Radial Basis Function Approximation Method Research Articles

A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.

Multiquadric radial basis function approximation method and asymptotic numerical method for nonlinear and linear static analysis of single-walled carbon nanotubes

In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Kármán type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.

Two-Grid Iterative Method for a Class of Fredholm Functional Integral Equations based on the Radial Basis Function Interpolation

In this paper, we discuss a two-grid iterative method for solving a class of Fredholm functional integral equations based on the radial basis function interpolation. Firstly, the existence and uniqueness of the solution are proved by Banach fixed point theorem. Secondly, the algorithm and convergence analysis of the radial basis function approximation method is given on the coarse grid. Thirdly, the fine grid iterative solution and convergence results are obtained. Finally, the validity and reliability of the theoretical analysis are verified by two numerical experiments.

A boundary element and radial basis function approximation method for a second order elliptic partial differential equation with general variable coefficients

A numerical method based on boundary integral equation and radial basis function approximation is presented for solving boundary value problems governed by a second‐order elliptic partial differential equation with variable coefficients. The equation arises in the analysis of steady‐state anisotropic heat or mass diffusion in nonhomogeneous media with properties that vary according to general smoothly varying functions of space. The method requires only the boundary of the solution domain to be discretized into elements. To check the validity and accuracy of the numerical solution, some specific problems with known solutions are solved.

A meshless symplectic method for two-dimensional nonlinear Schrödinger equations based on radial basis function approximation

For two-dimensional nonlinear Schrödinger equations, we propose a meshless symplectic method based on radial basis function interpolation. With the method of lines, we first discretize the equation in spatial domain by using the radial basis function approximation method and obtain a finite-dimensional Hamiltonian system. Then appropriate time integrator is employed to derive the full-discrete symplectic scheme. Compared with the classical conservative methods that are only valid on uniform grids, our meshless method is conservative for both uniform grids and nonuniform nodes. The accuracy and conservation properties are analyzed in detail. Several numerical experiments are presented to demonstrate the accuracy and the conservation properties of our approach.

Modeling high temperature deformation characteristics of AA7020 aluminum alloy using substructure-based constitutive equations and mesh-free approximation method

This research was aimed to assess the potential of a radial basis function (RBF) approximation method against the dislocation substructure-based constitutive model in predicting high-temperature deformation behavior of the AA7020 aluminum alloy. Hot compression tests were performed over a range of strain rate of 0.1–100 s−1 and a range of temperature of 350–500 °C up to a strain of 0.6. The hot deformation behavior of the alloy was first described by a substructure kinetic-based constitutive equation, with the effects of strain, strain rate and temperature together with dynamic recovery parameters taken into consideration. A RBF approximation method was then developed to model the flow behavior of the material. The RBF model, as a kind of novel mesh-free function estimation approach, was trained and tested with the obtained datasets from the hot compression tests. The performance of the developed analytical and neural computational models was evaluated using statistical criteria. The results showed that the RBF model was more proficient and accurate in predicting the hot deformation behavior of this aluminum alloy than the substructure-based constitutive model.

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.

Big geo data surface approximation using radial basis functions: A comparative study

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 based on the distance between two points. This method leads to the solution of an overdetermined linear system of equations.In this paper the RBF approximation methods are briefly described, a new approach to the RBF approximation of big datasets is presented, and a comparison for different Compactly Supported RBFs (CS-RBFs) is made with respect to the accuracy of the computation. The proposed approach uses symmetry of a matrix, partitioning the matrix into blocks and data structures for storage of the sparse matrix. The experiments are performed for synthetic and real datasets.

Radial basis function approximations: comparison and applications

• We propose the radial basis function (RBF) approximation of scattered data. • Approximation is based on the distance between a given point and a reference point. • The set of reference points is specified by a user. • Method returns the approximating function which is represented as a sum of M RBFs. • Experimental results show that proposed RBF approximation gives the best results. 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 dimension d > 2, because the other methods require the conversion of a scattered dataset to an ordered dataset (i.e. a semi-regular mesh is obtained by using some tessellation techniques), which is computationally expensive. The RBF approximation is non-separable, as it is based on the distance between two points. This method leads to a solution of linear system of equations (LSE) Ac = h . In this paper several RBF approximation methods are briefly introduced and a comparison of those is made with respect to the stability and accuracy of computation. The proposed RBF approximation offers lower memory requirements and better quality of approximation.

Educating local radial basis functions using the highest gradient of interest in three dimensional geometries

SummaryWe present a novel methodology to effectively localize radial basis function approximation methods in three dimensions. The local scheme requires shape parameter‐dependent functions that can be used to approximate gradients of scattered data and to solve partial differential operators. The optimum shape parameter is obtained from the highest gradient of interest, where a known analytical function, when boundary conditions are not present, or a shape parameter‐free global approximation are used to educate the localized scheme. The later option is applicable to problems where the operator needs to be solved multiple times, like in time evolution or stochastic integration. Past shape parameter's optimizations, for two‐dimensional domains, based on the condition number of the interpolant matrix, were unable to provide satisfactory approximations. The applicability of our method is illustrated in the context of an analytical expression interpolation and during a Ginzburg–Landau relaxation of a free energy functional. In general, the optimum shape parameter depends on geometry, node distribution, and density, whereas the approximation errors decrease as the node density and the local stencil size increase. The effective localization of radial basis functions motivates its use in moving boundary problems and accelerates solutions through sparse matrix solvers. Copyright © 2016 John Wiley & Sons, Ltd.

