A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space dimension. To achieve this, the exterior calculus is reformulated entirely in terms of the eigenvalues and eigenfunctions of the Laplacian operator on functions. The exterior derivatives of these eigenfunctions (and their wedge products) are shown to form a frame (a type of spanning set) for appropriate $L^2$ spaces of $k$-forms, as well as higher-order Sobolev spaces. Formulas are derived to express the Laplace-de Rham operators on forms in terms of the eigenfunctions and eigenvalues of the Laplacian on functions. By representing the Laplace-de Rham operators in this frame, spectral convergence results are obtained via Galerkin approximation techniques. Numerical examples demonstrate accurate recovery of eigenvalues and eigenforms of the Laplace-de Rham operator on 1-forms. The correct Betti numbers are obtained from the kernel of this operator approximated from data sampled on several orientable and non-orientable manifolds, and the eigenforms are visualized via their corresponding vector fields. These vector fields form a natural orthonormal basis for the space of square-integrable vector fields, and are ordered by a Dirichlet energy functional which measures oscillatory behavior. The spectral framework also shows promising results on a non-smooth example (the Lorenz 63 attractor), suggesting that a spectral formulation of exterior calculus may be feasible in spaces with no differentiable structure.
Exterior differential calculus in generalized Lie algebras(algebroids) category with applications to interior and exterior algebraic(differential) systems
Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
Institute of Scientific Computing, TU Dresden（德累斯顿工业大学科学计算研究所）；Dresden Center for Computational Materials Science (DCMS), TU Dresden（德累斯顿工业大学德累斯顿计算材料科学中心）；Center for Systems Biology Dresden (CSBD)（德累斯顿系统生物学中心）
Whitney Forms for Spherical Triangles I: The Euler, Cagnoli, and Tuynman Area Formulas, Barycentric Coordinates, and Construction with the Exterior Calculus
The exterior derivative and the mean value equality in $\mathbb{R}^n$
Instituto de Ciências Matemáticas e de Computação (ICMC), Universidade de São Paulo (USP)（圣保罗大学数学与计算科学学院）；Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学数学、统计与科学计算研究所）
