Iterative methods based on spline approximations to detect discontinuities from Fourier data

Recently, spline approximations have been proposed for the reconstruction of piecewise smooth functions from Fourier data. That approach makes possible to retrieve the functions from their Fourier coefficients for any given degree of accuracy when the discontinuity points are known. In this paper we present iterative methods based on those spline approximations, for several degrees, to find locations and amplitudes of the jumps of a piecewise smooth function, given its Fourier coefficients. We also present numerical experiments comparing with different previous approaches.     

On inverse methods for the resolution of the Gibbs phenomenon

When Fourier expansions, or more generally spectral methods, are used for the representation of nonsmooth functions, then one has to face the so-called Gibbs phenomenon. Considerable progresses have been made these last years to overcome the Gibbs phenomenon, using direct or inverse approaches, both in the discrete or continuous framework. A discrete inverse method for the global or local reconstruction of a non-smooth function starting from its oscillating (trigonometric) polynomial interpolant is introduced and both its capabilities and limits are emphasized.     

Preconditioning for radial basis functions with domain decomposition methods

In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N²) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c~1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.     

Numerical solution of the two-dimensional Fredholm integral equations using Gaussian radial basis function

In this paper, we introduce a numerical method for the solution of two-dimensional Fredholm integral equations. The method is based on interpolation by Gaussian radial basis function based on Legendre-Gauss-Lobatto nodes and weights. Numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the method.     

Enhancing credit default swap valuation with meshfree methods

In this paper, we apply the meshfree radial basis function (RBF) interpolation to numerically approximate zero-coupon bond prices and survival probabilities in order to price credit default swap (CDS) contracts. We assume that the interest rate follows a Cox-Ingersoll-Ross process while the default intensity is described by the Exponential-Vasicek model. Several numerical experiments are conducted to evaluate the approximations by the RBF interpolation for one- and two-factor models. The results are compared with those estimated by the finite difference method (FDM). We find that the RBF interpolation achieves more accurate and computationally efficient results than the FDM. Our results also suggest that the correlation between factors does not have a significant impact on CDS spreads.     

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function (x;α,h)≡exp(-[α²/h²]x²) where α is a user-choosable constant. Unfortunately, computing the coefficients of (x-jh;α,h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C_j(x;α,h) is defined by the set of linear combinations of the Gaussians such that C_j(kh)=1 when k=j and C_j(kh)=0 for all integers . We show that the cardinal functions for the uniform grid are C_j(x;h,α)=C(x/h-j;α) where C(X;α)≈(α²/π)sin(πX)/sinh(α²X). The relative error is only about 4exp(-2π²/α²) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h→0, but only to an “error saturation” proportional to exp(-π²/α²). Because the error in our approximation to the master cardinal function C(X;α) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f(x). The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.     

Parallel radial basis function methods for the global optimization of expensive functions

We introduce a master–worker framework for parallel global optimization of computationally expensive functions using response surface models. In particular, we parallelize two radial basis function (RBF) methods for global optimization, namely, the RBF method by Gutmann [Gutmann, H.M., 2001a. A radial basis function method for global optimization. Journal of Global Optimization 19(3), 201–227] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [Regis, R.G., Shoemaker, C.A., 2005. Constrained global optimization of expensive black box functions using radial basis functions, Journal of Global Optimization 31, 153–171] (CORS-RBF). We modify these algorithms so that they can generate multiple points for simultaneous evaluation in parallel. We compare the performance of the two parallel RBF methods with a parallel multistart derivative-based algorithm, a parallel multistart derivative-free trust-region algorithm, and a parallel evolutionary algorithm on eleven test problems and on a 6-dimensional groundwater bioremediation application. The results indicate that the two parallel RBF algorithms are generally better than the other three alternatives on most of the test problems. Moreover, the two parallel RBF algorithms have comparable performances on the test problems considered. Finally, we report good speedups for both parallel RBF algorithms when using a small number of processors.     

Spectral collocation and radial basis function methods for one-dimensional interface problems

Differential equations with singular sources or discontinuous coefficients yield non-smooth or even discontinuous solutions. This problem is known as the interface problem. High-order numerical solutions suffer from the Gibbs phenomenon in that the accuracy deteriorates if the discontinuity is not properly treated. In this work, we use the spectral and radial basis function methods and present a least squares collocation method to solve the interface problem for one-dimensional elliptic equations. The domain is decomposed into multiple sub-domains; in each sub-domain, the collocation solution is sought. The solution should satisfy more conditions than the given conditions associated with the differential equations, which makes the problem over-determined. To solve the over-determined system, the least squares method is adopted. For the spectral method, the weighted norm method with different scaling factors and the mixed formulation are used. For the radial basis function method, the weighted shape parameter method is presented. Numerical results show that the least squares collocation method provides an accurate solution with high efficacy and that better accuracy is obtained with the spectral method.     

A hybrid algorithm to optimize RBF network architecture and parameters for nonlinear time series prediction

This paper proposes a novel hybrid algorithm for automatic selection of the proper input variables, the number of hidden nodes of the radial basis function (RBF) network, and optimizing network parameters (weights, centers and widths) simultaneously. In the proposed algorithm, the inputs and the number of hidden nodes of the RBF network are represented by binary-coded strings and evolved by a genetic algorithm (GA). Simultaneously, for each chromosome with fixed inputs and number of hidden nodes, the corresponding parameters of the network are real-coded and optimized by a gradient-based fast-converging parameter estimation method. Performance of the presented hybrid approach is evaluated by several benchmark time series modeling and prediction problems. Experimental results show that the proposed approach produces parsimonious RBF networks, and obtains better modeling accuracy than some other algorithms.     

