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July  2021, 41(7): 3319-3341. doi: 10.3934/dcds.2020407

Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

Adrian Viorel 1,, , Cristian D. Alecsa 2, and  Titus O. Pinţa 3,

1. 

Babeş-Bolyai University Cluj-Napoca, Str. Kogalniceanu 3, 400084 Cluj-Napoca, Romania
2. 

Romanian Institute of Science and Technology, 400022 Cluj-Napoca, Romania and, Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, 400320 Cluj-Napoca, Romania
3. 

Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany

* Corresponding author: A. Viorel

Received  May 2020 Revised  September 2020 Published  July 2021 Early access  December 2020

The present work deals with the numerical long-time integration of damped Hamiltonian systems. The method that we analyze combines a specific Strang splitting, that separates linear dissipative effects from conservative ones, with an energy-preserving averaged vector field (AVF) integrator for the Hamiltonian subproblem. This construction faithfully reproduces the energy-dissipation structure of the continuous model, its equilibrium points and its natural Lyapunov function. As a consequence of these structural similarities, both the convergence to equilibrium and, more interestingly, the energy decay rate of the continuous dynamical system are recovered at a discrete level. The possibility of replacing the implicit AVF integrator by an explicit Störmer-Verlet one is also discussed, while numerical experiments illustrate and support the theoretical findings.

Keywords: Structure-preserving integrator, dissipative system, optimization, Łojasiewicz inequality.
Mathematics Subject Classification: Primary: 37M15, 65P10; Secondary: 34D05.
Citation: Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407
References:
[1]

C. D. Alecsa, S. C. László and A. Viorel, A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem, Numer. Algorithms, 84 (2020), 485-512.  doi: 10.1007/s11075-019-00765-z.

[2]

H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method. I. The continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system, Commun. Contemp. Math., 2 (2000), 1-34.  doi: 10.1142/S0219199700000025.

[3]

H. Attouch, Z. Chbani, J. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math. Program., 168 (2018), 123-175.  doi: 10.1007/s10107-016-0992-8.

[4]

P. Bégout, J. Bolte and M. A. Jendoubi, On damped second-order gradient systems, J. Differential Equations, 259 (2015), 3115-3143.  doi: 10.1016/j.jde.2015.04.016.

[5]

A. Bhatt, D. Floyd and B. E. Moore, Second order conformal symplectic schemes for damped Hamiltonian systems, J. Sci. Comput., 66 (2016), 1234-1259.  doi: 10.1007/s10915-015-0062-z.

[6]

R. I. Boţ, E. R. Csetnek and S. C. László, Approaching nonsmooth nonconvex minimization through second-order proximal-gradient dynamical systems, J. Evol. Equ., 18 (2018), 1291-1318.  doi: 10.1007/s00028-018-0441-7.

[7]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998, Translated from the 1990 French original by Yvan Martel and revised by the authors.

[8]

E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren, G. R. W. Quispel and W. M. Wright, Energy-preserving Runge-Kutta methods, M2AN Math. Model. Numer. Anal., 43 (2009), 645-649.  doi: 10.1051/m2an/2009020.

[9]

J. Diakonikolas and M. I. Jordan, Generalized momentum-based methods: A hamiltonian perspective, preprint, arXiv: 1906.00436.

[10]

G. Dujardin and P. Lafitte, Asymptotic behaviour of splitting schemes involving time-subcycling techniques, IMA J. Numer. Anal., 36 (2016), 1804-1841.  doi: 10.1093/imanum/drv059.

[11]

M. J. Ehrhardt, E. S. Riis, T. Ringholm and C.-B. Schönlieb, A geometric integration approach to smooth optimisation: Foundations of the discrete gradient method, preprint, arXiv: 1805.06444.

[12]

L. Einkemmer and A. Ostermann, Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions, SIAM J. Sci. Comput., 37 (2015), A1577–A1592. doi: 10.1137/140994204.

[13]

E. Emmrich, Discrete versions of gronwall's lemma and their application to the numerical analysis of parabolic problems, Preprint No. 637, URL https://www.math.uni-bielefeld.de/~emmrich/public/prepA.pdf.

