Radial Basis Function Approximation Research Articles

A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems

A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite-type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff's plate theory. The degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions. Numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency, accuracy, and robustness of the method. Copyright © 2005 John Wiley & Sons, Ltd.

Solving biharmonic problems with scattered-point discretization using indirect radial-basis-function networks

This paper reports a new indirect radial-basis-function (RBF) collocation method for numerically solving biharmonic boundary-value problems. The RBF approximations are constructed through an integration process. The new feature here is that integration constants are eliminated pointwise using the prescribed boundary conditions rather than employed as network weights. This treatment of integration constants is particularly well suited for the case of discretizing the governing differential equations on a set of scattered points. The proposed method is verified through the solution of benchmark test problems. Accurate results and high convergence rates are obtained.

A meshless Galerkin method for Dirichlet problems using radial basis functions

In this paper, a numerical method is given for partial differential equations, which combines the use of Lagrange multipliers with radial basis functions. It is a new method to deal with difficulties that arise in the Galerkin radial basis function approximation applied to Dirichlet (also mixed) boundary value problems. Convergence analysis results are given. Several examples show the efficiency of the method using TPS or Sobolev splines.

Dual bases and discrete reproducing kernels: a unified framework for RBF and MLS approximation

Moving least squares (MLS) and radial basis function (RBF) methods play a central role in multivariate approximation theory. In this paper we provide a unified framework for both RBF and MLS approximation. This framework turns out to be a linearly constrained quadratic minimization problem. We show that RBF approximation can be considered as a special case of MLS approximation. This sheds new light on both MLS and RBF approximation. Among the new insights are dual bases for the approximation spaces and certain discrete reproducing kernels.

RADIAL BASIS FUNCTION-ENHANCED DOMAIN-FREE DISCRETIZATION METHOD AND ITS APPLICATIONS

The recently proposed domain-free discretization (DFD) method is based on the Lagrange interpolation and polynomial-based differential quadrature (PDQ) method. In this article, the radial basis function (RBF) approximation is used in the DFD method as the interpolation scheme for function approximation, and the RBF-DQ method is applied to derivative approximation. The new variant of DFD method is then applied to simulation of natural convection in both concentric and eccentric annuli between inner circular and outer square cylinders. It is found from the numerical examples that the usage of radial basis function can greatly enhance the DFD method in terms of the computational stability and convergence speed.

Computing eigenmodes ofelliptic operators using radial basis functions

Radial basis function (RBF) approximations have been successfully used to solve boundary-value problems numerically. We show that RBFs can also be used to compute eigenmodes of elliptic operators. Particular attention is given to the Laplacian operator in two dimensions, including techniques to avoid degradation of the solution near the boundaries. For regions with corner singularities, special functions must be added to the basis to maintain good convergence. Numerical results compare favorably to basic finite-element methods.

Comparison of radial basis function approximation techniques

This paper compares three different radial basis function neural networks, as well as the diffuse element method, according to their ability of approximation. This is very useful for the optimization of electromagnetic devices. Tests are done on several analytical functions and on the TEAM workshop problem 25.

Polynomial particular solutions for certain partial differential operators

AbstractIn this article, we consider a variant of the Dual Reciprocity Method (DRM) for solving boundary value problems based on approximating source terms by polynomials other than the traditional basis functions. The use of pseudo‐spectral approximations and symbolic methods enables us to obtain highly accurate results without solving the often ill‐conditioned equations that occur when radial basis function approximations are used. When the given partial differential equation is either Poisson's equation or an inhomogeneous Helmholtz‐type equation, we are able to obtain either closed form particular solutions or efficient recursive algorithms. Using the particular solutions, we convert the inhomogeneous equations to homogeneous. The resulting homogeneous equations are then amenable to solution by boundary‐type methods such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Using the MFS, we provide numerical solutions to a variety of boundary value problems in R2 and R3. Using this approach, we can achieve high accuracy with a modest number of interpolation and collocation points. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 112–133, 2003

