标题： 样条插值在紧致黎曼流形上 显示英文标题

标题： Spline Interpolation on Compact Riemannian Manifolds

摘要： 样条插值是一种广泛使用的解决插值问题的方法，通过构造平滑的插值函数来最小化涉及拉普拉斯算子的正则化能量泛函。 虽然许多现有方法专注于欧几里得域或球面，并依赖于拉普拉斯算子的谱特性，但本文提出了一种在一般流形上进行样条插值的方法，利用其与克里金法的等价性。 具体而言，所提出的方法基于定义在流形上的随机场的有限元近似，结合高斯马尔可夫随机场和三角剖分网格上拉普拉斯-贝尔特拉米算子的离散化。 该框架可以通过域变形对具有局部各向异性的空间场进行建模。 该方法首先在球面上进行了验证，使用了分析测试用例和一个与污染相关的研究，并与经典的基于球面谐波的方法进行了比较。 在圆柱体表面的额外实验进一步说明了该方法的通用性。 显示英文摘要

摘要： Spline interpolation is a widely used class of methods for solving interpolation problems by constructing smooth interpolants that minimize a regularized energy functional involving the Laplacian operator. While many existing approaches focus on Euclidean domains or the sphere, relying on the spectral properties of the Laplacian, this work introduces a method for spline interpolation on general manifolds by exploiting its equivalence with kriging. Specifically, the proposed approach uses finite element approximations of random fields defined over the manifold, based on Gaussian Markov Random Fields and a discretization of the Laplace-Beltrami operator on a triangulated mesh. This framework enables the modeling of spatial fields with local anisotropies through domain deformation. The method is first validated on the sphere using both analytical test cases and a pollution-related study, and is compared to the classical spherical harmonics-based method. Additional experiments on the surface of a cylinder further illustrate the generality of the approach.