Application of Fredholm integral equations inverse theory to the radial basis function approximation problem

This paper reveals and examines the relationship between the solution and stability of Fredholm integral equations and radial basis function approximation or interpolation. The underlying system (kernel) matrices are shown to have a smoothing property which is dependent on the choice of kernel. Instead of using the condition number to describe the ill-conditioning, hence only looking at the largest and smallest singular values of the matrix, techniques from inverse theory, particularly the Picard condition, show that it is understanding the exponential decay of the singular values which is critical for interpreting and mitigating instability. Results on the spectra of certain classes of kernel matrices are reviewed, verifying the exponential decay of the singular values. Numerical results illustrating the application of integral equation inverse theory are also provided and demonstrate that interpolation weights may be regarded as samplings of a weighted solution of an integral equation. This is then relevant for mapping from one set of radial basis function centers to another set. Techniques for the solution of integral equations can be further exploited in future studies to find stable solutions and to reduce the impact of errors in the data

Spatiospectral concentration on a sphere

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200

KSOR iterative method with quadrature scheme for solving system of Fredholm integral equations of second kind

In this study, the system of second kind Fredholm integral equations has been discretized by using the first order quadrature scheme namely trapezoidal rule in order to construct the first order quadrature approximation equation. Next, the quadrature approximation equation obtained has been used to construct a system of linear equations. Three types of iterative methods were used to solve the system of linear equations such as Gauss-Seidel (GS), Successive Over Relaxation (SOR) and Kaudd Successive Over Relaxation (KSOR). For comparison purpose, two problems have been considered in this study in order to analyze the efficiency of these three proposed iterative methods for solving the problems. Based on the numerical results, it can be pointed out that KSOR is similar as SOR but both of these iterative methods are more efficient than GS method.Keywords: quadrature scheme; system of Fredholm integral equations; KSOR iterative metho

On polynomial collocation for second kind integral equations with fixed singularities of Mellin type

We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gauss quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with O(n^2[log n]^2) resp. O(n^2) operations, where n is the dimension of the polynomial trial space

Averaged Nystr\"om interpolants for the solution of Fredholm integral equations of the second kind

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nystr\"om methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose. New stability properties of the quadrature rules used are shown.Comment: 26 pages, 2 figure