diffusion models can be computed using meshless methods based on Radial Basis Function (RBF) interpolation.  ...  The RBF technique is demonstrated by solving the partial integro-differential equation (PIDE) in one-dimension for the American put and the European vanilla call/put options on dividend-paying stocks in  ...  Fausshauer et al., 2004a; Hon and Mao, 1999; Larsson et al., 2008) RBF-approximation methods with Multiquadric (MQ) as a basis function have been proposed for numerically solving the classical Black and  ... 
arXiv:1011.5650v4 fatcat:vk2zevmcgvek3pauyrmngt3l3i

Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions [article]

Jan Glaubitz, Anne Gelb
2021 arXiv   pre-print
Based on this observation we propose a technique to weakly enforce BCs in RBF methods.  ...  It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs).  ...  The authors would like to thank Simon-Christian Klein for helpful advice.  ... 
arXiv:2102.00967v1 fatcat:5ry5m734xnbk5ciz4musvqjqay
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