Iterative Techniques for Radial Basis Function Interpolation

Journal of Computational and Applied Mathematics, 2013

It is well-known that radial basis function interpolants suffer of bad conditioning if the basis of translates is used. In the recent work [12], the authors gave a quite general way to build stable and orthonormal bases for the native space N Φ (Ω) associated to a kernel Φ on a domain Ω ⊂ R s. The method is simply based on the factorization of the corresponding kernel matrix. Starting from that setting we describe a particular basis which turns out to be orthonormal in N Φ (Ω) and in 2,w (X), where X is a set of data sites of the domain Ω. The basis arises from a weighted singular value decomposition of the kernel matrix. This basis is also related to a discretization of the compact operator T Φ : N Φ (Ω) → N Φ (Ω), T Φ [f ](x) = Ω Φ(x, y)f (y)dy ∀x ∈ Ω and provides a connection with the continuous basis that arises from an eigen-decomposition of T Φ. Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.

BIT Numerical Mathematics, 2013

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial order. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given order defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any order of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness, including an application to the Best Linear Unbiased Estimator regression problem.

Interpolation and approximation methods are used across many fields. Standard interpolation and approximation methods rely on "ordering" that actually means tessellation in-dimensional space in general, like sorting, triangulation, generating of tetrahedral meshes etc. Tessellation algorithms are quite complex in-dimensional case. On the other hand, interpolation and approximation can be made using meshfree (meshless) techniques using Radial Basis Function (RBF). The RBF interpolation and approximation methods lead generally to a solution of linear system of equations. However, a similar approach can be taken for a reconstruction of a surface of scanned objects, etc. In this case this leads to a linear system of homogeneous equations, when a different approach has to be taken. In this paper we describe novel approaches based on RBFs for data interpolation and approximation generally in d-dimensional space. We will show properties and differences of "global" and "Compactly Supported RBF (CSRBF)", run-time and memory complexities. As the RBF interpolation and approximation naturally offer smoothness, we will analyze such properties as well as approaches how to decrease computational expenses. The proposed meshless interpolation and approximation will be demonstrated on different problems, e.g. inpainting removal, restoration of corrupted images with high percentage of corrupted pixels, digital terrain interpolation and approximation for GIS applications and methods for decreasing computational complexity.

Computer Methods in Applied Mechanics and Engineering, 2010

We have developed a parallel algorithm for radial basis function (rbf) interpolation that exhibits O (N) complexity, requires O (N) storage, and scales excellently up to a thousand processes. The algorithm uses a gmres iterative solver with a restricted additive Schwarz method (rasm) as a preconditioner and a fast matrix-vector algorithm. Previous fast rbf methods—achieving at most O (NlogN) complexity—were developed using multiquadric and polyharmonic basis functions. In contrast, the present method uses Gaussians with a small ...

The problem of choosing "good" nodes is a central one in polynomial interpolation. Made curious from this problem, in this work we present some results concerning the computation of optimal points sets for interpolation by radial basis functions. Two algorithms for the construction of near-optimal set of points are considered. The first, that depends on the radial function, compute optimal points by adding one of the maxima of the power function with respect to the preceding set. The second, which is independent of the radial function, is shown to generate near-optimal sets which correspond to Leja extremal points. Both algorithms produce point sets almost similar, in the sense of their mutual separation distances. We then compare the interpolation errors and the growth of the Lebesgue constants for both point sets. * This work has been done with the support of ex-60% funds of the University of Verona, year 2002.

Computers & Mathematics with Applications, 2005

Multivariate interpolation of smooth data using smooth radial basis functions is considered. The behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically. Explicit criteria for different types of limits are given. Using the results for the limits, the dependence of the error on the shape parameter of the radial basis function is investigated. The mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error.

Journal of Computational and Applied Mathematics, 2002

An e cient method for the multivariate interpolation of very large scattered data sets is presented. It is based on the local use of radial basis functions and represents a further improvement of the well known Shepard's method. Although the latter is simple and well suited for multivariate interpolation, it does not share the good reproduction quality of other methods widely used for bivariate interpolation. On the other hand, radial basis functions, which have proven to be highly useful for multivariate scattered data interpolation, have a severe drawback. They are unable to interpolate large sets in an e cient and numerically stable way and maintain a good level of reproduction quality at the same time. Both problems have been circumvented using radial basis functions to evaluate the nodal function of the modiÿed Shepard's method. This approach exploits the exibility of the method and improves its reproduction quality. The proposed algorithm has been implemented and numerical results conÿrm its e ciency.

Interpolation or approximation of scattered data is very often task in engineering problems. The Radial Basis Functions (RBF) interpolation is convenient for scattered (un-ordered) data sets in k-dimensional space, in general. This approach is convenient especially for a higher dimension k > 2 as the conversion to an ordered data set, e.g. using tessellation, is computationally very expensive. The RBF interpolation is not separable and it is based on distance of two points. It leads to a solution of a Linear System of Equations (LSE) =. There are two main groups of interpolating functions: 'global" and "local". Application of "local" functions, called Compactly Supporting RBF (CSFBF), can significantly decrease computational cost as they lead to a system of linear equations with a sparse matrix. In this paper the RBF interpolation theory is briefly introduced at the "application level" including some basic principles and computational issues and an incremental RBF computation is presented and approximation RBF as well. The RBF interpolation or approximation can be used also for image reconstruction, inpainting removal, for solution of Partial Differential Equations (PDE), in GIS systems, digital elevation model DEM etc.

Advances in Computational Mathematics, 1995

For interpolation of scattered multivariate data by radial basis functions, an \uncertainty relation" between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich{Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use.

Computers & Mathematics with Applications, 2002

This paper deals with vector field interpolation, i.e., the data are W3 values located in scattered W3 points, while the interpolating function is a function from W3 into W3. In order to take into account possible connections between the components of the interpolant, we derive it by solving a variational spline problem involving the rotational and the divergence of the interpolant, and depending on a parameter p significative of the balance of the rotational part and of the divergence part, and on the order m of derivatives of the rotational and divergence involved in the minimized seminorm. We so obtain interpolants whose expression is a(r) = Cbi +(z,-zi)ai + p,_r(r)! where @ is some 3 x 3 matrix function, p,-r is a degree m-1 vectorial polynomial, and where the uz are W3-vectors. Besides, the ai meet a relation generalizing the usual orthogonality to all polynomials of degree at most m-1. For p = 1, we find the usual m-harmonic splines in each component of o. Numerical examples show the interest of the method, and we compare the so-obtained functions with the ones obtained by Matlab's procedures.