The evaluation of compound options based on RBF approximation methods

Recently, real options have gained more importance in computational finance studies. It has already been shown that the compound option pricing can be formulated as a two-pass boundary PDE arising from Black–Scholes model. Radial basis function (RBF) as a meshfree approximation method is widely used for numerical study of the time dependent PDEs. In this paper, the aim is to introduce the robust numerical approach based on RBF-QR to compute the price of European compound options such as the popular put on put options. We also extend the proposed approach to American compound option pricing. The numerical experiments will show the efficiency of the performance for European and American compound option with single asset and multi-asset cases.

Computing Deformable Image Registration by Inverse Multiquadric Demons Algorithm

Radiation therapy is the mainstay of cancer treatment. However, significant weight loss and tumor shrinkage during treatment could affect the medical outcome. Advanced forms of adaptive radiotherapy use deformable image registration (DIR) accounting for such anatomical changes. Accurately defining the exact location and extent of the target with respect to surrounding organs at risk are difficult tasks. DIR model is commonly used as an alternative to measure how the tumor changes in size and location. In this paper, a reciprocal Multiquadric radial basis function approximation method is incorporated with the Demons algorithm for solving the DIR model. The proposed scheme could enhance the accuracy and efficiency of the deformable image registration. The numerical performance is compared with the classical DIR Demons algorithms. The result demonstrated that the combination of the reciprocal Multiquadric approximation and the Demons algorithm is useful in image registration.

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

AbstractWe propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

RBFs approximation method for Kawahara equation

In this paper, we apply radial basis function (RBF) method for the numerical solution of Kawahara equation. Derivatives of interpolation are used to approximate the spatial derivatives and the temporal derivative approximated by a low order forward difference scheme. Computational experiments are performed for single soliton, two solitons, and three solitons interaction to examine accuracy of the method. The results obtained by the present method are in good agreement with exact solutions and some earlier work in literature. Accuracy of the method is estimated in terms of the error norms L 2, L ∞ , the three invariants, number of nodes in the domain of influence, time step size, parameter dependent, and parameter independent RBFs.

Radial basis function approximation methods with extended precision floating point arithmetic

Radial basis function (RBF) methods that employ infinitely differentiable basis functions featuring a shape parameter are theoretically spectrally accurate methods for scattered data interpolation and for solving partial differential equations. It is also theoretically known that RBF methods are most accurate when the linear systems associated with the methods are extremely ill-conditioned. This often prevents the RBF methods from realizing spectral accuracy in applications. In this work we examine how extended precision floating point arithmetic can be used to improve the accuracy of RBF methods in an efficient manner. RBF methods using extended precision are compared to algorithms that evaluate RBF methods by bypassing the solution of the ill-conditioned linear systems.

A random variable shape parameter strategy for radial basis function approximation methods

Several variable shape parameter methods have been successfully used in radial basis function approximation methods. In many cases variable shape parameter strategies produced more accurate results than if a constant shape parameter had been used. We introduce a new random variable shape parameter strategy and give numerical results showing that the new random strategy often outperforms both existing variable shape and constant shape strategies.

Digital Total Variation Filtering as Postprocessing for Radial Basis Function Approximation Methods

Digital total variation (DTV) filtering techniques, that originated in the field of image processing, are adapted to postprocess radial basis function approximations of piecewise continuous functions. Through numerical examples, we show that DTV filtering is a fast, robust, postprocessing method that can be used to remove Gibbs oscillations while sharply resolving discontinuities. The method is applicable for arbitrarily located data points. A postprocessing method for scattered data had not been given previously.

Integrated multiquadric radial basis function approximation methods

Promising numerical results using once and twice integrated radial basis functions have been recently presented. In this work we investigate the integrated radial basis function (IRBF) concept in greater detail, connect to the existing RBF theory, and make conjectures about the properties of IRBF approximation methods. The IRBF methods are used to solve PDEs.

A comparison of freehand three-dimensional ultrasound reconstruction techniques

Three-dimensional freehand ultrasound imaging produces a set of irregularly spaced B-scans, which are typically reconstructed on a regular grid for visualization and data analysis. Most standard reconstruction algorithms are designed to minimize computational requirements and do not exploit the underlying shape of the data. We investigate whether an approximation with splines holds any promise as a better reconstruction method. A radial basis function approximation method is implemented and compared with three standard methods. While the radial basis approach is computationally expensive, it produces accurate reconstructions without the kind of visible artefacts common with the standard methods. The other potential advantages of radial basis functions, such as the direct computation of derivatives, make further investigation worthwhile.