Positive Definite Radial Basis Functions Research Articles

Scattered data interpolation: Strictly positive definite radial basis/cardinal functions

Radial basis functions are a simple and accurate method for multivariate interpolation but the ill-conditioning situation due to their interpolation matrices, discourages an acceptable approximation for both large number of nodes or flat function interpolation. In current work, a new type of basis named well-conditioned RBFs (WRBFs) was created by adding the strictly positive definite Radial Basis Functions (SPD-RBFs) to cardinal functions, was introduced and applied for interpolation. To light up this manner, two classes of global cardinal functions were used for adding by SPD-RBFs. These cardinal functions are Shepard functions and Quasi-cardinal RBFs. Theoretical and numerical analyses prove that utilizing WRBFs has some advantages such as eliminating the ill-conditioning system which has arisen from the global positive definite RBFs interpolation, improving the convergence of pure cardinal functions interpolation and also working better than pure RBFs for small shape parameters.

Unconditional convergence of iterative approximation methods

In this paper, we propose several iterative formats for quasi-interpolation by using positive definite radial basis functions. Firstly, we present the methodology of approximate interpolation, and then employ the diagonally compensated reduction to preprocess the iterative method. We show that the proposed iterative methods are convergent no matter which shape parameter is used, therefore it can transform a nonconvergent iterative format into a convergent one. The proposed iterative methods can approximate the corresponding exact interpolant with arbitrary precision. Numerical results illustrate the effectiveness of our methods.

RBF Interpolation Algorithm for FTS Tool Path Generation

Freeform optics are defined as nonrotational symmetric optical surfaces in the manufacturing industry. Freeform optics are extensively applied to many areas in order to improve system performance. Fast tool servo (FTS) assisting single-point diamond turning technology has high application prospects in freeform optics machining. This paper discusses the interpolation algorithm for tool path generation of FTS through the application of a radial basis function (RBF) algorithm. For this purpose, a positive definite RBF with compact support was employed as the interpolant. The existence is mathematically proven. Numerical simulations were performed to compare the performances of the RBF algorithm and commonly used algorithms for satisfying the requirements of existence, smoothness, and accuracy. Machining experiments were also conducted to validate the applicability of the algorithm. The simulation results showed that the RBF interpolation algorithm outperformed other algorithms in terms of smoothness. The RBF algorithm also provided the highest interpolation accuracy. Furthermore, the RBF interpolation algorithm exhibited the highest accuracy for error distribution, with large errors distributed mainly in transition areas. The machining results were also in general agreement with the simulation results although obvious practical errors were observed. Overall, RBF interpolation can provide higher accuracy and better smoothness in the tool path generation of FTS.

Uniformly Stable Parameterized Macromodeling Through Positive Definite Basis Functions

Reduced-order models are widely used to reduce the computational cost required by the numerical assessment of electrical performance during the design cycle of electronic circuits and systems. Although standard macromodeling algorithms can be considered to be well consolidated, the generation of macromodels that embed in a closed-form some dependence on the design variables still presents considerable margins for improvement. One of these aspects is the enforcement of uniform stability throughout the parameter space of interest. This article proposes a novel parameterized macromodeling strategy, which enforces by construction that all macromodel poles are stable for any combination of possibly several independent design variables. The key enabling factor is the adoption of positive definite multivariate basis functions for the representation of model variations induced by the parameters. This representation leads to robust model generation from tabulated frequency responses, at a computational cost that is dramatically reduced with respect to competing approaches. This result arises from a number of algebraic constraints for stability enforcement that depends on the model complexity (number of basis functions) and not on the model behavior as a function of the parameters. As a byproduct, the proposed strategy lends itself to a much improved scaling with the dimension of parameter space, allowing to circumvent the curse of dimensionality that may occur when the number of independent parameters grows beyond few units. To this end, we exploit representations based on positive definite radial basis functions. The benefits of the proposed approach are demonstrated through an extensive experimental campaign applied to both passive and active devices and components, comparing the performance of different model parameterizations in terms of accuracy, time requirements, and model compactness.

Multilevel interpolation of scattered data using ${\mathcal{H}}$-matrices

Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of ${\mathcal{H}}$ -matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using ${\mathcal{H}}$ -matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using ${\mathcal{H}}$ -matrices is of almost linear complexity with respect to the number of centers.

Iterative Solution of Saddle-Point Systems from Radial Basis Function (RBF) Interpolation

Scattered data interpolation using conditionally positive definite radial basis functions typically leads to large, dense, and indefinite systems of saddle-point type. Due to ill-conditioning, the ...

Domain decomposition methods in scattered data interpolation with conditionally positive definite radial basis functions

Scattered data interpolation using conditionally positive definite radial basis functions (RBFs) requires the solution of a symmetric saddle-point system. Based on an approximation of the system matrix as a hierarchical matrix, we solve the system iteratively using the GMRes algorithm and a domain decomposition preconditioner. The novelty of our work lies in the proposed solution of the subdomain problems using the nullspace method with an orthogonal basis represented as a sequence of Householder reflectors. The resulting positive definite subdomain systems are solved either directly or using an inner GMRes iteration with H-Cholesky preconditioning. Numerical tests demonstrate the effectiveness of this solution process for up to N=160000 centers in two and three dimensions.

