To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method that typically includes other conventional particle methods, such as smoothed particle hydrodynamics or moving particle semi-implicit methods. A new regularity of discrete parameters is proposed via two new indicators based on the Voronoi decomposition of the domain along with two hypotheses of reference weight functions. Then, truncation error estimates are derived for an interpolant, approximate gradient operator, and approximate Laplace operator in the generalized particle method. The convergence rates for these estimates are determined based on the frequency with which they appear in the regularity and hypotheses. Finally, the estimates are computed numerically and the results are shown to be in good agreement with the theoretical results.
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。
Numerical computation and new output bounds for time-limited balanced truncation of discrete-time systems
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；KU Leuven, Electrical Engineering (ESAT), Kulak Kortrijk Campus（荷语鲁汶大学，电气工程（ESAT），库拉克科特赖克校区）
Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh$\unicode{x2013}$Ritz procedure for generalized Hermitian eigenvalue problems
