We present a practical implementation of an optimal first-order method, due to Nesterov, for large-scale total variation regularization in tomographic reconstruction, image deblurring, etc. The algorithm applies to $μ$-strongly convex objective functions with $L$-Lipschitz continuous gradient. In the framework of Nesterov both $μ$ and $L$ are assumed known -- an assumption that is seldom satisfied in practice. We propose to incorporate mechanisms to estimate locally sufficient $μ$ and $L$ during the iterations. The mechanisms also allow for the application to non-strongly convex functions. We discuss the iteration complexity of several first-order methods, including the proposed algorithm, and we use a 3D tomography problem to compare the performance of these methods. The results show that for ill-conditioned problems solved to high accuracy, the proposed method significantly outperforms state-of-the-art first-order methods, as also suggested by theoretical results.
Heuristic parameter-choice rules for convex variational regularization based on error estimates
Zentrum für Technomathematik, Universität Bremen（不来梅大学技术数学中心）；Institute for Analysis and Algebra, Pockelsstr. 14, D-38118 Braunschweig, Germany（不伦瑞克工业大学分析与代数研究所）
Ground States and Singular Vectors of Convex Variational Regularization Methods
Magnetic Resonance Research Centre, Department of Chemical Engineering and Biotechnology（磁共振研究中心，化学工程与生物系）；Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik（明斯特西法利希威廉大学，数值与应用数学研究所）
