161 research outputs found
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
In this paper we investigate the approximation properties of kernel
interpolants on manifolds. The kernels we consider will be obtained by the
restriction of positive definite kernels on , such as radial basis
functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d.
For restricted kernels having finite smoothness, we provide a complete
characterization of the native space on \M. After this and some preliminary
setup, we present Sobolev-type error estimates for the interpolation problem.
Numerical results verifying the theory are also presented for a one-dimensional
curve embedded in and a two-dimensional torus
Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
The purpose of this paper is to establish that for any compact, connected
C^{\infty} Riemannian manifold there exists a robust family of kernels of
increasing smoothness that are well suited for interpolation. They generate
Lagrange functions that are uniformly bounded and decay away from their center
at an exponential rate. An immediate corollary is that the corresponding
Lebesgue constant will be uniformly bounded with a constant whose only
dependence on the set of data sites is reflected in the mesh ratio, which
measures the uniformity of the data.
The analysis needed for these results was inspired by some fundamental work
of Matveev where the Sobolev decay of Lagrange functions associated with
certain kernels on \Omega \subset R^d was obtained. With a bit more work, one
establishes the following: Lebesgue constants associated with surface splines
and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi
are quasi-uniformly distributed. The non-Euclidean case is more involved as the
geometry of the underlying surface comes into play. In addition to establishing
bounded Lebesgue constants in this setting, a "zeros lemma" for compact
Riemannian manifolds is established.Comment: 33 pages, 2 figures, new title, accepted for publication in SIAM J.
on Math. Ana
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
Analysis of moving least squares approximation revisited
In this article the error estimation of the moving least squares
approximation is provided for functions in fractional order Sobolev spaces. The
analysis presented in this paper extends the previous estimations and explains
some unnoticed mathematical details. An application to Galerkin method for
partial differential equations is also supplied.Comment: Journal of Computational and Applied Mathematics, 2015 Journal of
Computational and Applied Mathematic
RBF multiscale collocation for second order elliptic boundary value problems
In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. We are able to show that the convergence is linear in the number of levels. We also discuss the condition numbers of the arising systems and the effect of simple, diagonal preconditioners, now proving rigorously previous numerical observations
A Radial Basis Function Method for Computing Helmholtz-Hodge Decompositions
A radial basis function (RBF) method based on matrix-valued kernels is presented and analyzed for computing two types of vector decompositions on bounded domains: one where the normal component of the divergence-free part of the field is specified on the boundary, and one where the tangential component of the curl-free part of the field specified. These two decompositions can then be combined to obtain a full Helmholtz-Hodge decomposition of the field, i.e. the sum of divergence-free, curl-free, and harmonic fields. All decompositions are computed from samples of the field at (possibly scattered) nodes over the domain, and all boundary conditions are imposed on the vector fields, not their potentials, distinguishing this technique from many current methods. Sobolev-type error estimates for the various decompositions are provided and demonstrated with numerical examples
- …