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Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation

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Abstract

We describe a systematic mathematical approach to the geometric discretization of gauge field theories that is based on Dirac and multi-Dirac mechanics and geometry, which provide a unified mathematical framework for describing Lagrangian and Hamiltonian mechanics and field theories, as well as degenerate, interconnected, and nonholonomic systems. Variational integrators yield geometric structure-preserving numerical methods that automatically preserve the symplectic form and momentum maps, and exhibit excellent long-time energy stability. The construction of momentum-preserving variational integrators relies on the use of group-equivariant function spaces, and we describe a general construction for functions taking values in symmetric spaces. This is motivated by the geometric discretization of general relativity, which is a second-order covariant gauge field theory on the symmetric space of Lorentzian metrics.

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References

  1. Abbott, B.P., Abbott, R., Abbott, T.D., Abernathy, M.R., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R.X., et al.: GW151226: Observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116, 241103 (2016)

    Article  Google Scholar 

  2. Abbott, B.P., Abbott, R., Abbott, T.D., Abernathy, M.R., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R.X., et al.: Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016)

    Article  MathSciNet  Google Scholar 

  3. Abbott, B.P., Abbott, R., Abbott, T.D., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R.X., Adya, V.B., et al.: GW170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2. Phys. Rev. Lett. 118, 221101 (2017)

    Article  Google Scholar 

  4. Abbott, B.P., Abbott, R., Abbott, T.D., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R.X., Adya, V.B., et al.: GW170814: A three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett. 119, 141101 (2017)

    Article  Google Scholar 

  5. Abbott, B.P., Abbott, R., Abbott, T.D., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R.X., Adya, V.B., et al.: GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101 (2017)

    Article  Google Scholar 

  6. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2008)

    Book  MATH  Google Scholar 

  7. Al-Mohy, A.H., Higham, N.J.: Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation. SIAM J. Matrix Anal. Appl. 30(4), 1639–1657 (2009)

    Article  MATH  Google Scholar 

  8. Anderson, M., Matzner, R.A.: Extended lifetime in computational evolution of isolated black holes. Foundations of Physics 35(9), 1477–1495 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Arnold, D.N.: Numerical problems in general relativity. In: P. Neittaanmki, T. Tiihonen, P. Tarvainen (eds.) Numerical Mathematics and Advanced Applications, pp. 3–15. World Scientific, River Edge, NJ (2000)

    Google Scholar 

  10. Arnowitt, R., Deser, S., Misner, C.: Dynamical structure and definition of energy in general relativity. Phys. Rev. (2) 116, 1322–1330 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  11. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411–421 (2006)

    Article  Google Scholar 

  12. Campos, C.M., de León, M., de Diego, D., Vankerschaver, J.: Unambiguous formalism for higher order lagrangian field theories. Journal of Physics A: Mathematical and Theoretical 42(47), 475207 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Castrillón López, M., Gotay, M., Marsden, J.: Parametrization and stress-energy-momentum tensors in metric field theories. J. Phys. A 41(34), 344002, 10 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Chang, J.M., Peterson, C., Kirby, M.: Feature patch illumination spaces and Karcher compression for face recognition via Grassmannians. Adv. Pure Math. 2(04), 226 (2012)

    Article  Google Scholar 

  15. Dal Maso, G.: An introduction to \(\varGamma \)-convergence. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Basel, Switzerland (1993)

    Google Scholar 

  16. de Goes, F., Liu, B., Budninskiy, M., Tong, Y., Desbrun, M.: Discrete 2-tensor fields on triangulations. In: Computer Graphics Forum, vol. 33, pp. 13–24. Wiley Online Library (2014)

  17. Demoures, F., Gay-Balmaz, F., Leyendecker, S., Ober-Blöbaum, S., Ratiu, T., Weinand, Y.: Discrete variational Lie group formulation of geometrically exact beam dynamics. Numer. Math. 130(1), 73–123 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dirac, P.: Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series, vol. 2. Belfer Graduate School of Science, New York (1964)

    Google Scholar 

  19. Einstein, A.: Näherungsweise Integration der Feldgleichungen der Gravitation. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 688–696. (1916)

  20. Ellison, C.L.: Development of multistep and degenerate variational integrators for applications in plasma physics. Ph.D. thesis, Princeton University (2016)

