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Foundations of Computational Mathematics

, Volume 18, Issue 3, pp 757–788 | Cite as

Interpolation on Symmetric Spaces Via the Generalized Polar Decomposition

  • Evan S. Gawlik
  • Melvin Leok
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 

Notes

Acknowledgements

EG has been supported in part by NSF under Grants DMS-1411792, DMS-1345013. ML has been supported in part by NSF under Grants DMS-1010687, CMMI-1029445, DMS-1065972, CMMI-1334759, DMS-1411792, DMS-1345013.

References

  1. 1.
    P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Mani- folds. Princeton University Press, 2009.Google Scholar
  2. 2.
    P.-A. Absil, R. Mahony, and R. Sepulchre. Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica 80.2 (2004), pp. 199–220.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. H. Al-Mohy and N. J. Higham. Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation. SIAM Journal on Matrix Analysis and Applications 30.4 (2009), pp. 1639–1657CrossRefzbMATHGoogle Scholar
  4. 4.
    D. Amsallem and C. Farhat. Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA Journal 46.7 (2008), pp. 1803–1813.CrossRefGoogle Scholar
  5. 5.
    D. N. Arnold. Numerical problems in general relativity. Numerical Mathematics and Advanced Applications (P. Neittaanmki, T. Tiihonen, and P. Tarvainen, eds.), World Scientific (2000), pp. 3–15.Google Scholar
  6. 6.
    V. Arsigny, P. Fillard, X. Pennec, and N. Ayache. Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications 29.1 (2007), pp. 328–347.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. Arsigny, P. Fillard, X. Pennec, and N. Ayache. Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine 56.2 (2006), pp. 411–421.CrossRefGoogle Scholar
  8. 8.
    E. Begelfor and M. Werman. Affine invariance revisited. Conference on Computer Vision and Pattern Recognition. IEEE. 2006, pp. 2087–2094.Google Scholar
  9. 9.
    R. Bhatia. The Riemannian mean of positive matrices. Matrix Information Geometry. Springer, 2013, pp. 35–51.Google Scholar
  10. 10.
    S. M. Carroll. Spacetime and Geometry: An Introduction to General Relativity. San Francisco, CA, USA: Addison Wesley, 2004.zbMATHGoogle Scholar
  11. 11.
    E. Celledoni, M. Eslitzbichler, and A. Schmeding. Shape Analysis on Lie Groups with Applications in Computer Animation. arXiv preprint arXiv:1506.00783 (2015).
  12. 12.
    E. Celledoni and A. Iserles. Approximating the exponential from a Lie algebra to a Lie group. Math. Comp. 69.232 (2000), pp. 1457–1480.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J.-M. Chang, C. Peterson, M. Kirby, et al. Feature patch illumination spaces and Karcher compression for face recognition via Grassmannians. Advances in Pure Mathematics 2.04 (2012), p. 226.Google Scholar
  14. 14.
    F Demoures et al. Discrete variational Lie group formulation of geometrically exact beam dynamics. Numerische Mathematik 130.1 (2015), pp. 73-123.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    I. L. Dryden and K. V. Mardia. Statistical Shape Analysis: With Applications in R. Wiley, 2016.Google Scholar
  16. 16.
    T. Duchamp, G. Xie, and T. P.-Y. Yu. Single basepoint subdivision schemes for manifold-valued data: time-symmetry without space-symmetry. Foundations of Com- putational Mathematics 13.5 (2013), pp. 693–728. doi: 10.1007/s10208-013-9144-1.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Edelman, T. A. Arias, and S. T. Smith. The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications 20.2 (1998), pp. 303–353.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    P. T. Fletcher, C. Lu, and S. Joshi. Statistics of shape via principal geodesic analysis on Lie groups. 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Vol. 1. IEEE. 2003, pp. 1–7.Google Scholar
  19. 19.
    K. A. Gallivan, A. Srivastava, X. Liu, and P. Van Dooren. Efficient algorithms for inferences on grassmann manifolds. 2003 IEEE Workshop on Statistical Signal Processing. IEEE. 2003, pp. 315–318.Google Scholar
  20. 20.
    F. de Goes, B. Liu, M. Budninskiy, Y. Tong, and M. Desbrun. Discrete 2-Tensor Fields on Triangulations. Computer Graphics Forum. Vol. 33. 5.Wiley Online Library. 2014, pp. 13–24.Google Scholar
  21. 21.
    P. Grohs. Quasi-interpolation in Riemannian manifolds. IMA Journal of Numerical Analysis 33.3 (2013), pp. 849–874.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    P. Grohs, H. Hardering, and O. Sander. Optimal a priori discretization error bounds for geodesic finite elements. Foundations of Computational Mathematics 15.6 (2015), pp. 1357–1411.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    P. Grohs, M. Sprecher, and T. Yu. Scattered manifold-valued data approximation. Numerische Mathematik 135.4 (2017), pp. 987–1010. doi: 10.1007/s00211-016-0823-0.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    J. Hall and M. Leok. Lie group spectral variational integrators. Foundations of Computational Mathematics, pp. 1–59.Google Scholar
