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Energy decay for the damped wave equation on an unbounded network
Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient
1. | IMAG, Univ. Montpellier, CNRS, Montpellier, France |
2. | Institut de Mathématiques de Bourgogne, UMR 5584, Université Bourgogne Franche-Comté, 21000 Dijon, France |
3. | Faculty of Sciences Semlalia, Mathematics, Cadi Ayyad university, 40000 Marrakech, Morroco |
$\mathcal H$ |
$t→+∞$ |
$\begin{equation*}\ddot{x}(t) + γ(t) \dot{x}(t) + \nabla Φ (x(t)) = 0, \;\;\;\;\;\;\;\;{{\rm (IGS)}_{γ}}\end{equation*}$ |
$\nablaΦ$ |
$Φ: \mathcal H→\mathbb R$ |
$γ(t)$ |
$Φ(x(t))-{\rm{inf}}_\mathcal{H} Φ$ |
$γ(·)$ |
$γ(t) = \frac{α}{t}$ |
$α$ |
$α<3$ |
References:
[1] |
V. Apidopoulos, J.-F. Aujol and Ch. Dossal,
The differential inclusion modeling FISTA algorithm and optimality of convergence rate in the case $b \le 3$, SIAM J. Optim., 28 (2018), 551-574.
doi: 10.1137/17M1128642. |
[2] |
H. Attouch and A. Cabot,
Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity, J. Differential Equations, 263 (2017), 5412-5458.
doi: 10.1016/j.jde.2017.06.024. |
[3] |
H. Attouch, A. Cabot and P. Redont,
The dynamics of elastic shocks via epigraphical regularization of a differential inclusion, Adv. Math. Sci. Appl., 12 (2002), 273-306.
|
[4] |
H. Attouch, Z. Chbani, J. Peypouquet and P. Redont,
Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping, Math. Program. Ser. B, 168 (2018), 123-175.
doi: 10.1007/s10107-016-0992-8. |
[5] |
H. Attouch, Z. Chbani and H. Riahi, Rate of convergence of the Nesterov accelerated gradient method in the subcritical case $\alpha \le 3$, arXiv: 1706.05671v1, [math. OC], accepted in $COCV$.
doi: 10.1051/cocv/2017083. |
[6] |
H. Attouch and J. Peypouquet,
The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than $\frac{1}{k^2}$, SIAM J. Optim., 26 (2016), 1824-1834.
doi: 10.1137/15M1046095. |
[7] |
J.-F. Aujol and Ch. Dossal,
Stability of over-relaxations for the Forward-Backward algorithm, application to FISTA, SIAM J. Optim., 25 (2015), 2408-2433.
doi: 10.1137/140994964. |
[8] |
J.-F. Aujol and Ch. Dossal, Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for $b>0$, HAL Preprint 01547251, 2017. |
[9] |
M. Balti and R. May,
Asymptotic for the perturbed heavy ball system with vanishing damping term, Evolution Equations and Control Theory, 6 (2017), 177-186.
doi: 10.3934/eect.2017010. |
[10] |
A. Beck and M. Teboulle,
A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.
doi: 10.1137/080716542. |
[11] |
R. I. Bot and E. R. Csetnek,
Second order forward-backward dynamical systems for monotone inclusion problems, SIAM J. Control Optim., 54 (2016), 1423-1443.
doi: 10.1137/15M1012657. |
[12] |
A. Cabot, H. Engler and S. Gadat,
On the long time behavior of second order differential equations with asymptotically small dissipation, Transactions of the American Mathematical Society, 361 (2009), 5983-6017.
doi: 10.1090/S0002-9947-09-04785-0. |
[13] |
A. Cabot, H. Engler and S. Gadat,
Second order differential equations with asymptotically small dissipation and piecewise flat potentials, Electronic Journal of Differential Equations, 17 (2009), 33-38.
|
[14] |
A. Cabot and P. Frankel,
Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, Journal of Differential Equations, 252 (2012), 294-322.
doi: 10.1016/j.jde.2011.09.012. |
[15] |
A. Chambolle and Ch. Dossal,
On the convergence of the iterates of the "Fast Iterative Shrinkage/Thresholding Algorithm", Journal of Optimization Theory and Applications, 166 (2015), 968-982.
