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In this paper a novel variable selection method based on Radial Basis Function (RBF) neural networks and genetic algorithms is presented. The fuzzy means algorithm is utilized as the training method for the RBF networks, due to its inherent speed, the deterministic approach of selecting the hidden node centers and the fact that it involves only a single tuning parameter. The trade-off between the accuracy and parsimony of the produced model is handled by using Final Prediction Error criterion, based on the RBF training and validation errors, as a fitness function of the proposed genetic algorithm. The tuning parameter required by the fuzzy means algorithm is treated as a free variable by the genetic algorithm. The proposed method was tested in benchmark data sets stemming from the scientific communities of time-series prediction and medicinal chemistry and produced promising results.

The numerical solutions of the unsteady transient-convective diffusion problems are investigated by using multiquadric (MQ) and thin-plate spline (TPS) radial basis functions (RBFs) based on mesh-free collocation methods with global basis functions. The results of radial basis functions are compared with the mesh-dependent boundary element and finite difference methods as well as the analytical solution for high Péclet numbers. It is reported that for low Péclet numbers, MQ-RBF provides excellent agreement, while for high Péclet numbers, TPS-RBF is better than MQ-RBF.

The moving boundary conditions are implemented into the Fourier pseudo-spectral solution of the two-dimensional incompressible Navier–Stokes equations (NSE) in the vorticity-velocity form, using the radial basis functions (RBF). Without explicit definition of an external forcing function, the desired immersed boundary conditions are imposed by direct modification of the convection and diffusion terms. At the beginning of each time-step the solenoidal velocities, satisfying the desired moving boundary conditions, along with a modified vorticity are obtained and used in modification of the convection and diffusion terms of the vorticity evolution equation. Time integration is performed by the explicit fourth-order Runge–Kutta method and the boundary conditions are set at the beginning of each sub-step. The method is applied to a couple of moving boundary problems and more than second-order of accuracy in space is demonstrated for the Reynolds numbers up to Re = 550. Moreover, performance of the method is shown in comparison with the classical Fourier pseudo-spectral method.

In this paper, a meshless spectral radial point interpolation (MSRPI) method using weighted $\theta$-scheme is formulated for the numerical solutions of a class of nonlinear Kawahara-type evolutionary equations. The formulated method is applied for simulation of single and double solitary waves motion, wave generation and oscillatory shock waves propagation. Quality of approximation is measured via discrete $L_{\infty }$, $L_2$ and $L_{\mathrm{\text{rms}}}$ error norms. Three invariant quantities corresponding to mass, momentum and energy are also computed for the method validation. Stability analysis of the proposed method is briefly discussed and verified computationally. Comparison of the obtained results are made with other existing results in the literature revealing the method superiority.

A space-time collocation method (STCM) using asymptotically-constant basis functions is proposed and applied to the quantum Hamiltonian constraint for a loop-quantized treatment of the Schwarzschild interior. Canonically, these descriptions take the form of a partial difference equation (PDE). The space-time collocation approach presents a computationally efficient, convergent, and easily parallelizable method for solving this class of equations, which is the main novelty of this study. Results of the numerical simulations will demonstrate the benefit from a parallel computing approach; and show general flexibility of the framework to handle arbitrarily-sized domains. Computed solutions will be compared, when applicable, to a solution computed in the conventional method via iteratively stepping through a predefined grid of discrete values, computing the solution via a recursive relationship.

In this paper, we present a numerical meshless method, which is based on the radial basis functions (RBF), to solve the morphology-dependent resonance (MDR) modes of a dielectric micro-sphere. To check the accuracy of the considered RBF method, we calculated the transverse electric (TE) and transverse magnetic (TM) modes of a perfect sphere that has an analytical characteristic equation. Besides, we employed the traditional mesh-dependent finite difference (FD) method to study the same problem. When we compared the results obtained by the three methods, we observe that the RBF method’s results are in very good agreement with the analytical and FD method’s results. We conclude that the presented meshless method may also be applicable to prolate and oblate spheroidal geometries, which do not have an analytical equation to calculate the modes.

