We study the design and implementation of numerical methods to solve the generalized Langevin equation (GLE) focusing on canonical sampling properties of numerical integrators. For this purpose, we cast the GLE in an extended phase space formulation and derive a family of splitting methods which generalize existing Langevin dynamics integration methods. We show exponential convergence in law and the validity of a central limit theorem for the Markov chains obtained via these integration methods, and we show that the dynamics of a suggested integration scheme is consistent with asymptotic limits of the exact dynamics and can reproduce (in the short memory limit) a superconvergence property for the analogous splitting of underdamped Langevin dynamics. We then apply our proposed integration method to several model systems, including a Bayesian inference problem. We demonstrate in numerical experiments that our method outperforms other proposed GLE integration schemes in terms of the accuracy of sampling. Moreover, using a parameterization of the memory kernel in the GLE as proposed by Ceriotti et al [9], our experiments indicate that the obtained GLE-based sampling scheme outperforms state-of-the-art sampling schemes based on underdamped Langevin dynamics in terms of robustness and efficiency.
翻译:我们研究了解决通用兰格文方程式(GLE)的数字方法的设计和实施,重点是数字融合器的卡通抽样特性。为此,我们将GLE投入一个延长的阶段空间配方,并形成一个将现有的朗格文动力集成法加以推广的分解方法组合。我们展示了法律上的指数趋同和通过这些集成法获得的马尔科夫链中央界限理论的有效性。我们显示,建议的集成办法的动态符合确切动态的无约束限度,并可以(在短记忆限度内)复制(在短短的内存限度内)一个超同质属性,以类似地分解未充分铺设的兰格文动态。我们随后将我们提议的集成方法应用于几个模型系统,包括一个巴耶斯人推断问题。我们通过数字实验表明,从采样的准确性来看,我们的方法比其他拟议的GLEE一体化办法比其他拟议的GLE集集集集成计划要好。此外,使用Cerovti等人提议的GLE内记忆核心的参数参数[9],我们的实验表明,获得的GLE基采样制的采样计划超越了在兰基条件下的可靠状态。