[14]

G. França, J. Sulam, D. P. Robinson and R. Vidal, Conformal symplectic and relativistic optimization, preprint, arXiv: 1903.04100.

[15]

L. Gauckler, E. Hairer and C. Lubich, Dynamics, numerical analysis, and some geometry, in Proceedings of the International Congress of Mathematicians–-Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018,453–485.

[16]

E. Hairer, Energy-preserving variant of collocation methods, JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5 (2010), 73-84. 

[17]

E. Hairer and C. Lubich, Energy-diminishing integration of gradient systems, IMA J. Numer. Anal., 34 (2014), 452-461.  doi: 10.1093/imanum/drt031.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.

[19]

E. Hansen, F. Kramer and A. Ostermann, A second-order positivity preserving scheme for semilinear parabolic problems, Appl. Numer. Math., 62 (2012), 1428-1435.  doi: 10.1016/j.apnum.2012.06.003.

[20]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320.  doi: 10.1006/jdeq.1997.3393.

[21]

A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36. 

[22]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, vol. 17 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1991.

[23]

A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems, Classical Methods and Recent Advances, BCAM SpringerBriefs. SpringerBriefs in Mathematics, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2015 doi: 10.1007/978-3-319-23407-6.

[24]

M. I. Jordan, Dynamical symplectic and stochastic perspectives on gradient-based optimization, in Proceedings of the International Congress of Mathematicians–-Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018,523–549.

[25]

J. P. LaSalle, The Stability and Control of Discrete Processes, With a foreword by Jack K. Hale and Kenneth R. Meyer, vol. 62 of Applied Mathematical Sciences, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-1076-4.

[26]

S. C. László, Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimization, preprint, arXiv: 1811.09616.

[27]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in Les Équations aux Dérivées Partielles (Paris, 1962), Éditions du Centre National de la Recherche Scientifique, Paris, 1963, 87–89.

[28]

S. MacNamara and G. Strang, Operator splitting, in Splitting Methods in Communication, Imaging, Science, and Engineering, Sci. Comput., Springer, Cham, 2016, 95–114.

[29]

C. J. Maddison, D. Paulin, Y. W. Teh, B. O'Donoghue and A. Doucet, Hamiltonian descent methods, preprint, arXiv: 1809.05042.

[30]

R. McLachlan and M. Perlmutter, Conformal Hamiltonian systems, J. Geom. Phys., 39 (2001), 276-300.  doi: 10.1016/S0393-0440(01)00020-1.

[31]

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

[32]

K. Modin and G. Söderlind, Geometric integration of Hamiltonian systems perturbed by Rayleigh damping, BIT, 51 (2011), 977-1007.  doi: 10.1007/s10543-011-0345-1.

[33]

Y. Nesterov, Lectures on Convex Optimization, Second edition of [MR2142598], vol. 137 of Springer Optimization and Its Applications, Springer, Cham, 2018. doi: 10.1007/978-3-319-91578-4.

[34]

B. T. Polyak, Some methods of speeding up the convergence of iterative methods, Ž. Vyčisl. Mat. i Mat. Fiz., 4 (1964), 791–803.

[35]

A. Quaini and R. Glowinski, Splitting methods for some nonlinear wave problems, in Splitting Methods in Communication, Imaging, Science, and Engineering, Sci. Comput., Springer, Cham, 2016,643–676.

[36]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7 pp. doi: 10.1088/1751-8113/41/4/045206.

[37]

I. Segal, Non-linear semi-groups, Ann. of Math., 78 (1963), 339-364.  doi: 10.2307/1970347.

[38]

X. Shang and H. C. Öttinger, Structure-preserving integrators for dissipative systems based on reversible–irreversible splitting, Proceedings of the Royal Society A, 476 (2020), 20190446, 25 pp. doi: 10.1098/rspa.2019.0446.

[39]

B. Shi, S. S. Du, W. Su and M. I. Jordan, Acceleration via symplectic discretization of high-resolution differential equations, in Advances in Neural Information Processing Systems, 2019, 5745–5753.

[40]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.

[41]

W. Su, S. Boyd and E. J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, J. Mach. Learn. Res., 17 (2016), Paper No. 153, 43 pp.

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition doi: 10.1007/978-1-4612-0645-3.

[43]

J. Zhang, A. Mokhtari, S. Sra and A. Jadbabaie, Direct runge-kutta discretization achieves acceleration, in Advances in Neural Information Processing Systems, 2018, 3900–3909.

show all references

References:
[1]

C. D. Alecsa, S. C. László and A. Viorel, A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem, Numer. Algorithms, 84 (2020), 485-512.  doi: 10.1007/s11075-019-00765-z.

[2]

H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method. I. The continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system, Commun. Contemp. Math., 2 (2000), 1-34.  doi: 10.1142/S0219199700000025.

[3]

H. Attouch, Z. Chbani, J. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math. Program., 168 (2018), 123-175.  doi: 10.1007/s10107-016-0992-8.

[4]

P. Bégout, J. Bolte and M. A. Jendoubi, On damped second-order gradient systems, J. Differential Equations, 259 (2015), 3115-3143.  doi: 10.1016/j.jde.2015.04.016.

[5]

A. Bhatt, D. Floyd and B. E. Moore, Second order conformal symplectic schemes for damped Hamiltonian systems, J. Sci. Comput., 66 (2016), 1234-1259.  doi: 10.1007/s10915-015-0062-z.

[6]

R. I. Boţ, E. R. Csetnek and S. C. László, Approaching nonsmooth nonconvex minimization through second-order proximal-gradient dynamical systems, J. Evol. Equ., 18 (2018), 1291-1318.  doi: 10.1007/s00028-018-0441-7.

[7]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998, Translated from the 1990 French original by Yvan Martel and revised by the authors.

[8]

E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren, G. R. W. Quispel and W. M. Wright, Energy-preserving Runge-Kutta methods, M2AN Math. Model. Numer. Anal., 43 (2009), 645-649.  doi: 10.1051/m2an/2009020.

[9]

J. Diakonikolas and M. I. Jordan, Generalized momentum-based methods: A hamiltonian perspective, preprint, arXiv: 1906.00436.

[10]

G. Dujardin and P. Lafitte, Asymptotic behaviour of splitting schemes involving time-subcycling techniques, IMA J. Numer. Anal., 36 (2016), 1804-1841.  doi: 10.1093/imanum/drv059.

[11]

M. J. Ehrhardt, E. S. Riis, T. Ringholm and C.-B. Schönlieb, A geometric integration approach to smooth optimisation: Foundations of the discrete gradient method, preprint, arXiv: 1805.06444.

[12]

L. Einkemmer and A. Ostermann, Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions, SIAM J. Sci. Comput., 37 (2015), A1577–A1592. doi: 10.1137/140994204.

[13]

E. Emmrich, Discrete versions of gronwall's lemma and their application to the numerical analysis of parabolic problems, Preprint No. 637, URL https://www.math.uni-bielefeld.de/~emmrich/public/prepA.pdf.

[14]

G. França, J. Sulam, D. P. Robinson and R. Vidal, Conformal symplectic and relativistic optimization, preprint, arXiv: 1903.04100.

[15]

L. Gauckler, E. Hairer and C. Lubich, Dynamics, numerical analysis, and some geometry, in Proceedings of the International Congress of Mathematicians–-Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018,453–485.

[16]

E. Hairer, Energy-preserving variant of collocation methods, JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5 (2010), 73-84. 

[17]

E. Hairer and C. Lubich, Energy-diminishing integration of gradient systems, IMA J. Numer. Anal., 34 (2014), 452-461.  doi: 10.1093/imanum/drt031.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.

[19]

E. Hansen, F. Kramer and A. Ostermann, A second-order positivity preserving scheme for semilinear parabolic problems, Appl. Numer. Math., 62 (2012), 1428-1435.  doi: 10.1016/j.apnum.2012.06.003.

[20]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320.  doi: 10.1006/jdeq.1997.3393.

[21]

A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36. 

[22]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, vol. 17 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1991.

[23]

A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems, Classical Methods and Recent Advances, BCAM SpringerBriefs. SpringerBriefs in Mathematics, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2015 doi: 10.1007/978-3-319-23407-6.

[24]

M. I. Jordan, Dynamical symplectic and stochastic perspectives on gradient-based optimization, in Proceedings of the International Congress of Mathematicians–-Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018,523–549.

