In this paper, a numerical method is proposed for solving optimal control problem of Volterra integral equations using radial basis functions (RBFs) for approximating unknown function. Actually, the method is based on interpolation by radial basis functions including multiquadrics (MQs), to determine the control vector and the corresponding state vector in linear dynamic system while minimizing...

Motivated by 5] we describe a method related to scattered Hermite interpolation for which the solution of elliptic partial diierential equations by collocation is well-posed. We compare the method of 5] with our method. x1. Introduction In this paper we discuss the numerical solution of elliptic partial diierential equations using a collocation approach based on radial basis functions. To make ...

Interpolation by analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become “increasingly flat”, or the data points “coalesce” in the limit while the radial basis functions stays fixed. Both cases call for a careful regularization. If carried out explicitly, this yields a preconditioning technique fo...

Interpolation by translates of suitable radial basis functions is an important approach towards solving the scattered data problem. However, for a large class of smooth basis functions (including multiquadrics f(x)=(|x|+l), m > d/2, 2m−d ̈ 2Z), the existing theories guarantee the interpolant to approximate well only for a very small class of very smooth approximands. The approximands f need to b...

Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only non-oscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as their limiting case) leads to non-singular interpolants with in...

Common elastic registration schemes based on landmarks and radial basis functions (RBFs) such as thin-plate splines or multiquadrics are global. Here, we introduce radial basis functions with compact support for elastic registration of medical images which have an improved locality, i.e. which allow to constrain elastic deformations to image parts where required. We give the theoretical backgro...

Common elastic registration schemes based on landmarks and radial basis functions (RBFs) such as thin-plate splines or multiquadrics are global. Here, we introduce radial basis functions with compact support for elastic registration of medical images which have an improved locality, i.e. which allow to constrain elastic deformations to image parts where required. We give the theoretical backgro...

In this paper we report on two diierent experiments dealing with the numerical solution of diierential equations by radial basis functions: 1) the solution of a two-point boundary value problem; 2) the solution of a two-dimensional Poisson equation. In the second experiment we contrast a mul-tilevel collocation algorithm based on locally supported basis functions with two diierent direct soluti...

Many types of radial basis functions, such as multiquadrics, contain a free parameter. In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge. Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials. In the 1-D case, we prove that with s...

In this study, we are mainly interested in error estimates of interpolation, using smooth radial basis functions such as multiquadrics. The current theories of radial basis function interpolation provide optimal error bounds when the basis function φ is smooth and the approximand f is in a certain reproducing kernel Hilbert space Fφ. However, since the space Fφ is very small when the function φ...