  21. Farr, W.M.: Numerical relativity from a gauge theory perspective. Ph.D. thesis, Massachusetts Institute of Technology (2010)

  22. Gallivan, K.A., Srivastava, A., Liu, X., Van Dooren, P.: Efficient algorithms for inferences on Grassmann manifolds. In: IEEE Workshop on Statistical Signal Processing, pp. 315–318. IEEE (2003)

  23. Gawlik, E.S., Leok, M.: Interpolation on symmetric spaces via the generalized polar decomposition. Found. Comput. Math. 18(3), 757–788. (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gotay, M., Isenberg, J., Marsden, J., Montgomery, R.: Momentum maps and classical relativistic fields. Part I: Covariant field theory (1998). (preprint, arXiv:physics/9801019 [math-ph])

  25. Gotay, M., Marsden, J.: Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula. In: Mathematical aspects of classical field theory (Seattle, WA, 1991), Contemp. Math., vol. 132, pp. 367–392. Amer. Math. Soc., Providence, RI (1992)

  26. Gotay, M., Nester, J.: Presymplectic Lagrangian systems. I. The constraint algorithm and the equivalence theorem. Ann. Inst. H. Poincaré Sect. A (N.S.) 30(2), 129–142 (1979)

    MathSciNet  MATH  Google Scholar 

  27. Grohs, P.: Quasi-interpolation in Riemannian manifolds. IMA J. Numer. Anal. 33(3), 849–874 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Grohs, P., Hardering, H., Sander, O.: Optimal a priori discretization error bounds for geodesic finite elements. Found. Comput. Math. 15(6), 1357–1411 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gundlach, C., Calabrese, G., Hinder, I., Martín-Garcí-a, J.M.: Constraint damping in the Z4 formulation and harmonic gauge. Classical and Quantum Gravity 22(17), 3767 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics, vol. 31, second edn. Springer-Verlag, Berlin (2006)

  31. Hall, J., Leok, M.: Spectral variational integrators. Numer. Math. 130(4), 681–740 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hall, J., Leok, M.: Lie group spectral variational integrators. Found. Comput. Math. 17(1), 199–257 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces, vol. 80. Academic Press, New York-London (1979)

    MATH  Google Scholar 

  34. Higham, N.J.: Functions of matrices: theory and computation. SIAM, Philadelphia, PA (2008)

    Book  MATH  Google Scholar 

  35. Hong, Y., Kwitt, R., Singh, N., Davis, B., Vasconcelos, N., Niethammer, M.: Geodesic regression on the Grassmannian. In: Computer Vision–ECCV 2014, pp. 632–646. Springer (2014)

  36. Hydon, P.E., Mansfield, E.L.: Extensions of Noether’s second theorem: from continuous to discrete systems. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 467(2135), 3206–3221 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Jacobs, H., Yoshimura, H.: Tensor products of Dirac structures and interconnection in Lagrangian mechanical systems. J. Geom. Mech. 6(1), 67–98 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Jiang, T., Fang, X., Huang, J., Bao, H., Tong, Y., Desbrun, M.: Frame field generation through metric customization. ACM Trans. Graph. 34(4), 40 (2015)

    MATH  Google Scholar 

  39. Leok, M., Ohsawa, T.: Discrete Dirac structures and implicit discrete Lagrangian and Hamiltonian systems. AIP Conference Proceedings 1260(1), 91–102 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Leok, M., Ohsawa, T.: Variational and geometric structures of discrete Dirac mechanics. Found. Comput. Math. 11(5), 529–562 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Leok, M., Shingel, T.: General techniques for constructing variational integrators. Front. Math. China 7(2), 273–303 (2012). (Special issue on computational mathematics, invited paper)

