For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. %
By exploiting operator monotonicity of certain functions and interpolation techniques in variable Hilbert scales, we study these methods under general smoothness conditions. Order optimal error bounds are given in case the regularization parameter is chosen either {\it a priori} or {\it a posteriori} by the discrepancy principle. For realizing the discrepancy principle, some fast algorithm is proposed which is based on Newton's method applied to some properly transformed equations.
Proximal Implicit ODE Solvers for Accelerating Learning Neural ODEs
University of Utah（犹他大学）；University of California, Los Angeles（加州大学洛杉矶分校）；Illinois Institute of Technology（伊利诺伊理工学院）；University of California, Irvine（加州大学欧文分校）；Scientific Computing and Imaging (SCI) Institute（科学计算与成像研究所）
Accelerated Implicit GDA Schemes: Theoretical Guarantees and Application to Proximal Augmented Lagrangian Methods
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究所）；Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉学科中心）
An implicit multiscale solver for steady kinetic equations based on wave-particle decomposition
Hong Kong University of Science and Technology（香港科技大学）；Shenzhen Research Institute, Hong Kong University of Science and Technology（香港科技大学深圳研究院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）
