Global Radial Basis Functions Research Articles

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 orders) and time (up to 4th order) for some model problems in two dimensional irregular geometries.

A New Stable Local Radial Basis Function Approach for Option Pricing

In this paper, we develop a new local meshless approach based on radial basis functions (RBFs) to solve the Black---Scholes equation. The global RBF approximations derived from conventional global collocation method usually lead to ill-conditioned matrices. The new scheme employs the idea of the finite difference method to localize them. It removes the difficulty of ill-conditioning of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. As well as it is fast and it produces accurate results as shown in numerical experiments.

A greedy meshless local Petrov-Galerkin methodbased on radial basis functions

The meshless local Petrov–Galerkin (MLPG) method with global radial basis functions (RBF) as trial approximation leads to a full final linear system and a large condition number. This makes MLPG less efficient when the number of data points is increased. We can overcome this drawback if we avoid using more points from the data site than absolutely necessary. In this article, we equip the MLPG method with the greedy sparse approximation technique of (Schaback, Numercail Algorithms 67 (2014), 531–547) and use it for numerical solution of partial differential equations. This scheme uses as few neighbor nodal values as possible and allows to control the consistency error by explicit calculation. Whatever the given RBF is, the final system is sparse and the algorithm is well-conditioned. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 847–861, 2016

Estimation of river pollution source using the space-time radial basis collocation method

River contaminant source identification problems can be formulated as an inverse model to estimate the missing source release history from the observed contaminant plume. In this study, the identification of pollution sources in rivers, where strong advection is dominant, is solved by the global space-time radial basis collocation method (RBCM). To search for the optimal shape parameter and scaling factor which strongly determine the accuracy of the RBCM method, a new cost function based on the residual errors of not only the observed data but also the specified governing equation, the initial and boundary conditions, was constructed for the k-fold cross-validation technique. The performance of three global radial basis functions, Hardy's multiquadric, inverse multiquadric and Gaussian, were also compared in the test cases. The numerical results illustrate that the new cost function is a good indicator to search for near-optimal solutions. Application to a real polluted river shows that the source release history is reasonably recovered, demonstrating that the RBCM with the k-fold cross-validation is a powerful tool for source identification problems in advection-dominated rivers.

Localized radial basis function scheme for multidimensional transient generalized newtonian fluid dynamics and heat transfer

The local radial basis function (RBF) scheme is developed to simulate 2D and 3D heat transfer and flow dynamics of generalized Newtonian fluids (GNF). The local RBF scheme is a meshless numerical method based on radial basis functions with localized technique. The procedure of localization reduces the computational cost more efficiently than the traditional global RBF method. This meshless method does not require mesh generation, numerical integration and only needs point collocation. Besides, it is very easy to interpolate physical values and its derivatives everywhere in the domain. We consider one isothermal and three non-isothermal multidimensional transient GNF fluid and heat problems in this paper. The dynamic viscosity of the GNF is specified as two different models: the power law model (temperature independent) or Cross model (temperature dependent). The viscous heating is also considered in this work. Numerical results show that the local RBF scheme is stable and accurate as far as the four tested cases are concerned.

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

Meshfree methods based on radial basis function (RBF) approximation are becoming widely used for solving PDE problems. They are flexible with respect to the problem geometry and highly accurate. A disadvantage of these methods is that the linear system to be solved becomes dense for globally supported RBFs. A remedy is to introduce localisation techniques such as partition of unity. RBF partition of unity methods (RBF–PUM) allow for a significant sparsification of the linear system and lower the computational effort. 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. In order to price American options we employ a penalty approach. A penalty term, suitable for American multi-asset call options, has been designed. RBF–PUM is shown to be competitive compared with a finite difference method and a global RBF method. It is as accurate as the global RBF method, but significantly faster. The results for RBF–PUM look promising for extension to higher-dimensional problems.

Preconditioning for Radial Basis Function Partition of Unity Methods

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF---PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner.

