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We describe an efficient method for the approximation of functions using radial basis functions (RBFs), and extend this to a solver for boundary value problems on irregular domains. The method is based on RBFs with centers on a regular grid…

Numerical Analysis · Mathematics 2024-03-05 Yiqing Zhou , Daan Huybrechs

Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for big scattered datasets in $n-$dimensional space. It is a non-separable approximation, as it is…

Computational Engineering, Finance, and Science · Computer Science 2018-06-22 Zuzana Majdisova , Vaclav Skala

In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method is benefited from a direct discretization approach and…

Numerical Analysis · Mathematics 2020-10-28 Davoud Mirzaei

This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in…

Numerical Analysis · Mathematics 2024-12-20 Víctor Bayona , Argyrios Petras , Cécile Piret , Steven J. Ruuth

Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness…

Numerical Analysis · Mathematics 2024-03-07 Michele Bucelli , Francesco Regazzoni , Luca Dede' , Alfio Quarteroni

In this paper a numerical meshless method for solving the radiative transfer equations in a slab medium with an isotropic scattering is considered. The method is based on radial basis functions to approximate the solution of an…

Numerical Analysis · Computer Science 2014-08-12 J. A. Rad , S. Kazem , K. Parand

We consider how some methods of uniform and nonuniform interpolation by translates of radial basis functions -- specifically the so-called general multiquadrics -- perform in the presence of certain types of noise. These techniques provide…

Classical Analysis and ODEs · Mathematics 2018-02-14 Jean-Luc Bouchot , Keaton Hamm

In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs)…

Numerical Analysis · Mathematics 2018-11-15 R. Cavoretto , S. De Marchi , A. De Rossi , E. Perracchione , G. Santin

In this paper, we discuss the problem of constructing Radial Basis In this paper, we discuss the problem of constructing Radial Basis Function (RBF)-based Partition of Unity (PU) interpolants that are positive if data values are positive.…

Numerical Analysis · Mathematics 2017-01-26 Alessandra De Rossi , Emma Perracchione

We mainly concerned with a decoupled fractional Laplacian wave equation in this paper. A new time-space domain radial basis function (RBF) collocation method is introduced to solve the fractional wave equation, which describes seismic wave…

Computational Physics · Physics 2018-06-07 Yiran Xu , Jingye Li , Guofei Pang , Zhikai Wang , Xiaohong Chen , Benfeng Wang

Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in d-dimensional space. This approach is useful for a higher…

Numerical Analysis · Computer Science 2018-06-21 Zuzana Majdisova , Vaclav Skala

We introduce and investigate matrix approximation by decomposition into a sum of radial basis function (RBF) components. An RBF component is a generalization of the outer product between a pair of vectors, where an RBF function replaces the…

Machine Learning · Computer Science 2021-06-25 Elizaveta Rebrova , Yu-Hang Tang

Semi-lagrangian schemes for discretization of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for high-dimensional problems…

Numerical Analysis · Mathematics 2023-06-09 Alessandro Alla , Hugo Oliveira , Gabriele Santin

Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since the meshless method does not need an underlying connectivity in the form of control volumes or…

Numerical Analysis · Mathematics 2021-09-15 Shantanu Shahane , Anand Radhakrishnan , Surya Pratap Vanka

Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted…

Numerical Analysis · Mathematics 2024-02-27 Jennifer E. Fromm , Nils Wunsch , Kurt Maute , John A. Evans , Jiun-Shyan Chen

Meshfree methods, including the reproducing kernel particle method (RKPM), have been widely used within the computational mechanics community to model physical phenomena in materials undergoing large deformations or extreme topology…

Numerical Analysis · Mathematics 2025-06-19 Jennifer E. Fromm , John A. Evans , J. S. Chen

In this paper, the accuracy of two mesh-free approximation approaches, the Gravity model and Radial Basis Function, are compared. The two schemes' convergence behaviors prove that RBF is faster and more accurate than the Gravity model. As a…

Machine Learning · Computer Science 2021-03-23 Amirehsan Ghasemi , Kelvin J Msechu , Arash Ghasemi , Mbakisya A. Onyango , Ignatius Fomunung , Joseph Owino

The purpose of this study is to apply some new RBF collocation schemes and recently-developed kernel RBFs to various types of partial differential equation systems. By analogy with the Fasshauer's Hermite interpolation, we recently…

Numerical Analysis · Mathematics 2025-10-20 W. Chen

In this paper we present a new fast and accurate method for Radial Basis Function (RBF) approximation, including interpolation as a special case, which enables us to effectively find the optimal value of the RBF shape parameter. In…

Numerical Analysis · Mathematics 2023-11-09 Roberto Cavoretto , Alessandra De Rossi , Sandro Lancellotti

The accurate robust and efficient transfer of the deformation gradient tensor between meshes of different resolution is crucial in cardiac electromechanics simulations. We present a novel method that combines rescaled localized Radial Basis…

Numerical Analysis · Mathematics 2023-06-02 Michele Bucelli , Francesco Regazzoni , Luca Dede' , Alfio Quarteroni