A simple embedding method for solving partial differential equations on surfaces

Ruuth, Steven J.; Merriman, Barry

doi:10.1016/j.jcp.2007.10.009

Cited by 226 publications

(307 citation statements)

References 29 publications

(59 reference statements)

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Section: Introductionmentioning

confidence: 99%

“…It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space.…”

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confidence: 99%

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Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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The Laplace-Beltrami operator, a fundamental object associated with Riemannian manifolds, encodes all intrinsic geometry of manifolds and has many desirable properties. Recently, we proposed the point integral method (PIM), a novel numerical method for discretizing the Laplace-Beltrami operator on point clouds (Li et al. in Commun Comput Phys 22(1):228-258, 2017). In this paper, we analyze the convergence of PIM for Poisson equation with Neumann boundary condition on submanifolds that are isometrically embedded in Euclidean spaces. Keywords: Laplace-Beltrami operator, Neumann boundary, Point cloud, Point integral method, Convergence analysisMathematics Subject Classification: 65N12, 65N25, 65N75 BackgroundThe partial differential equations on manifolds arise in a wide variety of applications. In many problems, including material science [10,20], fluid flow [22,25], biology and biophysics [2,3,21,37], people need to study the physical process, for instance diffusion and convection, in curved surfaces which introduce different kinds of PDEs in surfaces. It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space. One fundamental assumption in the manifold model is that the point cloud samples a smooth manifold. Thus, the information of the manifold is very useful to understand the data or images. PDEs on the manifold, particularly the Laplace-Beltrami equation, encode several intrinsic information of the manifold, thus helping reveal the underlying structures in the data or images. To get the information encoded in PDEs, we need to solve them in the unstructured point cloud. Given that the point cloud is embedded in a high-dimensional space, the traditional methods for PDEs on 2D surfaces do not work.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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“…We present a simple illustration of motion by mean curvature as an annealing technique. This is work in progress with Shahriari and Ruuth [64] which numerically addresses (NLCH) and (MCH) on surfaces using the Closest Point Method of Ruuth and Merriman [61]. On any given surface, we combine the time integration of the (MCH) with motion by mean curvature (viewing u as a level set function) to arrive at low energy states.…”

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confidence: 99%

On global minimizers for a variational problem with long-range interactions

Choksi

2012

Quart. Appl. Math.

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Abstract. Energy-driven pattern formation induced by competing short and longrange interactions is common in many physical systems. In these proceedings we report on certain rigorous asymptotic results concerning global minimizers of a nonlocal perturbation to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. We also discuss two hybrid numerical methods for accessing the ground states of these functionals.

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“…For wider stencils, it is necessary to consider one-sided non-oscillatory approximations of the derivatives in order to minimize errors due to differencing over kinks of the distance function. The closest point mapping is used in [22] as an Eulerian method to track interfaces and in [14,20] for solving PDEs on surfaces. In the case that Γ is given as a collection of parameterized patches, one may use the fast algorithm proposed in [23] to compute the closest point mappings.…”

Section: The Implicit Boundary Integral Methodsmentioning

confidence: 99%

Implicit boundary integral methods for the Helmholtz equation in exterior domains

Chen

Tsai

2017

Res Math Sci

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We propose a new algorithm for solving Helmholtz equations in exterior domains with implicitly represented boundaries. The algorithm not only combines the advantages of implicit surface representation and the boundary integral method, but also provides a new way to compute a class of the so-called hypersingular integrals. The keys to the proposed algorithm are the derivation of the volume integrals which are equivalent to any given integrals on smooth closed hypersurfaces, and the ability to approximate the natural limit of the singular integrals via seamless extrapolation. We present numerical results for both two-and three-dimensional scattering problems at near resonant frequencies as well as with non-convex scattering surfaces. Keywords: Helmholtz equation, Hypersingular integrals, Level set methods, Closest point projection, Boundary integrals BackgroundLet Γ be a closed and compact C 2 hypersurface that separates R m , m = 2, 3, into a simply connected and bounded open region Ω and its complement. We consider the solution of the following Neumann boundary value problem for the Helmholtz equation:(1.1)For wave scattering by a sound hard boundary ∂Ω, a total wave field u tot (x) is a function satisfying(1.2) This total wave field is then written artificially as the sum u tot = u inc + u sc , where u inc is a known function that is used to model the far-field condition of u tot and u sc is the solution of (1

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A simple embedding method for solving partial differential equations on surfaces

Cited by 226 publications

References 29 publications

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

On global minimizers for a variational problem with long-range interactions

Implicit boundary integral methods for the Helmholtz equation in exterior domains

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