9,766 research outputs found
Order-Preserving Derivative Approximation with Periodic Radial Basis Functions
In this exploratory paper we study the convergence rates of an iterated method for approximating derivatives of periodic functions using radial basis function (RBF) interpolation. Given a target function sampled on some node set, an approximation of the m th derivative is obtained by m successive applications of the operator “interpolate, then differentiate”- this process is known in the spline community as successive splines or iterated splines. For uniformly spaced nodes on the circle, we give a sufficient condition on the RBF kernel to guarantee that, when the error is measured only at the nodes, this iterated method approximates all derivatives with the same rate of convergence. We show that thin-plate spline, power function, and Matérn kernels restricted to the circle all satisfy this condition, and numerical evidence is provided to show that this phenomena occurs for some other popular RBF kernels. Finally, we consider possible extensions to higher-dimensional periodic domains by numerically studying the convergence of an iterated method for approximating the surface Laplace (Laplace-Beltrami) operator using RBF interpolation on the unit sphere and a torus
On the spectral distribution of kernel matrices related to\ud radial basis functions
This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants
Computing with functions in spherical and polar geometries I. The sphere
A collection of algorithms is described for numerically computing with smooth
functions defined on the unit sphere. Functions are approximated to essentially
machine precision by using a structure-preserving iterative variant of Gaussian
elimination together with the double Fourier sphere method. We show that this
procedure allows for stable differentiation, reduces the oversampling of
functions near the poles, and converges for certain analytic functions.
Operations such as function evaluation, differentiation, and integration are
particularly efficient and can be computed by essentially one-dimensional
algorithms. A highlight is an optimal complexity direct solver for Poisson's
equation on the sphere using a spectral method. Without parallelization, we
solve Poisson's equation with million degrees of freedom in one minute on
a standard laptop. Numerical results are presented throughout. In a companion
paper (part II) we extend the ideas presented here to computing with functions
on the disk.Comment: 23 page
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Shape reconstruction from gradient data
We present a novel method for reconstructing the shape of an object from
measured gradient data. A certain class of optical sensors does not measure the
shape of an object, but its local slope. These sensors display several
advantages, including high information efficiency, sensitivity, and robustness.
For many applications, however, it is necessary to acquire the shape, which
must be calculated from the slopes by numerical integration. Existing
integration techniques show drawbacks that render them unusable in many cases.
Our method is based on approximation employing radial basis functions. It can
be applied to irregularly sampled, noisy, and incomplete data, and it
reconstructs surfaces both locally and globally with high accuracy.Comment: 16 pages, 5 figures, zip-file, submitted to Applied Optic
Are ghost surfaces quadratic-flux-minimizing?
Two candidates for "almost-invariant" toroidal surfaces passing through
magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost
surfaces, use families of periodic pseudo-orbits (i.e. paths for which the
action is not exactly extremal). QFMin pseudo-orbits, which are
coordinate-dependent, are field lines obtained from a modified magnetic field,
and ghost-surface pseudo-orbits are obtained by displacing closed field lines
in the direction of steepest descent of magnetic action, . A generalized Hamiltonian definition of ghost
surfaces is given and specialized to the usual Lagrangian definition. A
modified Hamilton's Principle is introduced that allows the use of Lagrangian
integration for calculation of the QFMin pseudo-orbits. Numerical calculations
show QFMin and Lagrangian ghost surfaces give very similar results for a
chaotic magnetic field perturbed from an integrable case, and this is explained
using a perturbative construction of an auxiliary poloidal angle for which
QFMin and Lagrangian ghost surfaces are the same up to second order. While
presented in the context of 3-dimensional magnetic field line systems, the
concepts are applicable to defining almost-invariant tori in other
degree-of-freedom nonintegrable Lagrangian/Hamiltonian systems.Comment: 8 pages, 3 figures. Revised version includes post-publication
corrections in text, as described in Appendix C Erratu
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