  25. 25.
    S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. Vol. 80. Academic press, 1979.Google Scholar
  26. 26.
    N. J. Higham. Functions of Matrices: Theory and Computation. SIAM, 2008.Google Scholar
  27. 27.
    N. J. Higham. J-orthogonal matrices: Properties and generation. SIAM review 45.3 (2003), pp. 504–519.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    N. J. Higham, C. Mehl, and F. Tisseur. The canonical generalized polar decomposition. SIAM Journal on Matrix Analysis and Applications 31.4 (2010), pp. 2163–2180.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    J. Hilgert and K.-H. Neeb. Structure and Geometry of Lie Groups. Springer, 2011.Google Scholar
  30. 30.
    Y. Hong et al. Geodesic regression on the Grassmannian. Computer Vision-ECCV 2014. Springer, 2014, pp. 632–646.Google Scholar
  31. 31.
    A. Iserles and A. Zanna. Efficient computation of the matrix exponential by generalized polar decompositions. SIAM Journal on Numerical Analysis 42.5 (2005), pp. 2218–2256.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    T. Jiang et al. Frame field generation through metric customization. ACM Transactions on Graphics (TOG) 34.4 (2015), p. 40.Google Scholar
  33. 33.
    H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30.5 (1977), pp. 509–541.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    A. W. Knapp. Lie Groups: Beyond an Introduction. Vol. 140. Springer, 2013.Google Scholar
  35. 35.
    S. Kobayashi and K. Nomizu. Foundations of Differential Geometry. Vol. 2.Wiley New York, 1969.zbMATHGoogle Scholar
  36. 36.
    A. Marthinsen. Interpolation in Lie groups. SIAM Journal on Numerical Analysis 37.1 (1999), pp. 269–285.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    R. Mathias. A chain rule for matrix functions and applications. SIAM Journal on Matrix Analysis and Applications 17.3 (1996), pp. 610–620.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    M. Moakher. A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications 26.3 (2005), pp. 735–747.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    A. Mota, W. Sun, J. T. Ostien, J. W. Foulk, and K. N. Long. Lie-group interpolation and variational recovery for internal variables. Computational Mechanics 52.6 (2013), pp. 1281–1299.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    H. Z. Munthe-Kaas, G. R.W. Quispel, and A. Zanna. Symmetric spaces and Lie triple systems in numerical analysis of differential equations. BIT Numerical Mathematics 54.1 (2014), pp. 257–282.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    H. Z. Munthe-Kaas, G. Quispel, and A. Zanna. Generalized polar decompositions on Lie groups with involutive automorphisms. Foundations of Computational Mathematics 1.3 (2001), pp. 297–324.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    O. Sander. Geodesic finite elements for Cosserat rods. International Journal for Numerical Methods in Engineering 82.13 (2010), pp. 1645–1670.MathSciNetzbMATHGoogle Scholar
  43. 43.
    O. Sander. Geodesic finite elements of higher order. IMA J. Numer. Anal. 36.1 (2016), pp. 238–266.MathSciNetzbMATHGoogle Scholar
  44. 44.
    O. Sander. Geodesic finite elements on simplicial grids. International Journal for Numerical Methods in Engineering 92.12 (2012), pp. 999–1025.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    T. Shingel. Interpolation in special orthogonal groups. IMA Journal of Numerical Analysis (2008).Google Scholar
  46. 46.
    L. N. Trefethen and D. Bau III. Numerical Linear Algebra. Vol. 50. SIAM, 1997.Google Scholar
  47. 47.
    P. Turaga, A. Veeraraghavan, A. Srivastava, and R. Chellappa. Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 33.11 (2011), pp. 2273–2286.CrossRefGoogle Scholar
  48. 48.
    F. Vetrano, C. Le Garrec, G. D. Mortchelewicz, and R. Ohayon. Assessment of strategies for interpolating POD based reduced order models and application to aeroelasticity. Journal of Aeroelasticity and Structural Dynamics 2.2 (2012).Google Scholar
  49. 49.
    J. Wallner, E. N. Yazdani, and A. Weinmann. Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces. Advances in Computational Mathematics 34.2 (2011), pp. 201–218.MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    E. N. Yazdani and T. P.-Y. Yu. On Donoho’s Log-Exp subdivision scheme: choice of retraction and time-symmetry. Multiscale Modeling and Simulation 9.4 (2011), pp. 1801–1828.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SFoCM 2017

Authors and Affiliations

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

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