Interpolation on Symmetric Spaces Via the Generalized Polar Decomposition

Article

Abstract

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.

Keywords

Interpolation Manifold-valued data Symmetric space Generalized polar decomposition Grassmannian Lorentzian metric Lie triple system Geodesic finite element Karcher mean Log-Euclidean mean 

Mathematics Subject Classification

Primary 65D05 53C35 Secondary 65N30 58J70 53B30 

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Copyright information

© SFoCM 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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