Related Experiment Video
Updated: Jul 1, 2025
09:39
Characterizing Dissipative Elastic Metamaterials Produced by Additive Manufacturing
Published on: June 28, 2024
914
Second Order Splitting Dynamics with Vanishing Damping for Additively Structured Monotone Inclusions
Radu Ioan Boţ1, David Alexander Hulett1
1Faculty of Mathematics, University of Vienna, Vienna, Austria.
Journal of Dynamics and Differential Equations
|March 4, 2024
Summary
This study introduces a novel splitting system for finding operator zeros in Hilbert spaces. The method ensures convergence to solutions and fast velocity reduction, with applications in convex optimization.
Area of Science:
- Optimization Theory
- Functional Analysis
- Numerical Analysis
Background:
- Addresses the challenge of finding zeros for the sum of maximally monotone and cocoercive operators.
- Builds upon existing methods by introducing a novel second-order dynamical system with vanishing damping.
Purpose of the Study:
- To analyze the asymptotic behavior of trajectories generated by a time-dependent forward-backward splitting system.
- To establish weak convergence of trajectories to the solution set of the operator equation.
- To demonstrate fast convergence of velocities to zero.
Main Methods:
- Utilizes a second-order dynamical system with vanishing damping.
- Employs a time-dependent forward-backward splitting operator.
- Analyzes the system's behavior in a real Hilbert space framework.
Main Results:
- Proves weak convergence of generated trajectories to the set of zeros of A + B.
- Demonstrates that the velocities of the trajectories converge rapidly to zero.
- Derives fast convergence rates for a specific convex optimization problem as a special case.
Conclusions:
- The proposed splitting system is effective for finding zeros of sums of monotone and cocoercive operators.
- The system offers theoretical guarantees for convergence and fast velocity reduction.
- Numerical experiments validate the theoretical findings and demonstrate practical applicability.
Related Concept Videos
01:20Types of Damping
6.4K
If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...
6.4K
01:18Second Order systems II
109
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
109
01:07Damped Oscillations
5.7K
In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...
Although friction and other non-conservative...
5.7K
01:13Parameters Affecting Nonlinear Elimination: Zero-Order Input, First-Order Absorption and Two-Compartment Model
66
Drugs administered through various routes can lead to nonlinear elimination, resulting in complex pharmacokinetic behaviors crucial to understanding efficacious drug dosing.
When a drug is administered through a constant intravenous infusion and eliminated via nonlinear pharmacokinetics, it follows zero-order input. For example, oral drugs undergo first-order absorption upon administration and are eliminated through nonlinear pharmacokinetics.
In the case of subcutaneously administered drugs,...
When a drug is administered through a constant intravenous infusion and eliminated via nonlinear pharmacokinetics, it follows zero-order input. For example, oral drugs undergo first-order absorption upon administration and are eliminated through nonlinear pharmacokinetics.
In the case of subcutaneously administered drugs,...
66
01:17Second-Order Circuits
1.4K
Integrating two fundamental energy storage elements in electrical circuits results in second-order circuits, encompassing RLC circuits and circuits with dual capacitors or inductors (RC and RL circuits). Second-order circuits are identified by second-order differential equations that link input and output signals.
Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in...
Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in...
1.4K
01:14Stability of structures
165
In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical,...
165