We describe experiments with a versatile pictorial prototype-based learning scheme for 3-D object recognition. The Generalized Radial Basis Function (GRBF) scheme seems to be amenable to realization in biophysical hardware because the only kind of computation it involves can be effectively carried out by combining receptive fields. Furthermore, the scheme is computationally attractive because it brings together the old notion of a "grandmother" cell and the rigorous approximation methods of regularization and splines.

It has been argued that Lyapunov exponents as a measure of predictability are of limited value because they only provide a global average. Characterizing an attractor by a distribution of times for initial uncertainties to increase by a factor of q has been suggested as a more useful alternative. These have found favor in some applications, despite assumptions of the fictitious perfect model scenario. Here, an electronic circuit, which offers a good test-bed for addressing predictability in the imperfect model scenario, is presented. A novel measure of predictability is presented and implications of model imperfection on characterizing the dynamics of chaotic systems are discussed. It is a case for a multiple model approach.

We describe a heuristic method for reconstructing a region in the plane from a noisy sample of points. The method uses radial basis functions with Gaussian kernels to compute a fuzzy membership function which provides an implicit approximation for the region. We also evaluate our reconstruction method for several sampling conditions.

In this paper, a BEM-based meshless method is developed for buckling analysis of elastic plates with various boundary conditions that include elastic supports and restraints. The proposed method is based on the concept of the Analog Equation Method (AEM) of Katsikadelis. According to this method, the original eigenvalue problem for a governing differential equation of buckling is replaced by an equivalent plate bending problem subjected to an appropriate fictitious load under the same boundary conditions. The fictitious load is established using a technique based on BEM and approximated by using the radial basis functions. The eigenmodes of the actual problem are obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The method has all the advantages of the pure BEM. To validate its effectiveness, accuracy as well as applicability of the proposed method, numerical results of various problems are presented.

Ray tracing of implicit surfaces based on radial basis functions can demand high computational cost in the presence of a large number of radial centers. Recently, it was presented the least squares hermite radial basis functions (LS-HRBF) Implicits, a method for implicit surface reconstruction from Hermitian data (points equipped with their normal vectors) which makes use of iterative center selection in order to reduce the number of centers. In the present work, we propose an antialiazed sphere tracing algorithm fully implemented in OpenGL Shader Language for ray tracing LS-HRBF Implicits, which exploits a regular partition of unity for strong parallelization. We show that interactive frame rates can be achieved for surfaces composed of thousands of centers even when rendering effects such as cube mapping, soft shadows and ambient occlusion are used.

Medical image registration is the process of aligning two or more images in such a way that its anatomical structures properly overlap each other in a common spatial domain and resultant 3D images can be used for diagnosis and therapy by surgeons. A number of nonlinear methods have been developed for inter-subject and intra-subject 3D medical image registration. This paper is a part of research experiments which uses the Fast Radial Basis Function (RBF) technique for nonrigid registration of 3D medical images. The technique is a point-based registration evaluation algorithm which registers MR or CT images in less than a second with no compromise on accuracy as compared to standard RBF-based methods. Further we demonstrate that the accuracy of the registration improves when using increasingly more salient feature points (i.e. point landmarks and a few external curves) without affecting the speed of the algorithm. External curves are extracted using a combined watershed and active contours algorithm. Our results show that both accuracy and speed of the Fast Radial Basis algorithm is improved on intra-subject registration of MR image datasets obtained from the Vanderbilt Database as compared to the standard competing methods.

This paper describes the Fast Radial Basis Function (RBF) method for cardiac motion tracking in 3D CT using non-rigid medical image registration based on parameterized (regular) surfaces. The technique is a point-based registration evaluation algorithm which does register 3D MR or CT images in real time. We first extract the surface of the whole heart 3D CT and its contrast enhanced part (left ventricle (LV) blood cavity) of each dataset with a semiautomatic contouring and a fully-automatic triangulation method followed by a global surface parameterization and optimization algorithm. In second step, a set of registration experiments are run to calculate the deformation field at various phases of cardiac motion or cycle from CT images, which results into significant deformation during each phase of a cycle. The surface points of the whole heart and LV are used to register the source systole image to various diastole target images taken at different phases during a heart beat. Our registration accuracy improves with the increase in number of salient feature points (i.e. optimized parameterized surfaces) and it has no effect on the speed of the algorithm (i.e. still less than a second). The results show that the target registration error is less than 3mm (2.53) and the performance of the Fast RBF algorithm is less than a second using a whole heart CT dataset of a single patient taken over the course of the entire cardiac cycle. At the end, the results for recovery (or analysis) of bigger deformation in heart CT images using the Fast RBF algorithm is compared to the state-of-the-art Free Form Deformation (FFD) registration technique. It is proved that the Fast RBF method is performing better in speed and slightly less accurate than the FFD (when measured in terms of NMI) due to iterative nature of the latter.