doi: 10.1007/s10957-015-0746-4. |
[16] |
Y. Drori and M. Teboulle,
Performance of first-order methods for smooth convex minimization: A novel approach, Mathematical Programming, Series A, 145 (2014), 451-482.
doi: 10.1007/s10107-013-0653-0. |
[17] |
M. Ghisi, M. Gobbino and A. Haraux,
The remarkable effectiveness of time-dependent damping terms for second order evolution equations, SIAM J. Control Optim., 54 (2016), 1266-1294.
doi: 10.1137/15M1029485. |
[18] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Masson, Paris, 1991. |
[19] |
A. Haraux and M. A. Jendoubi,
Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evolution Equations and Control Theory, 2 (2013), 461-470.
doi: 10.3934/eect.2013.2.461. |
[20] |
M. A. Jendoubi and R. May,
Asymptotics for a second-order differential equation with nonautonomous damping and an integrable source term, Applicable Analysis, 94 (2015), 436-444.
doi: 10.1080/00036811.2014.903569. |
[21] |
R. May,
Asymptotic for a second order evolution equation with convex potential and vanishing damping term, Turk. J. Math., 41 (2017), 681-685.
doi: 10.3906/mat-1512-28. |
[22] |
Y. Nesterov,
A method of solving a convex programming problem with convergence rate $O(1/k^2)$, Soviet Mathematics Doklady, 27 (1983), 372-376.
|
[23] |
Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, volume 87 of Applied Optimization. Kluwer Academic Publishers, Boston, MA, 2004.
doi: 10.1007/978-1-4419-8853-9. |
[24] |
M. Schmidt, N. Le Roux and F. Bach, Convergence rates of inexact proximal-gradient methods for convex optimization, NIPS'11 - 25 th Annual Conference on Neural Information Processing Systems, Dec 2011, Grenada, Spain. (2011) HAL inria-00618152v3. |
[25] |
M. V. Solodov and B. F. Svaiter,
A unified framework for some inexact proximal point algorithms, Numer. Funct. Anal. Optim., 22 (2001), 1013-1035.
doi: 10.1081/NFA-100108320. |
[26] |
W. Su, S. Boyd and E. J. Candès,
A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, Journal of Machine Learning Research, 17 (2016), 1-43.
|
[27] |
S. Villa, S. Salzo, L. Baldassarres and A. Verri,
Accelerated and inexact forward-backward, SIAM J. Optim., 23 (2013), 1607-1633.
doi: 10.1137/110844805. |
show all references
References:
[1] |
V. Apidopoulos, J.-F. Aujol and Ch. Dossal,
The differential inclusion modeling FISTA algorithm and optimality of convergence rate in the case $b \le 3$, SIAM J. Optim., 28 (2018), 551-574.
doi: 10.1137/17M1128642. |
[2] |
H. Attouch and A. Cabot,
Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity, J. Differential Equations, 263 (2017), 5412-5458.
doi: 10.1016/j.jde.2017.06.024. |
[3] |
H. Attouch, A. Cabot and P. Redont,
The dynamics of elastic shocks via epigraphical regularization of a differential inclusion, Adv. Math. Sci. Appl., 12 (2002), 273-306.
|
[4] |
H. Attouch, Z. Chbani, J. Peypouquet and P. Redont,
Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping, Math. Program. Ser. B, 168 (2018), 123-175.
doi: 10.1007/s10107-016-0992-8. |
[5] |
H. Attouch, Z. Chbani and H. Riahi, Rate of convergence of the Nesterov accelerated gradient method in the subcritical case $\alpha \le 3$, arXiv: 1706.05671v1, [math. OC], accepted in $COCV$.
doi: 10.1051/cocv/2017083. |
[6] |
H. Attouch and J. Peypouquet,
The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than $\frac{1}{k^2}$, SIAM J. Optim., 26 (2016), 1824-1834.
doi: 10.1137/15M1046095. |
[7] |
J.-F. Aujol and Ch. Dossal,
Stability of over-relaxations for the Forward-Backward algorithm, application to FISTA, SIAM J. Optim., 25 (2015), 2408-2433.