    Article  MathSciNet  MATH  Google Scholar 

  42. Leok, M., Zhang, J.: Discrete Hamiltonian variational integrators. IMA J. Numer. Anal. 31(4), 1497–1532 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. Marsden, J., Patrick, G., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199(2), 351–395 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. Marsden, J., Pekarsky, S., Shkoller, S., West, M.: Variational methods, multisymplectic geometry and continuum mechanics. J. Geom. Phys. 38(3-4), 253–284 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  45. Marsden, J., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 317–514 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  46. Mathias, R.: A chain rule for matrix functions and applications. SIAM J. Matrix Anal. Appl. 17(3), 610–620 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  47. McLachlan, R.I., Modin, K., Verdier, O., Wilkins, M.: Geometric generalisations of SHAKE and RATTLE. Found. Comput. Math. 14(2), 339–370 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Müller, S., Ortiz, M.: On the \(\Gamma \)-convergence of discrete dynamics and variational integrators. J. Nonlinear Sci. 14(3), 279–296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. Munthe-Kaas, H.Z., Quispel, G.R.W., Zanna, A.: Generalized polar decompositions on Lie groups with involutive automorphisms. Found. Comput. Math. 1(3), 297–324 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  50. Munthe-Kaas, H.Z., Quispel, G.R.W., Zanna, A.: Symmetric spaces and Lie triple systems in numerical analysis of differential equations. BIT Numer. Math. 54(1), 257–282 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Palatini, A.: Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rendiconti del Circolo Matematico di Palermo (1884-1940) 43(1), 203–212 (1919)

    Article  MATH  Google Scholar 

  52. Pretorius, F.: Evolution of binary black-hole spacetimes. Phys. Rev. Lett. 95, 121101 (2005)

    Article  MathSciNet  Google Scholar 

  53. Sander, O.: Geodesic finite elements for Cosserat rods. Int. J. Numer. Meth. Eng. 82(13), 1645–1670 (2010)

    MathSciNet  MATH  Google Scholar 

  54. Sander, O.: Geodesic finite elements on simplicial grids. Int. J. Numer. Meth. Eng. 92(12), 999–1025 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. Sander, O.: Geodesic finite elements of higher order. IMA J. Numer. Anal. 36(1), 238–266 (2016)

    MathSciNet  MATH  Google Scholar 

  56. Skinner, R., Rusk, R.: Generalized Hamiltonian dynamics. I. Formulation on \(T^*Q\oplus TQ\). Journal of Mathematical Physics 24(11), 2589–2594 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  57. Skinner, R., Rusk, R.: Generalized Hamiltonian dynamics. II. Gauge transformations. Journal of Mathematical Physics 24(11), 2595–2601 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tulczyjew, W.M.: The Legendre transformation. Ann. Inst. H. Poincaré Sect. A (N.S.) 27(1), 101–114 (1977)

    MathSciNet  MATH  Google Scholar 

  59. Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition. IEEE T. Pattern Anal. 33(11), 2273–2286 (2011)

    Article  Google Scholar 

  60. Vankerschaver, J., Liao, C., Leok, M.: Generating functionals and Lagrangian partial differential equations. J. Math. Phys. 54(8), 082901 (22 pages) (2013)

    Article  MathSciNet  MATH  Google Scholar 

  61. Vankerschaver, J., Yoshimura, H., Leok, M.: On the geometry of multi-Dirac structures and Gerstenhaber algebras. J. Geom. Phys. 61(8), 1415–1425 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  62. Vankerschaver, J., Yoshimura, H., Leok, M.: The Hamilton–Pontryagin principle and multi-Dirac structures for classical field theories. J. Math. Phys. 53(7), 072903 (25 pages) (2012)

    Article  MathSciNet  MATH  Google Scholar 

  63. Weyhausen, A., Bernuzzi, S., Hilditch, D.: Constraint damping for the Z4c formulation of general relativity. Phys. Rev. D 85, 024038 (2012)

    Article  Google Scholar 

  64. Yoshimura, H., Marsden, J.: Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. J. Geom. Phys. 57(1), 133–156 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Yoshimura, H., Marsden, J.: Dirac structures in Lagrangian mechanics Part II: Variational structures. J. Geom. Phys. 57(1), 209–250 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Evan Gawlik for helpful discussions and comments. The author has been supported in part by the National Science Foundation under Grants DMS-1010687, CMMI-1029445, DMS-1065972, CMMI-1334759, DMS-1411792, DMS-1345013, DMS-1813635.

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  1. Department of Mathematics, University of California, 9500 Gilman Drive, La Jolla, San Diego, CA, 92093-0112, USA

    Melvin Leok

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Leok, M. Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation. Found Comput Math 19, 965–989 (2019). https://doi.org/10.1007/s10208-019-09420-4

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