Node adaptive domain decomposition method by radial basis functions

During the last years, there has been increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Asymptotic behavior of gibbs functions forM-band wavelet expansions

In this paper the asymptotic behavior of Gibbs function for a class ofM-band wavelet expansions is given. In particular, the Daubechies’ wavelets are included in this class. The authors are partially supported by the Chinese National Natural Science Foundation (19571972), the Key Project Foundation (69735020) and the Zhejiang Provincial Science Foundation of China (196083)     

Some issues on interpolation matrices of locally scaled radial basis functions

Radial basis function interpolation on a set of scattered data is constructed from the corresponding translates of a basis function, which is conditionally positive definite of order m ? 0, with the possible addition of a polynomial term. In many applications, the translates of a basis function are scaled differently, in order to match the local features of the data such as the flat region and the data density. Then, a fundamental question is the non-singularity of the perturbed interpolation (N × N) matrix. In this paper, we provide some counter examples of the matrices which become singular for N ? 3, although the matrix is always non-singular when N = 2. One interesting feature is that a perturbed matrix can be singular with rather small perturbation of the scaling parameter.     

Application of new advanced CNN structure with adaptive thresholds to color edge detection

Color edge detection is much more efficient than gray scale detection when edges exist at the boundary between regions of different colors with no change in intensity. This paper presents adaptive templates, which are capable of detecting various color and intensity changes in color image. To avoid conception of multilayer proposed in literatures, modification has been done to the CNN structure. This modified structure allows a matrix C, which carries the change information of pixels, to replace the control parts in the basic CNN equation. This modification is necessary because in multilayer structure, it faces the challenge of how to represent the intrinsic relationship among each primary layer. Additionally, in order to enhance the accuracy of edge detection, adaptive detection threshold is employed. The adaptive thresholds are considered to be alterable criteria in designing matrix C. The proposed synthetic system not only avoids the problem which is engendered by multi-layers but also exploits full information of pixels themselves. Experimental results prove that the proposed method is efficient.     

An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

The problem of interpolation of scattered data on the unit sphere has many applications in geodesy and Earth science in which the sphere is taken as a model for the Earth. Spherical radial basis functions provide a convenient tool for constructing the interpolant. However, the underlying linear systems tend to be ill-conditioned. In this paper, we present an additive Schwarz preconditioner for accelerating the solution process. An estimate for the condition number of the preconditioned system will be discussed. Numerical experiments using MAGSAT satellite data will be presented.     

Numerical analyses of the boundary effect of radial basis functions in 3D surface reconstruction

Surface reconstruction is very important for surface characterization and graph processing. Radial basis function has now become a popular method to reconstruct 3D surfaces from scattered data. However, it is relatively inaccurate at the boundary region. To solve this problem, a circle of new centres are added outside the domain of interest. The factors that influence the boundary behaviour are analyzed quantitatively via numerical experiments. It is demonstrated that if the new centres are properly located, the boundary problem can be effectively overcome whilst not reducing the accuracy at the interior area. A modified Graham scan technique is introduced to obtain the boundary points from a scattered point set. These boundary points are extended outside with an appropriate distance, and then uniformized to form the new auxiliary centres.      

A low-cost-memory CUDA implementation of the conjugate gradient method applied to globally supported radial basis functions implicits

Hermitian radial basis functions implicits is a method capable of reconstructing implicit surfaces from first-order Hermitian data. When globally supported radial functions are used, a dense symmetric linear system must be solved. In this work, we aim at exploring and computing a matrix-free implementation of the Conjugate Gradients Method on the GPU in order to solve such linear system. The proposed method parallelly rebuilds the matrix on demand for each iteration. As a result, it is able to compute the Hermitian-based interpolant for datasets that otherwise could not be handled due to the high memory demanded by their linear systems.     

An adaptive truncation criterion,for linesearch-based truncated Newton methods in large scale nonconvex optimization

Starting from the paper by Nash and Sofer (1990), we propose a heuristic adaptive truncation criterion for the inner iterations within linesearch-based truncated Newton methods. Our aim is to possibly avoid “over-solving” of the Newton equation, based on a comparison between the predicted reduction of the objective function and the actual reduction obtained. A numerical experience on unconstrained optimization problems highlights a satisfactory effectiveness and robustness of the adaptive criterion proposed, when a residual-based truncation criterion is selected.     

Digital total variation filtering as postprocessing for Chebyshev pseudospectral methods for conservation laws

Digital total variation filtering is analyzed as a fast, robust, post-processing method for accelerating the convergence of pseudospectral approximations that have been contaminated by Gibbs oscillations. The method, which originated in image processing, can be combined with spectral filters to quickly post-process large data sets with sharp resolution of discontinuities and with exponential accuracy away from the discontinuities.     

Best estimates of RBF-based meshless Galerkin methods for Dirichlet problem

In this paper, Galerkin methods based on the radial basis functions to deal with the partial differential equations are discussed. The best error estimates for this method are obtained.