Adaptive dynamic programming

Unlike the many soft computing applications where it suffices to achieve a "good approximation most of the time," a control system must be stable all of the time. As such, if one desires to learn a control law in real-time, a fusion of soft computing techniques to learn the appropriate control law with hard computing techniques to maintain the stability constraint and guarantee convergence is required. The objective of the paper is to describe an adaptive dynamic programming algorithm (ADPA) which fuses soft computing techniques to learn the optimal cost (or return) functional for a stabilizable nonlinear system with unknown dynamics and hard computing techniques to verify the stability and convergence of the algorithm. Specifically, the algorithm is initialized with a (stabilizing) cost functional and the system is run with the corresponding control law (defined by the Hamilton-Jacobi-Bellman equation), with the resultant state trajectories used to update the cost functional in a soft computing mode. Hard computing techniques are then used to show that this process is globally convergent with stepwise stability to the optimal cost functional/control law pair for an (unknown) input affine system with an input quadratic performance measure (modulo the appropriate technical conditions). Three specific implementations of the ADPA are developed for 1) the linear case, 2) for the nonlinear case using a locally quadratic approximation to the cost functional, and 3) the nonlinear case using a radial basis function approximation of the cost functional; illustrated by applications to flight control.

Observations on the behavior of radial basis function approximations near boundaries

RBF approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of PDEs. RBFs allow arbitrarily scattered data, generalize easily to several space dimensions, and can be spectrally accurate. However, accuracy degradations near boundaries in many cases severely limit the utility of this approach. With that as motivation, this study aims at gaining a better understanding of the properties of RBF approximations near the ends of an interval in 1-D and towards edges in 2-D.

Numerical Integration of Harmonic Functions with Restricted Sampling Data

Based on the idea of kriging and the radial basis function approximation, we develop in this paper a numerical scheme to integrate harmonic functions with restricted sampling data. To be more precise, the integration is performed by using the function values which are given as discrete sampling data on only part of the boundary. These problems often arise from non-destructive evaluation techniques in the engineering industry. The existence and uniqueness of the solution and the error estimation for the proposed numerical scheme are also discussed. Several numerical experimental results are presented for the verification on the accuracy and convergence of the method.

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.

Radial basis function approximations as smoothing splines

Radial basis function methods for interpolation can be interpreted as roughness-minimizing splines. Although this relationship has already been established for radial basis functions of the form g( r)= r α and g(r)=r α log (r) , it is extended here to include a much larger class of functions. This class includes the multiquadric g( r)=( r 2+ c 2) 1/2 and inverse multiquadric g( r)=( r 2+ c 2) −1/2 functions as well as the Gaussian exp (−r 2/D) . The crucial condition is that the Fourier transform of g(| x|) be positive, except possibly at the origin. The appropriate measure of roughness is defined in terms of this Fourier transform. To allow for possibility of noisy data, the analysis is presented within the general framework of smoothing splines, of which interpolation is a special case. Two diagnostic quantities, the cross-validation function and the sensitivity, indicate the accuracy of the approximation.

Geometric selection of centers for radial basis function approximations involved in intensive computer simulations

A method for the selection of centers for radial basis function (RBF) approximation is introduced, which reduces the computational cost of the evaluation of the approximating function. The method takes into consideration: 1. the geometric information (arc length and curvature) of the approximating RBF expansion with all data points as centers, in order to retain as centers just those data points that have relevant geometric characteristics; 2. the distribution of data points in the function domain do not leave rather large regions of the function domain without centers; 3. the approximation error to control the number of centers in such a way that a desired fitting accuracy can be achieved. Numerical experiments are carried out to illustrate the performance of the method on a number of test functions.

On the boundary element dual reciprocity method

In this paper, the boundary element dual reciprocity method (BEDRM) is introduced for the solution of a class of boundary value problems. Convergence analysis of the BEDRM is carried out for Poisson's equation with Dirichlet boundary condition, using some important results on radial basis function approximation.

