Interpolation on Symmetric Spaces Via the Generalized Polar Decomposition
Interpolation on Symmetric Spaces Via the Generalized Polar Decomposition
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DOI：
10.1007/s10208-017-9353-0
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发表时间：
2016-05
影响因子：
3
通讯作者：
Evan S. Gawlik;M. Leok
中科院分区：
文献类型：
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作者：
Evan S. Gawlik;M. Leok
我们构造了在对称空间中取值的函数的插值算子——一个关于每一点都具有逆对称的光滑流形。我们构造的关键是观察到每个对称空间都可以被实现为齐次空间，它的协集有正则表示，这是借助于广义极分解——将实非奇异矩阵分解为对称正定矩阵乘以正交矩阵的乘积的一种推广。通过插值这些正则协集表示，我们得到了对称空间值函数的一类保结构插值算子。作为应用，我们构造了洛伦兹度量空间、对称正定矩阵空间和格拉斯曼空间的插值算子。在洛伦兹度量的情况下，我们的插值算子为数值相对论提供了一组有限元素，它们是帧不变的，并且具有保证是洛伦兹点向的签名。我们通过数值插值史瓦西度规来说明它们的潜在效用。
We construct interpolation operators for functions taking values in a symmetric space—a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition—a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically.