| 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 |
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.
| [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
| [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. |
| [1] |
Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206 |
| [2] |
Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417 |
| [3] |
Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146 |
| [4] |
Lin Lu, Qi Wang, Yongzhong Song, Yushun Wang. Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4745-4765. doi: 10.3934/dcdsb.2020311 |
| [5] |
Makoto Okumura, Takeshi Fukao, Daisuke Furihata, Shuji Yoshikawa. A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition. Communications on Pure and Applied Analysis, 2022, 21 (2) : 355-392. doi: 10.3934/cpaa.2021181 |
| [6] |
Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018 |
| [7] |
Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093 |
| [8] |
Kongzhi Li, Xiaoping Xue. The Łojasiewicz inequality for free energy functionals on a graph. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022066 |
| [9] |
Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014 |
| [10] |
Jann-Long Chern, Zhi-You Chen, Yong-Li Tang. Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2299-2318. doi: 10.3934/dcds.2013.33.2299 |
| [11] |
Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic and Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373 |
| [12] |
Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109 |
| [13] |
Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 |
| [14] |
Hideo Kubo. Asymptotic behavior of solutions to semilinear wave equations with dissipative structure. Conference Publications, 2007, 2007 (Special) : 602-613. doi: 10.3934/proc.2007.2007.602 |
| [15] |
Gengen Zhang. Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2587-2606. doi: 10.3934/dcdsb.2021149 |
| [16] |
Jinying Ma, Honglei Xu. Empirical analysis and optimization of capital structure adjustment. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1037-1047. doi: 10.3934/jimo.2018191 |
| [17] |
Hong Zhang, Fei Yang. Optimization of capital structure in real estate enterprises. Journal of Industrial and Management Optimization, 2015, 11 (3) : 969-983. doi: 10.3934/jimo.2015.11.969 |
| [18] |
Ahmet Sahiner, Gulden Kapusuz, Nurullah Yilmaz. A new smoothing approach to exact penalty functions for inequality constrained optimization problems. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 161-173. doi: 10.3934/naco.2016006 |
| [19] |
Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895 |
| [20] |
Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 |
2020 Impact Factor: 1.392
[Back to Top]