Global Versus Localized RBF Meshless Methods for Solving Incompressible Fluid Flow with Heat Transfer

Global and localized radial basis function (RBF) meshless methods are compared for solving viscous incompressible fluid flow with heat transfer using structured multiquadratic RBFs. In the global approach, the collocation is made globally over the whole domain, so the size of the discretization matrices scales as the number of the nodes in the domain. The localized meshless method uses a local collocation defined over a set of overlapping domains of influence. Only small systems of linear equations need to be solved for each node. The computational effort thus grows linearly with the number of nodes—the localized approach is slightly more expensive on serial processors, but is highly parallelizable. Numerical results are presented for three benchmark problems—the lid-driven cavity, natural convection within an enclosure, and forced convective flow over a backward-facing step—and results are compared with the finite-element method (FEM) and experimental data.

Tailoring the Fundamental Frequency of Laminated Composite Panels Using Material Properties

Optimization of material properties is performed to maximize the fundamental frequency of the laminated composite panels by means of the genetic algorithm. The global radial basis function collocation method is used to calculate the fundamental frequency of clamped laminated composite panels. In this paper, the objective function of optimization problem is the maximum fundamental frequency; optimization variables are material properties of laminated panels. The results for the optimal material properties and the maximum fundamental frequencies of the 2-layer plates are presented to verify the validity of present method.

Layer Thickness Optimization of Laminated Composite Plates to Maximize the First-Order Natural Frequency

Layer thickness optimization is performed to maximize the first-order natural frequency of clamped laminated composite plates using the genetic algorithm and meshless global radial basis function collocation method. The objective function of optimization problem is the maximum first-order natural frequency; optimization variables are layer thickness. The optimal layer thickness and the maximum first-order natural frequency of the 2-layer plates are presented to demonstrate the accuracy of present method.

Shape Parameter Optimization of Gaussian Radial Basis Function Using Genetic Algorithm for Natural Frequencies of Laminated Composite Plates

Natural frequencies of simply supported laminated composite plates are calculated by the meshless global collocation method based on Gaussian radial basis function. The accuracy of meshless global radial basis function collocation method depends on the choice of shape parameter of radial basis function. In present paper, the shape parameter of Gaussian radial basis function is optimized using the genetic algorithm. Gaussian radial basis function with optimal shape parameter is utilized to analyze the natural frequencies of simply supported laminated composite plates. The present results are compared with the results of available literatures which verify the accuracy of present method.

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF–PUM) proposed here. The objective of this paper is to establish that RBF–PUM is viable for parabolic PDEs of convection–diffusion type. The stability and accuracy of RBF–PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection–diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF–PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF–PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

Improving variable-fidelity modelling by exploring global design space and radial basis function networks for aerofoil design

The global variable-fidelity modelling (GVFM) method presented in this article extends the original variable-complexity modelling (VCM) algorithm that uses a low-fidelity and scaling function to approximate a high-fidelity function for efficiently solving design-optimization problems. GVFM uses the design of experiments to sample values of high- and low-fidelity functions to explore global design space and to initialize a scaling function using the radial basis function (RBF) network. This approach makes it possible to remove high-fidelity-gradient evaluation from the process, which makes GVFM more efficient than VCM for high-dimensional design problems. The proposed algorithm converges with 65% fewer high-fidelity function calls for a one-dimensional problem than VCM and approximately 80% fewer for a two-dimensional numerical problem. The GVFM method is applied for the design optimization of transonic and subsonic aerofoils. Both aerofoil design problems show design improvement with a reasonable number of high- and low-fidelity function evaluations.

