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, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

最近，人们对动态流动的分析产生了极大的兴趣，其中静止极限是凸能量的最小值。引起极大兴趣的特定流分别是 Nesterov 算法和快速迭代收缩阈值算法的连续限制。在本文中，我们通过动态流来解决线性不适定问题。由于线性算子方程残差的平方范数是凸函数，因此能量最小化流的凸分析的理论结果是适用的。但是，在本文受限的情况下，它们往往可以得到显着改进。而且，