期刊
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
卷 22, 期 5, 页码 1567-1629出版社
SPRINGER
DOI: 10.1007/s10208-021-09536-6
关键词
Linear ill-posed problems; Regularisation theory; Dynamical regularisation; Optimal convergence rates; Showalter's method; Heavy ball method; Vanishing viscosity flow; Spectral analysis
资金
- Austrian Science Fund (FWF) [I2419-N32, SFB F68, F6804-N36, F6807-N36, S11704, I3661-N27]
- Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy The Berlin Mathematics Research CenterMATH [EXC-2046/1, 390685689]
This paper examines solutions to ill-posed linear problems using dynamical flows, leveraging results from convex analysis to improve these solutions and introduce optimal regularization methods and convergence rates. The proposed flows for minimizing the norm of the residual of a linear operator equation are shown to provide optimal convergence rates for regularized solutions, setting benchmarks for further studies in convex analysis.
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.
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