The radial basis functions meshfree method is a research method for thin plate problem which has gradually developed into a more mature meshfree method. It includes finite element, radial basis functions meshfree collocation method, etc. In this paper, we introduce the multilevel radial basis function collocation method for the fourth-order thin plate problem. We use nonsymmetric Kansa multilevel radial basis function collocation method to solve the fourth-order thin plate problem. Two numerical examples based on Wendland’s $C^6$ and $C^8$ functions are given to examine that the convergence of the multilevel radial basis function collocation method which is good for solving the fourth-order thin plate problem.

A study of free vibration of Timoshenko beams and Mindlin plates is presented whereby the analysis is based on the radial basis function method. Various approaches for computing eigenvalues using radial basis functions are also discussed together with the numerical results for various thickness-to-length ratios.

A Backus–Gilbert approximation method is introduced in this paper as a tool for numerically solving initial-boundary value problems. The formulation of the method with its connection to the standard moving least-squares formulation will be given along with some numerical examples including a numerical solution to the viscous nonlinear Burgers equation in two-dimensions. In addition, we highlight some of the main advantages of the method over previous numerical methods based on meshless collocation approximation in order to validate its robust approximating power and easy handling of initial-boundary value problems.

In this paper, we propose a simple and direct numerical procedure to obtain particular solutions for various types of differential equations. This procedure employs the power series expansion of a differential operator. Chebyshev polynomials are selected as basis functions for the approximation of the inhomogeneous terms of the given partial differential equation. This numerical scheme provides a highly efficient and accurate approximation for the evaluation of a particular solution for a variety of classes of partial differential equations. To demonstrate the effectiveness of the proposed scheme, we couple the method of fundamental solutions to solve a modified Helmholtz equation with irregular boundary configuration. The solutions were observed to have high accuracy.

It has been proven by the authors that both the upper and lower bounds in energy norm of the exact solution to elasticity problems can now be obtained by using the fully compatible finite element method (FEM) and linearly conforming point interpolation method (LC-PIM). This paper examines the upper bound property of the linearly conforming radial point interpolation method (LC-RPIM), where the Radial Basis Functions (RBFs) are used to construct shape functions and node-based smoothed strains are used to formulate the discrete system equations. It is found that the LC-RPIM also provides the upper bound of the exact solution in energy norm to elasticity problems, and it is much sharper than that of LC-PIM due to the decrease of stiffening effect. An effective procedure is also proposed to determine both upper and lower bounds for the exact solution without knowing it in advance: using the LC-RPIM to compute the upper bound, using the standard fully compatible FEM to compute the lower bound based on the same mesh for the problem domain. Numerical examples of 1D, 2D and 3D problems are presented to demonstrate these important properties of LC-RPIM.

Recently, particular solutions using radial basis functions have been used as a basis for solving inhomogeneous partial differential equations as a one-stage approach without the need of finding homogeneous solutions. In this paper, we adopt a newly developed adaptive greedy algorithm to enhance the performance of the one-stage method and alleviate the difficulty of ill-conditioning of the resultant matrix. To demonstrate the effectiveness of coupling these two methods, we give two 3D examples with excellent numerical results.

A new version of the method of approximate particular solutions (MAPSs) using radial basis functions (RBFs) has been proposed for solving a general class of elliptic partial differential equations. In the solution process, the Laplacian is kept on the left-hand side as a main differential operator. The other terms are moved to the right-hand side and treated as part of the forcing term. In this way, the close-form particular solution is easy to obtain using various RBFs. The numerical scheme of the new MAPSs is simple to implement and yet very accurate. Three numerical examples are given and the results are compared to Kansa's method and the method of fundamental solutions.