We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal eq...We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.展开更多
In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Ope...In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.展开更多
This contribution gives results on the action of the Laplace-Beltrami derivative on suffi- ciently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least c...This contribution gives results on the action of the Laplace-Beltrami derivative on suffi- ciently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.展开更多
We consider optimal control problems of elliptic PDEs on hypersurfaces F in R^n for n = 2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhed...We consider optimal control problems of elliptic PDEs on hypersurfaces F in R^n for n = 2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of F. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.展开更多
We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces.The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadratu...We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces.The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadrature coupled with standard finite element methods for parametric surfaces.Possibly up to a log term,optimal rate of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in L^(2)and H^(1).The performances of the algorithms are illustrated on different settings including the approximation of Gaussian fields on surfaces.展开更多
In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator po...In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.展开更多
0. Introduction Let X be a real separable Banach space and X~* be its dual space, Let B(X) be the Borel field, i.e., the topological σ-field. A functional u: X→R’ is called a bounded smooth functional, if n∈N, f1,...0. Introduction Let X be a real separable Banach space and X~* be its dual space, Let B(X) be the Borel field, i.e., the topological σ-field. A functional u: X→R’ is called a bounded smooth functional, if n∈N, f1, …, fn∈ X~* and φ∈Cb~∞(Rn), such that展开更多
Riemannian流形和Khler流形上Laplace-Beltrami算子谱的下界的估计是微分几何研究领域的热点问题.针对LiS和Tran M A得到的关于Laplace-Beltrami算子谱的下界的估计,利用华罗庚先生和陆启铿先生关于有界对称典型域的研究结论,得出了...Riemannian流形和Khler流形上Laplace-Beltrami算子谱的下界的估计是微分几何研究领域的热点问题.针对LiS和Tran M A得到的关于Laplace-Beltrami算子谱的下界的估计,利用华罗庚先生和陆启铿先生关于有界对称典型域的研究结论,得出了第一类有界对称典型域上Laplace-Beltrami算子谱的下界估计.展开更多
基金supported in part by the Hong Kong RGC 16302223.
文摘We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.
文摘In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.
基金supported by CAPES-Brasilsupported by FAPESP-Brasil(Grant No.2010/19734-6)
文摘This contribution gives results on the action of the Laplace-Beltrami derivative on suffi- ciently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.
文摘We consider optimal control problems of elliptic PDEs on hypersurfaces F in R^n for n = 2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of F. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.
基金supported by the NSF(Grants DMS-1817691,DMS-2110811).
文摘We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces.The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadrature coupled with standard finite element methods for parametric surfaces.Possibly up to a log term,optimal rate of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in L^(2)and H^(1).The performances of the algorithms are illustrated on different settings including the approximation of Gaussian fields on surfaces.
文摘In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.
文摘0. Introduction Let X be a real separable Banach space and X~* be its dual space, Let B(X) be the Borel field, i.e., the topological σ-field. A functional u: X→R’ is called a bounded smooth functional, if n∈N, f1, …, fn∈ X~* and φ∈Cb~∞(Rn), such that
文摘Riemannian流形和Khler流形上Laplace-Beltrami算子谱的下界的估计是微分几何研究领域的热点问题.针对LiS和Tran M A得到的关于Laplace-Beltrami算子谱的下界的估计,利用华罗庚先生和陆启铿先生关于有界对称典型域的研究结论,得出了第一类有界对称典型域上Laplace-Beltrami算子谱的下界估计.
