# Convergence Rates of First and Higher Order Dynamics for Solving Linear Ill-posed Problems

## TL;DR

This paper analyzes dynamical flows for solving linear ill-posed problems, demonstrating they are optimal regularization methods with convergence rates surpassing general convex analysis results by focusing on the squared norm of the residuum.

## Contribution

It introduces specific dynamical flows for linear ill-posed problems and proves their optimal regularization properties and enhanced convergence rates.

## Key findings

- Flows are proven to be optimal regularization methods.
- Convergence rates are significantly higher than general convex analysis results.
- Focus on the squared norm of the residuum enables improved rates.

## Abstract

Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov's algorithm and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), respectively. In this paper we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residuum of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. We prove that the proposed flows for minimising the residuum of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions. In particular we show that in comparison to convex analysis results the rates can be significantly higher, which is possible by constraining the investigations to the particular convex energy functional, which is the squared norm of the residuum.