doi: 10.3934/jcd.2022003
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High-order symplectic Lie group methods on $ SO(n) $ using the polar decomposition

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

* Corresponding author: Melvin Leok

Received  July 2018 Revised  January 2022 Early access February 2022

Fund Project: This research has been supported in part by NSF under grants DMS-1411792, DMS-1345013, DMS-1813635, CCF-2112665, by AFOSR under grant FA9550-18-1-0288, and by the DoD under grant HQ00342010023 (Newton Award for Transformative Ideas during the COVID-19 Pandemic)

A variational integrator of arbitrarily high-order on the special orthogonal group $ SO(n) $ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie–Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on $ SO(3) $ which demonstrate that they are comparable to variational Runge–Kutta–Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.

Citation: Xuefeng Shen, Khoa Tran, Melvin Leok. High-order symplectic Lie group methods on $ SO(n) $ using the polar decomposition. Journal of Computational Dynamics, doi: 10.3934/jcd.2022003
References:
[1] P.-A. AbsilR. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008.  doi: 10.1515/9781400830244.
[2]

R. H. Bartels and G. W. Stewart, Solution of the matrix equation $ax+ xb = c$ [f4], Communications of the ACM, 15 (1972), 820-826.  doi: 10.1145/361573.361582.

[3]

G. Bogfjellmo and H. Marthinsen, High-order symplectic partitioned Lie group methods, Found. Comput. Math., 16 (2016), 493-530.  doi: 10.1007/s10208-015-9257-9.

[4]

E. Celledoni and B. Owren, A class of intrinsic schemes for orthogonal integration, SIAM J. Numer. Anal., 40 (2002), 2069-2084.  doi: 10.1137/S0036142901385143.

[5]

V. Duruisseaux and M. Leok, A variational formulation of accelerated optimization on Riemannian manifolds, SIAM Journal on Mathematics of Data Science, Accepted.

[6]

V. DuruisseauxJ. Schmitt and M. Leok, Adaptive Hamiltonian variational integrators and applications to symplectic accelerated optimization, SIAM J. Sci. Comput., 43 (2021), A2949-A2980.  doi: 10.1137/20M1383835.

[7]

G. H. GolubS. Nash and C. Van Loan, A Hessenberg-Schur method for the problem $ax+ xb = c$, IEEE Trans. Automat. Control, 24 (1979), 909-913.  doi: 10.1109/TAC.1979.1102170.

[8]

J. Hall and M. Leok, Lie group spectral variational integrators, Found. Comput. Math., 17 (2017), 199-257.  doi: 10.1007/s10208-015-9287-3.

[9]

T. LeeM. Leok and N. McClamroch, Lie group variational integrators for the full body problem in orbital mechanics, Celestial Mech. Dynam. Astronom., 98 (2007), 121-144.  doi: 10.1007/s10569-007-9073-x.

[10]

M. Leok, Generalized Galerkin variational integrators, arXiv: math/0508360.

[11]

M. Leok and T. Shingel, General techniques for constructing variational integrators, Front. Math. China, 7 (2012), 273-303.  doi: 10.1007/s11464-012-0190-9.

[12]

J. E. MarsdenS. Pekarsky and S. Shkoller, Discrete Euler–Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.  doi: 10.1088/0951-7715/12/6/314.

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, 1999. doi: 10.1007/978-0-387-21792-5.

[14]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[15]

H. Z. Munthe-KaasG. R. W. Quispel and A. Zanna, Generalized polar decompositions on Lie groups with involutive automorphisms, Found. Comput. Math., 1 (2001), 297-324.  doi: 10.1007/s002080010012.

[16]

A. WibisonoA. C. Wilson and M. I. Jordan, A variational perspective on accelerated methods in optimization, Proc. Natl. Acad. Sci. USA, 113 (2016), E7351-E7358.  doi: 10.1073/pnas.1614734113.

show all references

References:
[1] P.-A. AbsilR. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008.  doi: 10.1515/9781400830244.
[2]

R. H. Bartels and G. W. Stewart, Solution of the matrix equation $ax+ xb = c$ [f4], Communications of the ACM, 15 (1972), 820-826.  doi: 10.1145/361573.361582.

[3]

G. Bogfjellmo and H. Marthinsen, High-order symplectic partitioned Lie group methods, Found. Comput. Math., 16 (2016), 493-530.  doi: 10.1007/s10208-015-9257-9.

[4]

E. Celledoni and B. Owren, A class of intrinsic schemes for orthogonal integration, SIAM J. Numer. Anal., 40 (2002), 2069-2084.  doi: 10.1137/S0036142901385143.

[5]

V. Duruisseaux and M. Leok, A variational formulation of accelerated optimization on Riemannian manifolds, SIAM Journal on Mathematics of Data Science, Accepted.

[6]

V. DuruisseauxJ. Schmitt and M. Leok, Adaptive Hamiltonian variational integrators and applications to symplectic accelerated optimization, SIAM J. Sci. Comput., 43 (2021), A2949-A2980.  doi: 10.1137/20M1383835.

[7]

G. H. GolubS. Nash and C. Van Loan, A Hessenberg-Schur method for the problem $ax+ xb = c$, IEEE Trans. Automat. Control, 24 (1979), 909-913.  doi: 10.1109/TAC.1979.1102170.

[8]

J. Hall and M. Leok, Lie group spectral variational integrators, Found. Comput. Math., 17 (2017), 199-257.  doi: 10.1007/s10208-015-9287-3.

[9]

T. LeeM. Leok and N. McClamroch, Lie group variational integrators for the full body problem in orbital mechanics, Celestial Mech. Dynam. Astronom., 98 (2007), 121-144.  doi: 10.1007/s10569-007-9073-x.

[10]

M. Leok, Generalized Galerkin variational integrators, arXiv: math/0508360.

[11]

M. Leok and T. Shingel, General techniques for constructing variational integrators, Front. Math. China, 7 (2012), 273-303.  doi: 10.1007/s11464-012-0190-9.

[12]

J. E. MarsdenS. Pekarsky and S. Shkoller, Discrete Euler–Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.  doi: 10.1088/0951-7715/12/6/314.

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, 1999. doi: 10.1007/978-0-387-21792-5.

[14]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[15]

H. Z. Munthe-KaasG. R. W. Quispel and A. Zanna, Generalized polar decompositions on Lie groups with involutive automorphisms, Found. Comput. Math., 1 (2001), 297-324.  doi: 10.1007/s002080010012.

[16]

A. WibisonoA. C. Wilson and M. I. Jordan, A variational perspective on accelerated methods in optimization, Proc. Natl. Acad. Sci. USA, 113 (2016), E7351-E7358.  doi: 10.1073/pnas.1614734113.

Figure 1.  Order comparison plots between VRKMK and VPD methods: The black-dashed lines are references for the corresponding orders
Figure 2.  Long-term energy error for second order VRKMK/VPD methods
Figure 3.  Long-term energy error for third order VRKMK/VPD methods
Figure 4.  Long-term energy error for fourth order VRKMK/VPD methods
Figure 5.  Long-term energy error for sixth order VRKMK/VPD methods
Figure 6.  Orthogonality error for sixth order VRKMK/VPD methods
Figure 7.  Run-time comparison for fourth and sixth order VRKMK/VPD methods
Table 1.  Butcher tableaux for the comparison tests
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