The structure-preserving doubling algorithm (SDA) is a fairly efficient method for solving problems closely related to Hamiltonian (or Hamiltonian-like) matrices, such as computing the required solutions to algebraic Riccati equations. However, for large-scale problems in $\mathbb{C}^n$ (also $\mathbb{R}^n$), the SDA with an $O(n^3)$ computational complexity does not work well. In this paper, we propose a new decoupled form of the SDA (we name it as dSDA), building on the associated Krylov subspaces thus leading to the inherent low-rank structures. Importantly, the approach decouples the original two to four iteration formulae. The resulting dSDA is much more efficient since only one quantity (instead of the original two to four) is computed iteratively. For large-scale problems, further efficiency is gained from the low-rank structures. This paper presents the theoretical aspects of the dSDA. A practical algorithm dSDA t with truncation and many illustrative numerical results will appear in a second paper.
Decoupled Structure-Preserving Doubling Algorithm with Truncation for Large-Scale Algebraic Riccati Equations
在\emph{Guo et al, arXiv:2005.08288}中，我们提出了一种解耦形式的保结构加倍算法（dSDA）。该方法简化了原来的两到四个耦合递归，使其能够解决大型代数Riccati方程和其他相关问题。在本文中，我们考虑的数值计算的新的dSDA求解大规模连续时间代数Riccati方程的低秩结构（从而拥有数值低秩的解决方案）。借助于一种新的截断策略，控制了近似解的秩。因此，可以有效地处理大规模问题。说明性的数值例子来证明和证实我们的要求。
A fully decoupled and structure-preserving relaxation Crank--Nicolson finite element method for Gross--Pitaevskii--Poisson model
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Xiangtan University（湘潭大学）；College of Mathematics and Statistics, Hengyang Normal University（衡阳师范学院）；Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University（湘潭大学）；Department of Mathematical Sciences, The University of Texas at El Paso（得克萨斯大学埃尔帕索分校）
Linear, decoupled, positivity preserving, positive-definiteness preserving and energy stable schemes for the diffusive Oldroyd-B coupled with PNP model
Fully decoupled, linear and structure-preserving block-centered finite difference methods for the Keller-Segel chemotaxis system on staggered non-uniform grids
Ocean University of China（中国海洋大学）；Laboratory of Marine Mathematics, Ocean University of China（中国海洋大学海洋数学实验室）
A unified framework of fully decoupled, bound-preserving and energy-dissipative schemes for two-phase flow in porous media
School of Mathematics, Shandong University（山东大学数学学院）；Mathematics Department, University of Massachusetts（马萨诸塞大学数学系）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
A linear, decoupled and positivity-preserving time-staggered block-centered finite difference method for the multi-species Keller-Segel chemotaxis system
Ocean University of China（中国海洋大学）；Laboratory of Marine Mathematics, Ocean University of China（中国海洋大学海洋数学实验室）
Decoupled Divergence-Free Neural Networks Basis Method for Incompressible Fluid Problems
Banma Network Technology Co., Ltd.（斑马网络技术有限公司）；School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University（上海交通大学数学科学学院与教育部计算数学重点实验室）；School of Mathematics and Statistics, Yunnan University（云南大学数学与统计学院）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
