A short note on plain convergence of adaptive least-squares finite element methods

A note on concatenation of quasi-Monte Carlo and plain Monte Carlo rules in high dimensions

In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the $s$-dimensional unit cube by using rank-1 lattice point sets only for the first $d\, (<s)$ coordinates and random points for the remaining $s-d$ coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as l…

非侵入式最小二乘泛函后验误差估计器：具有普通收敛性的线性和非线性问题

Non-intrusive Least-Squares Functional A Posteriori Error Estimator: Linear and Nonlinear Problems with Plain Convergence

本文提出一种非侵入式最小二乘泛函后验误差估计器，适用于非最小二乘有限元求解的线性和非线性问题，并证明了自适应算法的普通收敛性。

Further results on a space-time FOSLS formulation of parabolic PDEs

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer& Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.

关于随机线性二步方法对SDEs渐近均方稳定性的一个注记

A note on Asymptotic mean-square stability of stochastic linear two-step methods for SDEs

本文研究了二阶方法应用于标量随机微分方程及系统时的渐近均方稳定性条件，揭示了BDF2方法无论步长如何均稳定，而AB2和AM2方法无条件稳定，且高阶方法在标量情况下稳定性无提升，但在多维情况下表现不同。

关于缩放总最小二乘问题条件数的注记

A note on the condition number of the scaled total least squares problem

本文研究了缩放总最小二乘问题的规范条件数的显式表达式，通过引入技术简化了条件数的表达式，并推导出新的结果。基于这些结果，总最小二乘问题的条件数新表达式可以作为特殊情况得出。新形式的条件数在存储和计算上有优势，并提出了三种不同的方法来估计条件数。通过数值实验验证了结果的有效性。

关于预条件化加权线性最小二乘的注记，及其对弱约束变分数据同化的影响

A note on preconditioning weighted linear least squares, with consequences for weakly-constrained variational data assimilation

本文分析了使用模型矩阵近似进行线性加权最小二乘预条件化的影响，探讨了模型和加权矩阵的特征结构之间的相互作用，并通过一个小例子说明此类预条件器可能的低效性，最后讨论了这些结果在弱约束4D-Var数据同化问题中的影响。