Solving an eigenvalue problem on a periodic domain using a radial basis function finite difference scheme

Local radial basis functions (RBFs) are becoming increasingly popular as an alternative to global RBFs, as the latter suffer from ill-conditioning. In this paper, a local meshless method based on RBFs in a finite-difference (FD) mode with better conditioned matrices has been developed for solving an eigenvalue problem with a periodic domain. Through numerical experiments, we examine the accuracy of the method as a result of variation in the number and layout of nodes in the domain and the effects of shape parameter, using various globally supported RBFs. The presented scheme has been validated on two different types of nodal arrangement, namely uniform and non-uniform node distributions. The results obtained from the method are found to be in good agreement with the benchmark analytical solutions. In addition, a higher-order RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for solving the eigenvalue problem with a periodic domain. Tests show that both accuracy and convergence order can be improved dramatically by using higher-order RBF-FD formulae, which converge at a rate of O(h8.5) compared to the standard-order method which converges as O(h4.3) for uniformly distributed nodes with spacing h.

The localized RBFs collocation methods for solving high dimensional PDEs

Abstract In this paper we present a localized meshless method using radial basis functions (RBFs) for solving up to six dimensional problems. To improve the difficulty of selecting a shape parameter of RBF-MQ, a normalized scheme is introduced. We also make a comparison between the global and local RBF methods in terms of stability and accuracy. To demonstrate the applicability of the localized RBF method for high dimensional problems, two numerical examples with Dirichlet boundary conditions are given.

Estimation of Lyapunov spectrum and model selection for a chaotic time series

The estimation of the Lyapunov spectrum for a chaotic time series is discussed in this study. Three models: the local linear (LL) model; the local polynomial (LP) model and the global radial basis function (RBF) model, are compared for estimating the Lyapunov spectrum in this study. The number of neighbors for training the LL model and the LP model; the number of centers for building the RBF model, have been determined by the generalized degree of freedom for a chaotic time series. The above models have been applied to three artificial chaotic time series and two real-world time series, the numerical results show that the model-chosen LL model provides more accurate estimation than other models for clean data set while the RBF model behaves more robust to noise than other models for noisy data set.

An adaptive‐hybrid meshfree approximation method

SUMMARYIt is now commonly agreed that the global radial basis functions (GRBF) method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of ill‐conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriately shaped parameters were minimized. In contrast, the compactly supported radial basis functions (CSRBF) are more relaxing on the smoothness requirements. By settling with the algebraic order of convergence only, the CSRBF method, provided the support radii are properly chosen, can approximate functions with less smoothness. The reality is that end users must know the functions to be approximated a priori to decide which method to be used; this is not practical if one is solving a time‐evolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later. In this paper, we propose a hybrid algorithm making use of both GRBF and CSRBF with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Second, we apply the GRBFs (with adaptive subspace selection) on the adaptively generated data sites, and lastly, the CSRBF (with adaptive support selection) that can be used as a blackbox algorithm for robust approximations to a wider class of functions and for solving PDEs. Copyright © 2011 John Wiley & Sons, Ltd.

Magnetic Field Computation Using Compact Support Radial Basis

This paper describes a novel approach of using compact support radial basis functions for solving low frequency electromagnetic problems. The objective is to study its effectiveness compared to the techniques that use global radial basis functions (RBFs).

Meshless physical simulation of semiconductor devices using a wavelet-based nodes generator

This paper describes a meshless method with wavelet-based nodes for the two-dimensional time-dependent simulation of semiconductor devices. In this method the solution is approximated using global radial basis functions (RBF) and distributed wavelet-generated points. This allows the computation of problems with complex-shaped boundaries and forming fine and coarse points abundance in locations where variable solutions change rapidly and slowly, respectively. The method is suitable for the semiconductor part of very time consuming global modeling of microwave/millimeter wave circuits due to a large reduction of number of nodes with an acceptable results.

Dimension-splitting data points redistribution for meshless approximation

To better approximate nearly singular functions with meshless methods, we propose a data points redistribution method extended from the well-known one-dimensional equidistribution principle. With properly distributed data points, nearly singular functions can be well approximated by linear combinations of global radial basis functions. The proposed method is coupled with an adaptive trial subspace selection algorithm in order to reduce computational cost. In our numerical examples, clear exponential convergence (with respect to the numbers of data points) can be observed.