[25]

J. P. LaSalle, The Stability and Control of Discrete Processes, With a foreword by Jack K. Hale and Kenneth R. Meyer, vol. 62 of Applied Mathematical Sciences, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-1076-4.

[26]

S. C. László, Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimization, preprint, arXiv: 1811.09616.

[27]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in Les Équations aux Dérivées Partielles (Paris, 1962), Éditions du Centre National de la Recherche Scientifique, Paris, 1963, 87–89.

[28]

S. MacNamara and G. Strang, Operator splitting, in Splitting Methods in Communication, Imaging, Science, and Engineering, Sci. Comput., Springer, Cham, 2016, 95–114.

[29]

C. J. Maddison, D. Paulin, Y. W. Teh, B. O'Donoghue and A. Doucet, Hamiltonian descent methods, preprint, arXiv: 1809.05042.

[30]

R. McLachlan and M. Perlmutter, Conformal Hamiltonian systems, J. Geom. Phys., 39 (2001), 276-300.  doi: 10.1016/S0393-0440(01)00020-1.

[31]

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

[32]

K. Modin and G. Söderlind, Geometric integration of Hamiltonian systems perturbed by Rayleigh damping, BIT, 51 (2011), 977-1007.  doi: 10.1007/s10543-011-0345-1.

[33]

Y. Nesterov, Lectures on Convex Optimization, Second edition of [MR2142598], vol. 137 of Springer Optimization and Its Applications, Springer, Cham, 2018. doi: 10.1007/978-3-319-91578-4.

[34]

B. T. Polyak, Some methods of speeding up the convergence of iterative methods, Ž. Vyčisl. Mat. i Mat. Fiz., 4 (1964), 791–803.

[35]

A. Quaini and R. Glowinski, Splitting methods for some nonlinear wave problems, in Splitting Methods in Communication, Imaging, Science, and Engineering, Sci. Comput., Springer, Cham, 2016,643–676.

[36]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7 pp. doi: 10.1088/1751-8113/41/4/045206.

[37]

I. Segal, Non-linear semi-groups, Ann. of Math., 78 (1963), 339-364.  doi: 10.2307/1970347.

[38]

X. Shang and H. C. Öttinger, Structure-preserving integrators for dissipative systems based on reversible–irreversible splitting, Proceedings of the Royal Society A, 476 (2020), 20190446, 25 pp. doi: 10.1098/rspa.2019.0446.

[39]

B. Shi, S. S. Du, W. Su and M. I. Jordan, Acceleration via symplectic discretization of high-resolution differential equations, in Advances in Neural Information Processing Systems, 2019, 5745–5753.

[40]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.

[41]

W. Su, S. Boyd and E. J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, J. Mach. Learn. Res., 17 (2016), Paper No. 153, 43 pp.

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition doi: 10.1007/978-1-4612-0645-3.

[43]

J. Zhang, A. Mokhtari, S. Sra and A. Jadbabaie, Direct runge-kutta discretization achieves acceleration, in Advances in Neural Information Processing Systems, 2018, 3900–3909.

Figure 1.  The (eventual) exponential decay of the potential energy (a) as well as state-spaces trajectories (b) for decreasing step sizes $ h = 1 \text{ and } 0.1 $, both with identical initial conditions $ u_0 = 0.01, v_0 = 0 $. The double-well potential is depicted in c)
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Figure 2.  Capturing an energy plateau: a) the AVF Splitting algorithm (black) compared to the trapezoidal rule (blue) for $ h = 1 $ (and benchmark Runge-Kutta (red)); b) AVF Splitting algorithm (black) compared to the conformal symplectic algorithm (28) (green) for $ h = 0.1 $. The contour lines of the nonconvex potential (33) are depicted in c)
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Figure 3.  Erroneous total energy oscillations: a) AVF Splitting algorithm (black) compared to the trapezoidal rule (blue) for $ h = 0.1 $ (and benchmark Runge-Kutta (red)); b) AVF Splitting algorithm (black) compared to the conformal symplectic algorithm (28) (green) for $ h = 0.01 $. The contour lines of the Rosenbrock potential are depicted in c)
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