608 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
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Carbon reservoirs on Mars: Constraints from Martian meteorites
We have measured the abundance and stable isotopic composition of magmatic carbon extracted from a suite of shergottites. The results confirm previous findings that primordial carbon on Mars is isotopically lighter than that of the Earth
A Hybrid Radial Basis Function - Pseudospectral Method for Thermal Convection in a 3-D Spherical Shell
A novel hybrid spectral method that combines radial basis function (RBF) and Chebyshev pseudospectral (PS) methods in a “2+1” approach is presented for numerically simulating thermal convection in a 3-D spherical shell. This is the first study to apply RBFs to a full 3D physical model in spherical geometry. In addition to being spectrally accurate, RBFs are not defined in terms of any surface based coordinate system such as spherical coordinates. As a result, when used in the lateral directions, as in this study, they completely circumvent the pole issue with the further advantage that nodes can be “scattered” over the surface of a sphere. In the radial direction, Chebyshev polynomials are used, which are also spectrally accurate and provide the necessary clustering near the boundaries to resolve boundary layers. Applications of this new hybrid methodology are given to the problem of convection in the Earth’s mantle,which is modeled by a Boussinesq fluid at infinite Prandtl number. To see whether this numerical technique warrants further investigation, the study limits itself to an isoviscous mantle.Benchmark comparisons are presented with other currently used mantle convection codes for Rayleigh number 7 · 103 and 105. The algorithmic simplicity of the code (mostly due to RBFs)allows it to be written in less than 400 lines of Matlab and run on a single workstation. We find that our method is very competitive with those currently used in the literature
A Fast and Accurate Algorithm for Spherical Harmonic Analysis on HEALPix Grids with Applications to the Cosmic Microwave Background Radiation
The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used
extensively in astrophysics for data collection and analysis on the sphere. The
scheme was originally designed for studying the Cosmic Microwave Background
(CMB) radiation, which represents the first light to travel during the early
stages of the universe's development and gives the strongest evidence for the
Big Bang theory to date. Refined analysis of the CMB angular power spectrum can
lead to revolutionary developments in understanding the nature of dark matter
and dark energy. In this paper, we present a new method for performing
spherical harmonic analysis for HEALPix data, which is a central component to
computing and analyzing the angular power spectrum of the massive CMB data
sets. The method uses a novel combination of a non-uniform fast Fourier
transform, the double Fourier sphere method, and Slevinsky's fast spherical
harmonic transform (Slevinsky, 2019). For a HEALPix grid with pixels
(points), the computational complexity of the method is , with an initial set-up cost of . This compares
favorably with runtime complexity of the current methods
available in the HEALPix software when multiple maps need to be analyzed at the
same time. Using numerical experiments, we demonstrate that the new method also
appears to provide better accuracy over the entire angular power spectrum of
synthetic data when compared to the current methods, with a convergence rate at
least two times higher
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
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
Pattern Formation from the Gierer–Meinhardt Model on the Stanford Bunny
Snapshot from a simulation the Gierer–Meinhardt model on the Stanford Bunny computed using the radial basis function finite difference (RBF-FD) method. This model is consists of activator/inhibitor system of reaction diffusion equations and is known to generate Turing patterns that mimic the coats of some animals found in nature
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