标题： 向扩展拉格朗日分子动力学的精确误差分析 显示英文标题

标题： Towards sharp error analysis of extended Lagrangian molecular dynamics

摘要： 扩展拉格朗日分子动力学（XLMD）方法为减少具有约束隐变量的一类分子动力学模拟的计算成本提供了有用的框架。 XLMD方法通过为隐变量引入一个虚构质量$\varepsilon$来放松约束，求解一组奇异摄动常微分方程。 尽管在过去十年中XLMD在多个不同背景下展示了有利的数值性能，但该方法的数学分析仍然很少。 我们提出了在经典可极化力场模型背景下的XLMD方法的首次误差分析。 虽然相对于原子自由度的动力学是通用且非线性的，但可极化力场模型的关键数学简化在于隐变量的约束由一个线性方程组给出。 我们证明，当隐变量的初始值以我们定义的方式相容时，当虚构质量$\varepsilon$变得较小时，原子自由度的误差为$\mathcal{O}(\varepsilon)$，隐变量的误差为$\mathcal{O}(\sqrt{\varepsilon})$，当隐变量的维度$d'$为1时。 此外，当隐变量的初始值被改进为在某种意义上最优相容时，我们证明隐变量的收敛率可以提高到$\mathcal{O}(\varepsilon)$。 数值结果验证了这两个估计不仅对于$d' =1$是精确的，而且对于任意的$d'$也是精确的。 在一般的$d'$设置下，我们确实得到了收敛，但对于原子变量和潜在变量的收敛速率是非精确的$\mathcal{O}(\sqrt{\varepsilon})$。 显示英文摘要

摘要： The extended Lagrangian molecular dynamics (XLMD) method provides a useful framework for reducing the computational cost of a class of molecular dynamics simulations with constrained latent variables. The XLMD method relaxes the constraints by introducing a fictitious mass $\varepsilon$ for the latent variables, solving a set of singularly perturbed ordinary differential equations. While favorable numerical performance of XLMD has been demonstrated in several different contexts in the past decade, mathematical analysis of the method remains scarce. We propose the first error analysis of the XLMD method in the context of a classical polarizable force field model. While the dynamics with respect to the atomic degrees of freedom are general and nonlinear, the key mathematical simplification of the polarizable force field model is that the constraints on the latent variables are given by a linear system of equations. We prove that when the initial value of the latent variables is compatible in a sense that we define, XLMD converges as the fictitious mass $\varepsilon$ is made small with $\mathcal{O}(\varepsilon)$ error for the atomic degrees of freedom and with $\mathcal{O}(\sqrt{\varepsilon})$ error for the latent variables, when the dimension of the latent variable $d'$ is 1. Furthermore, when the initial value of the latent variables is improved to be optimally compatible in a certain sense, we prove that the convergence rate can be improved to $\mathcal{O}(\varepsilon)$ for the latent variables as well. Numerical results verify that both estimates are sharp not only for $d' =1$, but also for arbitrary $d'$. In the setting of general $d'$, we do obtain convergence, but with the non-sharp rate of $\mathcal{O}(\sqrt{\varepsilon})$ for both the atomic and latent variables.