Spectral Exterior Calculus
Spectral Exterior Calculus
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DOI：
10.1002/cpa.21885
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发表时间：
2018-02
影响因子：
3
通讯作者：
Tyrus Berry;D. Giannakis
中科院分区：
文献类型：
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作者：
Tyrus Berry;D. Giannakis
提出了一种建立流形学习问题外部微积分的谱方法。在大数据的限制下，谱法收敛于真正的外部微积分。同时，谱方法将内存需求与数据点数量和环境空间维度解耦。为了实现这一点，外部演算完全被重新表述为函数上拉普拉斯算子的特征值和特征函数。这些特征函数的外部导数（以及它们的楔积）被证明形成了一个坐标系（一种生成集），适用于适当的k -型L2空间，以及高阶Sobolev空间。导出了用拉普拉斯函数的本征函数和本征值表示形式上的拉普拉斯-德-朗算子的公式。通过在这个框架中表示拉普拉斯算子，利用伽辽金近似技术得到了谱收敛结果。数值例子证明了拉普拉斯-德-拉姆算子在1 - form上的特征值和特征形式的精确恢复。通过在若干可定向流形和不可定向流形上采样的数据逼近该算子的核，得到了正确的Betti数，并通过其对应的向量场将特征形式可视化。这些向量场构成了平方可积向量场空间的自然标准正交基，并由测量振荡行为的狄利克雷能量泛函排序。谱框架在一个非光滑的例子（洛伦兹63吸引子）上也显示出有希望的结果，表明外部微积分的谱公式在没有可微结构的空间中是可行的。©2020 Wiley期刊公司
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 L2 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. © 2020 Wiley Periodicals, Inc.