Tikhonov regularization of a second order dynamical system with Hessian driven damping
Radu Ioan Boţ1, Ernö Robert Csetnek1, Szilárd Csaba László2
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Mathematical Programming
|November 1, 2021
Summary
This study explores fast convergence for dynamical systems in Hilbert spaces. Tikhonov regularization ensures trajectories strongly converge to the minimum norm solution.
Area of Science:
- Optimization and Dynamical Systems
- Convex Analysis in Hilbert Spaces
Background:
- Dynamical systems are crucial for solving optimization problems.
- Understanding trajectory behavior is key to convergence analysis.
Purpose of the Study:
- To analyze asymptotic properties of second-order dynamical systems.
- To investigate the impact of Hessian-driven damping and Tikhonov regularization.
- To achieve fast convergence for function values and strong convergence for trajectories.
Main Methods:
- Utilizing a second-order dynamical system model.
- Incorporating Hessian-driven damping and Tikhonov regularization.
- Analyzing trajectories in Hilbert spaces for smooth convex functions.
Main Results:
- Demonstrated fast convergence rates for function values along trajectories.
- Established strong convergence of trajectories to the minimum norm minimizer.
- Showcased the effectiveness of Tikhonov regularization in achieving convergence.
Conclusions:
- The proposed dynamical system with Tikhonov regularization offers efficient minimization.
- Fast convergence and strong convergence properties are key contributions.
- This framework advances the study of convex function minimization in Hilbert spaces.
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