Sampled-Data Indirect Adaptive Control of Bioreactor Using Affine Radial Basis Function Network Architecture

This paper considers a problem of bioreactor control, which is formulated in Anderson and Miller (1990) and Ungar (1990) as a benchmark problem for application of neural network-based adaptive control algorithms. A completely adaptive control of this strongly nonlinear system is achieved with no a priori knowledge of its dynamics. This becomes possible thanks to a novel architecture of the controller, which is based on an affine Radial Basis Function network approximation of the sampled-data system mapping. Approximation with such net-work could be considered as a generalization of a standard practice to linearize a nonlinear system about the working regime. As the network is affine in the control components, it can be inverted with respect to the control vector by using fast matrix computations. The considered approach includes several features, recently introduced in some advanced process control algorithms. These features—multirate sampling, on-line adaptation, and Radial Basis Function approximation of the system nonlinearity—are crucial for the achieved high performance of the controller.

Calculation of the endurance limit and design of autofrettage pressurized components: Schon, M. and Seeger, T. (in German) Mater. wiss. Werkst. tech. (1995) 26 (7), 347–356

In this paper, a new local meshless approach based on radial basis functions (RBFs) is presented to price the options under the Black–Scholes model. The global RBF approximations derived from the conventional global collocation method usually lead to ill-conditioned matrices. Employing the idea of local approximants of the finite difference (FD) method and combining it with the radial basis function (RBF) method can result in a local meshless approach such as RBF-FD. It removes the difficulty of ill-conditionness of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. It is fast and produces high accurate results as shown in numerical experiments. Moreover, we took into account the variation of shape parameter and analyzed numerically the behavior of the RBF-FD method.

On Shifted Cardinal Interpolation by Gaussians and Multiquadrics

A radial basis function approximation is a linear combination of translates of a fixed functionϕ: Rd→R. Such functions possess many useful and interesting properties when the translates are integers andϕis radially symmetric. We study the closely related problem for which the fixed function is the shifted Gaussianϕ=G(·−α), whereG(x)=exp(−λ‖x‖22) andα∈Rd. Specifically, we exploit the theory of elliptic functions to establish the invertibility of the Toeplitz operator[formula]whenαhas no half-integer components; it is singular otherwise. This implies the existence of ashifted Gaussian cardinal function, that is, a linear combinationχof integer translates of the shifted Gaussian satisfyingχ(j)=δ0j. We also study shifted cardinal functions when the parameterλtends to zero. In particular, we discover their uniform convergence to the sinc function when the shift vectorαpossesses no half-integer components. Our methods are based in part on similar results established by the first author when the basis function is the Hardy multiquadric. Several intriguing links with the theory of shifted B-spline cardinal interpolation are described in the finale.

Radial basis function network architecture for nonholonomic motion planning and control of free-flying manipulators

This paper considers a problem of nonholonomic motion planning. A practical paradigm for planning and stabilization of motion in a class of multivariate nonlinear (nonholonomic) systems is presented and applied to a planar free-floating manipulator system. The controller architecture designed in the paper is based on the radial basis function approximation of an optimal control program for any desired motion. This architecture also incorporates a sampled-data feedback stabilization algorithm. The proposed control technique overcomes certain problems associated with other control approaches available for nonholonomic systems. The presented simulation results reveal a promising potential of the proposed control paradigm. This paradigm can be extended to a broader class of nonlinear control problems.

Adaptation of centers of approximation for nonlinear tracking control

Sufficient conditions for the asymptotic stability and exponential convergence of tracking control for nonlinear systems have proliferated in recent research. Various theories for stability have been related to, and rely upon, the functional form of the governing nonlinear equations, the linearizability of the governing equations by canonical (Lie) transformations, the linear appearance of adaptive parameters, and the richness of the input excitation. Previously, sufficient conditions have been derived for a class of Hamiltonian adaptive tracking control problems using radial basis function approximants. We demonstrate that a crucial element determining the performance of this class of tracking control methodologies is the location of the centers of approximation in the radial basis approximant/network. A learning vector quantization algorithm is employed to simultaneously adapt both the centers of approximation and the feedback gains. The simultaneous adaptation of both approximant centers in configuration space and feedback gains can demonstrate remarkable improvements in convergence rate, as compared to adaptation of feedback gains alone.