A dual reciprocity multiwavelet Galerkin method for the numerical solution of Poisson׳s equation

A dual reciprocity multiwavelet Galerkin method is developed for the solution of Poisson׳s equation in this paper, which combines the dual reciprocity boundary element method (DRBEM) and the multiwavelet Galerkin method (MGM). The DRBEM is used to transform the domain integral in the boundary element formulation of Poisson׳s equation into the boundary of the domain, which is based on compactly supported positive definite radial basis function. Then, the MGM is employed for solving the resulting boundary integral equation, in which Alpert multiwavelets are employed to construct the trial and test functions of Galerkin variational formulation. Because of the use of multiwavelets, the resulting system matrix can be approximated by a sparse matrix. Compared to the DRBEM based on radial basis functions, the present method reduces the memory spaces and computational costs of the system matrix significantly. Numerical results show the efficiency of the present method.

A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis

In this article, we present a numerical method to solve two-dimensional Fredholm integral equations of the second kind on general domains. The method utilizes meshless and spectral collocation techniques but it is not traditional meshless collocation method. The point interpolation method with the help of strictly positive definite radial basis functions is used to construct shape functions as a basis functions in the frame of spectral collocation methods. These basis functions (shape functions) have Kronecker delta function property. Since the proposed method is meshless, it does not need any domain element and so it is independent of the geometry of the domain. The method reduces the solution of the two-dimensional integral equation to the solution of a linear system of algebraic equations. Convergence analysis with error estimates are given with full discussion. Furthermore, some numerical examples are presented to show the validity and efficiency of SMRPI.

Closed form representations for a class of compactly supported radial basis functions

In this paper we re-examine Wendland's strategy for the construction of compactly supported positive definite radial basis functions. We acknowledge that this strategy can be modified to capture a much larger range of functions, including the so-called missing Wendland functions which have been the subject of a recent paper by Schaback (Adv Comput Math 34:67---81, 2011). Our approach is to focus on a general integral representation of such functions and we will show how a careful evaluation of this integral leads to new closed form expressions for both Wendland's original functions and the missing ones. The resulting expressions are easy to code and so provide the potential user with a quick way of accessing a desired example for a given application.

The capability of approximation for neural networks interpolant on the sphere

Compared with planar hyperplane, fitting data on the sphere has been an important and active issue in geoscience, metrology, brain imaging, and so on. In this paper, using a functional approach, we rigorously prove that for given distinct samples on the unit sphere there exists a feed-forward neural network with single hidden layer which can interpolate the samples, and simultaneously near best approximate the target function in continuous function space. Also, by using the relation between spherical positive definite radial basis functions and the basis function on the Euclidean space ℝd + 1, a similar result in a spherical Sobolev space is established. Copyright © 2010 John Wiley & Sons, Ltd.

Fast surface reconstruction and hole filling using positive definite radial basis functions

Surface reconstruction from large unorganized data sets is very challenging, especially if the data present undesired holes. This is usually the case when the data come from laser scanner 3D acquisitions or if they represent damaged objects to be restored. An attractive field of research focuses on situations in which these holes are too geometrically and topologically complex to fill using triangulation algorithms. In this work a local approach to surface reconstruction from point-clouds based on positive definite Radial Basis Functions (RBF) is presented that progressively fills the holes by expanding the neighbouring information. The method is based on the algorithm introduced in [7] which has been successfully tested for the smooth multivariate interpolation of large scattered data sets. The local nature of the algorithm allows for real time handling of large amounts of data, since the computation is limited to suitable small areas, thus avoiding the critical efficiency problem involved in RBF multivariate interpolation. Several tests on simulated and real data sets demonstrate the efficiency and the quality of the reconstructions obtained using the proposed algorithm.

Interpolation by basis functions of different scales and shapes

Under very mild additional assumptions, translates of conditionally positive definite radial basis functions allow unique interpolation to scattered multivariate data, because the interpolation matrices have a symmetric and positive definite dominant part. In many applications the data density varies locally according to the signal behaviour, and then the translates should get different scalings that match the local data density. Furthermore, if there is a local anisotropy in the data, the radial basis functions should possibly be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are no longer guaranteed. However, this brief note is the first paper to provide sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a simple matrix perturbation argument combined with the Ball-Narcowich-Ward stability results.

Solution of stokes flow using an iterative DRBEM based on compactly-supported, positive-definite radial basis function

This paper describes an iterative dual reciprocity boundary element method (DRBEM) based on the compactly-supported, positive definite radial basis function for the solution of Stokes flow problems. The method involves the solution of Laplace equations for vorticity, and Poisson equations for velocity with the solenoidal components of vorticity as right-hand sides. These equations are solved using iterative BEM and DRBEM. For the DRBEM, the compactly supported, positive definite radial basis function is used to approximate the body force term to convert it to boundary integrals. A relaxation method is devised such that the resulting algebraic equations are solved without the need of assembling a matrix. Hence, large systems of equations such as the fine discretization needed for Stokes flow problems can be handled easily. The results of Stokes flow in a square cavity and a circular cavity are presented and compared with other numerical model results. The iterative DRBEM solution is found to be satisfactory.

Solution of Poisson's equation by iterative DRBEM using compactly supported, positive definite radial basis function

In the numerical solution of three-dimensional boundary value problems, the matrix size can be so large that it is beyond a computer's capacity to solve it. To overcome this difficulty, an iterative dual reciprocity boundary element method (DRBEM) is developed to solve Poisson's equation without the need of assembling a matrix. The DRBEM procedure requires that the right hand side of Poisson's equation be approximated by a radial basis function interpolation. In the iterative solution, it is found that only compactly supported, positive definite radial basis functions lead to converged results.