doi: 10.1137/140994964. |
[8] |
J.-F. Aujol and Ch. Dossal, Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for $b>0$, HAL Preprint 01547251, 2017. |
[9] |
M. Balti and R. May,
Asymptotic for the perturbed heavy ball system with vanishing damping term, Evolution Equations and Control Theory, 6 (2017), 177-186.
doi: 10.3934/eect.2017010. |
[10] |
A. Beck and M. Teboulle,
A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.
doi: 10.1137/080716542. |
[11] |
R. I. Bot and E. R. Csetnek,
Second order forward-backward dynamical systems for monotone inclusion problems, SIAM J. Control Optim., 54 (2016), 1423-1443.
doi: 10.1137/15M1012657. |
[12] |
A. Cabot, H. Engler and S. Gadat,
On the long time behavior of second order differential equations with asymptotically small dissipation, Transactions of the American Mathematical Society, 361 (2009), 5983-6017.
doi: 10.1090/S0002-9947-09-04785-0. |
[13] |
A. Cabot, H. Engler and S. Gadat,
Second order differential equations with asymptotically small dissipation and piecewise flat potentials, Electronic Journal of Differential Equations, 17 (2009), 33-38.
|
[14] |
A. Cabot and P. Frankel,
Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, Journal of Differential Equations, 252 (2012), 294-322.
doi: 10.1016/j.jde.2011.09.012. |
[15] |
A. Chambolle and Ch. Dossal,
On the convergence of the iterates of the "Fast Iterative Shrinkage/Thresholding Algorithm", Journal of Optimization Theory and Applications, 166 (2015), 968-982.
doi: 10.1007/s10957-015-0746-4. |
[16] |
Y. Drori and M. Teboulle,
Performance of first-order methods for smooth convex minimization: A novel approach, Mathematical Programming, Series A, 145 (2014), 451-482.
doi: 10.1007/s10107-013-0653-0. |
[17] |
M. Ghisi, M. Gobbino and A. Haraux,
The remarkable effectiveness of time-dependent damping terms for second order evolution equations, SIAM J. Control Optim., 54 (2016), 1266-1294.
doi: 10.1137/15M1029485. |
[18] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Masson, Paris, 1991. |
[19] |
A. Haraux and M. A. Jendoubi,
Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evolution Equations and Control Theory, 2 (2013), 461-470.
doi: 10.3934/eect.2013.2.461. |
[20] |
M. A. Jendoubi and R. May,
Asymptotics for a second-order differential equation with nonautonomous damping and an integrable source term, Applicable Analysis, 94 (2015), 436-444.
doi: 10.1080/00036811.2014.903569. |
[21] |
R. May,
Asymptotic for a second order evolution equation with convex potential and vanishing damping term, Turk. J. Math., 41 (2017), 681-685.
doi: 10.3906/mat-1512-28. |
[22] |
Y. Nesterov,
A method of solving a convex programming problem with convergence rate $O(1/k^2)$, Soviet Mathematics Doklady, 27 (1983), 372-376.
|
[23] |
Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, volume 87 of Applied Optimization. Kluwer Academic Publishers, Boston, MA, 2004.
doi: 10.1007/978-1-4419-8853-9. |
[24] |
M. Schmidt, N. Le Roux and F. Bach, Convergence rates of inexact proximal-gradient methods for convex optimization, NIPS'11 - 25 th Annual Conference on Neural Information Processing Systems, Dec 2011, Grenada, Spain. (2011) HAL inria-00618152v3. |
[25] |
M. V. Solodov and B. F. Svaiter,
A unified framework for some inexact proximal point algorithms, Numer. Funct. Anal. Optim., 22 (2001), 1013-1035.
doi: 10.1081/NFA-100108320. |
[26] |
W. Su, S. Boyd and E. J. Candès,
A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, Journal of Machine Learning Research, 17 (2016), 1-43.
|
[27] |
S. Villa, S. Salzo, L. Baldassarres and A. Verri,
Accelerated and inexact forward-backward, SIAM J. Optim., 23 (2013), 1607-1633.
doi: 10.1137/110844805. |
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