Spherical Radial Basis Functions, Theory and Applications

Hubbert, Simon; Gia, Quôc Thông Lê; Morton, Tanya M.

doi:10.1007/978-3-319-17939-1

Cited by 29 publications

(20 citation statements)

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“…In general, the number of harmonics can be chosen independently based on the regularity of each term. We should note that one can select a different basis functions φ ν (θ, ϕ) too, for example, spherical radial basis functions (see [24]). Additionally, when the bounded domain is of a more general shape but smooth, instead of spherical harmonics as considered in this paper, one can investigate use of more general local radial basis functions to represent Cauchy data on the boundary of the domain as a part of the developed algorithms.…”

Section: Remark 6 (I)mentioning

confidence: 99%

Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

Epshteyn

Xia

2019

J Sci Comput

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In this work, we propose efficient and accurate numerical algorithms based on Difference Potentials Method for numerical solution of chemotaxis systems and related models in 3D. The developed algorithms handle 3D irregular geometry with the use of only Cartesian meshes and employ Fast Poisson Solvers. In addition, to further enhance computational efficiency of the methods, we design a Difference-Potentialsbased domain decomposition approach which allows mesh adaptivity and easy parallelization of the algorithm in space. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.In this work we develop efficient and accurate numerical algorithms based on Difference Potentials Method (DPM) for the Patlak-Keller-Segel (PKS) chemotaxis model and related problems in 3D. The proposed methods handle irregular geometry with the use of only Cartesian meshes, employ Fast Poisson Solvers, allow easy parallelization and adaptivity in space.Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, for instance a chemical one. Chemotaxis is an essential process in many medical and biological applications, including cell aggregation and pattern formation mechanisms and tumor growth. Modeling of chemotaxis dates back to the pioneering work by Patlak, Keller and Segel [28,29,38]. The PKS chemotaxis model consists of coupled convection-diffusion and reaction-diffusion partial differential equations:subject to zero Neumann boundary conditions and the initial conditions on the cell density ρ(x, y, z, t) and on the chemoattractant concentration c(x, y, z, t). In the system (1), the coefficient χ is a chemotactic sensitivity constant, γ ρ and γ c are the reaction coefficients. The parameter α is equal to either 1 or 0, which corresponds to the "parabolic-parabolic" or reduced "parabolic-elliptic" coupling, respectively. In this work, we will focus on the PKS chemotaxis model (1) with α = γ ρ = γ c = 1, but the developed methods are not restricted by this assumption and can be easily extended to more general chemotaxis systems and related models.In the past several years, chemotaxis models have been extensively studied (see, e.g., [19,20,21,22, 39] and references below). A known property of many chemotaxis systems is their ability to represent a concentration phenomenon that is mathematically described by fast growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular behavior. This blowup represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems, see, e.g., [1,5,6,7,11,12,36,40].Capturing blowing up or spiky solutions is a challenging task numerically, but at the same time design of robust numerical algorithms is crucial for the modeling and analysis of chemotaxis mechanisms. Let us briefly review some of the recent numerical methods that have been proposed for the "parabolic-parabolic" cou...

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Section: Remark 6 (I)mentioning

confidence: 99%

Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

Epshteyn

Xia

2019

J Sci Comput

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“…Such functions are usually introduced in the context of solving interpolation and approximation problems of scattered data on the sphere, leading to the use of positive definite kernels. The reader is referred to, for example, Hubbert et al (), Freeden et al (), and Schreiner (), for a detailed description; here we use the expression

W false(\overset{r}{true}, {\overset{r}{true}}_{0} false) = \sum_{ℓ = 0}^{\infty} k_{ℓ} \cdot Q_{ℓ} false(\overset{r}{true} \cdot {\overset{r}{true}}_{0} false) .

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Section: Gravity Gradients In Rotated Frames At Different Spatial Scalesmentioning

confidence: 99%

An Analysis of Gravitational Gradients in Rotated Frames and Their Relation to Oriented Mass Sources

Panet

2018

JGR Solid Earth

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Many mass sources within the Earth and its fluid envelopes show elongated geometries, aligning with the orientations of plate boundaries and plate motions, coastlines, rivers, and drainage basins for instance. To enhance their identification and separation in global or regional gravity observations and models, a dedicated method based on gravitational gradients analysis is presented here. This approach provides a detailed description of the geographic pattern of the gravity variations, which are accurately mapped thanks to the regular spatial coverage of high‐accuracy satellite data and arise from lateral density changes within the planet. First, gravity gradients are defined at different spatial scales in spherical frames, which are rotated along the radial axis according to the orientation of the source. The sensitivity of these gradients to the mass distribution inside a spherical Earth is described and analytical expressions relating the source to the observable are introduced. Then, the gravity gradients responses at different spatial scales to flat, elementary mass sources located at the surface and at increasing depth are studied. Specifically, the paper investigates how a source width and orientation can be determined, for localized and oscillatory mass anomalies with different width‐to‐length aspect ratios. This theoretical case study aims at providing a basis for the analysis of more complex mass structures, when applying the presented method to static or time‐varying satellite gravity field models. It may help deciphering the nature of the gravity sources by the detection of meaningful geometries and orientations in the gravity field.

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“…Though our study is generally devoted to solve "standard" 2D and 3D problems on scattered data, i.e., problems where the domain Ω is contained in R 2 or R 3 , we consider particular situations in which the domain Ω is the surface of a sphere such as the 2-sphere in R 3 [19,23], and we also comment on experiments in higher dimensions. Note that, for simplicity, we do not report thorough quantitative analysis in higher dimensions (s > 3), because from a numerical standpoint the experiments carried out for s ≤ 3 confirm both the theoretical analysis outlined in Section 2 and the practical study provided in Section 4.…”

Section: Numerical Experimentsmentioning

confidence: 99%