American Institute of Mathematical Sciences

Advanced
March  2011, 3(1): 81-111. doi: 10.3934/jgm.2011.3.81

A theoretical framework for backward error analysis on manifolds

Anders C. Hansen 1,

1. 

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, United Kingdom

Received  September 2010 Revised  March 2011 Published  April 2011

Backward Error Analysis (BEA) has been a crucial tool when analyzing long-time behavior of numerical integrators, in particular, one is interested in the geometric properties of the perturbed vector field that a numerical integrator generates. In this article we present a new framework for BEA on manifolds. We extend the previously known "exponentially close" estimates from $\mathbb{R}^n$ to smooth manifolds and also provide an abstract theory for classifications of numerical integrators in terms of their geometric properties. Classification theorems of type "symplectic integrators generate symplectic perturbed vector fields" are known to be true in $\mathbb{R}^n.$ We present a general theory for proving such theorems on manifolds by looking at the preservation of smooth $k$-forms on manifolds by the pull-back of a numerical integrator. This theory is related to classification theory of subgroups of diffeomorphisms. We also look at other subsets of diffeomorphisms that occur in the classification theory of numerical integrators. Typically these subsets are anti-fixed points of group homomorphisms.
Keywords: infinite-dimensional manifolds, diffeomorphism groups, Lie groups., Backward error analysis.
Mathematics Subject Classification: 34A26, 65L05, 65J99, 53A35, 53A4.
Citation: Anders C. Hansen. A theoretical framework for backward error analysis on manifolds. Journal of Geometric Mechanics, 2011, 3 (1) : 81-111. doi: 10.3934/jgm.2011.3.81
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences," Springer-Verlag, New York, second edition, 1988.  Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics," Springer-Verlag, New York, second edition, 1989, Translated from the Russian by K. Vogtmann and A. Weinstein.  Google Scholar

[3]

G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys., 74 (1994), 1117-1143. doi: 10.1007/BF02188219.  Google Scholar

[4]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, In "Chaotic Numerics (Geelong, 1993)," volume 172 of "Contemp. Math.," pages 63-74, Amer. Math. Soc., Providence, RI, 1994.  Google Scholar

[5]

E. Cartan, Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93-161.  Google Scholar

[6]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.  Google Scholar

[7]

O. Gonzalez, D. J. Higham and A. M. Stuart, Qualitative properties of modified equations, IMA J. Numer. Anal., 19 (1999), 169-190. doi: 10.1093/imanum/19.2.169.  Google Scholar

[8]

E. Hairer, Global modified Hamiltonian for constrained symplectic integrators, Numer. Math., 95 (2003), 325-336. doi: 10.1007/s00211-002-0428-7.  Google Scholar

[9]

E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math., 76 (1997), 441-462. doi: 10.1007/s002110050271.  Google Scholar

[10]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations.  Google Scholar

[11]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems.  Google Scholar

[12]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, In "Acta Numerica, 2000," volume 9 of "Acta Numer.," pages 215-365, Cambridge Univ. Press, Cambridge, 2000.  Google Scholar

[13]

J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics," Springer-Verlag, New York, 2003.  Google Scholar

[14]

R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434. doi: 10.1017/S0962492902000053.  Google Scholar

[15]

H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author.  Google Scholar

[16]

R. S. Palais, "Foundations of Global Non-linear Analysis," W. A. Benjamin, Inc., New York-Amsterdam, 1968.  Google Scholar

[17]

S. Reich, "Numerical Integration of the Generatized Euler Equations," Technical report, Vancouver, BC, Canada, Canada, 1993. Google Scholar

[18]

S. Reich, On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, Numer. Math., 76 (1997), 231-247. doi: 10.1007/s002110050261.  Google Scholar

[19]

S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570 (electronic). doi: 10.1137/S0036142997329797.  Google Scholar

[20]

R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54-120.  Google Scholar

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences," Springer-Verlag, New York, second edition, 1988.  Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics," Springer-Verlag, New York, second edition, 1989, Translated from the Russian by K. Vogtmann and A. Weinstein.  Google Scholar

[3]

G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys., 74 (1994), 1117-1143. doi: 10.1007/BF02188219.  Google Scholar

[4]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, In "Chaotic Numerics (Geelong, 1993)," volume 172 of "Contemp. Math.," pages 63-74, Amer. Math. Soc., Providence, RI, 1994.  Google Scholar

[5]

E. Cartan, Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93-161.  Google Scholar

[6]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.  Google Scholar

[7]

O. Gonzalez, D. J. Higham and A. M. Stuart, Qualitative properties of modified equations, IMA J. Numer. Anal., 19 (1999), 169-190. doi: 10.1093/imanum/19.2.169.  Google Scholar

[8]

E. Hairer, Global modified Hamiltonian for constrained symplectic integrators, Numer. Math., 95 (2003), 325-336. doi: 10.1007/s00211-002-0428-7.  Google Scholar

[9]

E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math., 76 (1997), 441-462. doi: 10.1007/s002110050271.  Google Scholar

[10]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations.  Google Scholar

[11]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems.  Google Scholar

[12]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, In "Acta Numerica, 2000," volume 9 of "Acta Numer.," pages 215-365, Cambridge Univ. Press, Cambridge, 2000.  Google Scholar

[13]

J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics," Springer-Verlag, New York, 2003.  Google Scholar

[14]

R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434. doi: 10.1017/S0962492902000053.  Google Scholar

[15]

H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author.  Google Scholar

[16]

R. S. Palais, "Foundations of Global Non-linear Analysis," W. A. Benjamin, Inc., New York-Amsterdam, 1968.  Google Scholar

[17]

S. Reich, "Numerical Integration of the Generatized Euler Equations," Technical report, Vancouver, BC, Canada, Canada, 1993. Google Scholar

[18]

S. Reich, On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, Numer. Math., 76 (1997), 231-247. doi: 10.1007/s002110050261.  Google Scholar

[19]

S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570 (electronic). doi: 10.1137/S0036142997329797.  Google Scholar

[20]

R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54-120.  Google Scholar

[1]

Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012
[2]

Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008
[3]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[4]

Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379
[5]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007
[6]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
[7]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
[8]

Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139-167. doi: 10.3934/jgm.2016001
[9]

Jan J. Dijkstra and Jan van Mill. Homeomorphism groups of manifolds and Erdos space. Electronic Research Announcements, 2004, 10: 29-38.

[10]

Benjamin Couéraud, François Gay-Balmaz. Variational discretization of thermodynamical simple systems on Lie groups. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1075-1102. doi: 10.3934/dcdss.2020064
[11]

Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347
[12]

M. F. Newman and Michael Vaughan-Lee. Some Lie rings associated with Burnside groups. Electronic Research Announcements, 1998, 4: 1-3.

[13]

Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323
[14]

Gregory S. Chirikjian. Information-theoretic inequalities on unimodular Lie groups. Journal of Geometric Mechanics, 2010, 2 (2) : 119-158. doi: 10.3934/jgm.2010.2.119
[15]

Nikolaos Karaliolios. Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups. Journal of Modern Dynamics, 2017, 11: 125-142. doi: 10.3934/jmd.2017006
[16]

Adriano Da Silva, Alexandre J. Santana, Simão N. Stelmastchuk. Topological conjugacy of linear systems on Lie groups. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3411-3421. doi: 10.3934/dcds.2017144
[17]

Robert L. Griess Jr., Ching Hung Lam. Groups of Lie type, vertex algebras, and modular moonshine. Electronic Research Announcements, 2014, 21: 167-176. doi: 10.3934/era.2014.21.167
[18]

Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021021
[19]

Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006
[20]

Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy