Related papers: A least-squares implicit RBF-FD closest point meth…
Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding…
Partial differential equations (PDEs) on surfaces arise in a wide range of applications. The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is a recent embedding method that has been used to solve a…
We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the…
The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study…
This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in…
Elliptic partial differential equations are important both from application and analysis points of views. In this paper we apply the Closest Point Method to solving elliptic equations on general curved surfaces. Based on the closest point…
The Closest Point method, initially developed by Ruuth and Merriman, allows for the numerical solution of surface partial differential equations without the need for a parameterization of the surface itself. Surface quantities are embedded…
Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions…
Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $\Omega$. One of the requirements leading to approximation robustness is to place…
The discretization of surface intrinsic PDEs has challenges that one might not face in the flat space. The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in the flat space to…
Maps from a source manifold $ {\mathcal M}$ to a target manifold ${\mathcal N}$ appear in liquid crystals, colour image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems…
Recently, collocation based radial basis function (RBF) partition of unity methods (PUM) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation…
Real-time fluid and aeroacoustic simulation on complex surfaces can have interactive applications - from globe-based weather visualizations to immersive computer games with physically accurate wind and sound. However, conventional…
Derivative boundary conditions introduce challenges for mesh-free discretizations of PDEs on surfaces, especially when the domain is represented by randomly sampled point clouds. The recently developed two-step tangent-space RBF-generated…
We generalize the closest point method (CPM) to solve surface partial differential equations with general boundary conditions. The proposed extrapolation method provides a unified framework for treating a broad class of inhomogeneous…
In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in $\mathbb{R}^d$. Our method…
We describe an efficient method for the approximation of functions using radial basis functions (RBFs), and extend this to a solver for boundary value problems on irregular domains. The method is based on RBFs with centers on a regular grid…
We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We…
The discretization of surface intrinsic elliptic partial differential equations (PDEs) poses interesting challenges not seen in flat space. The discretization of these PDEs typically proceeds by either parametrizing the surface,…
We demonstrate a method for filtering images defined on curved surfaces embedded in 3D. Applications are noise removal and the creation of artistic effects. Our approach relies on in-surface diffusion: we formulate Weickert's edge/coherence…