Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth
Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth
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DOI：
10.1090/mcom/3785
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发表时间：
2020-06
期刊：
影响因子：
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通讯作者：
Zhenning Cai;Jianfeng Lu;Siyao Yang
中科院分区：
文献类型：
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作者：
Zhenning Cai;Jianfeng Lu;Siyao Yang
我们考虑了最近提出的尺蠖蒙特卡罗方法的数值分析，以解决开放量子系统的数值符号问题。我们专注于增长的数值误差相对于模拟时间，尺蠖蒙特卡罗方法显示出一个平坦的曲线比直接应用蒙特卡罗方法的经典戴森系列。为了更好地理解尺蠖蒙特卡罗方法的基本机制，我们区分两种类型的指数误差增长，这被称为数值符号问题和误差放大。前者是由于随机方法中方差的快速增长，这可以从Dyson级数中观察到，而后者来自于数值解的演化。我们的分析表明，部分重算技术可以被认为是一种工具，以平衡这两种类型的错误，尺蠖蒙特卡罗方法是一个成功的情况下，数值符号问题是有效地抑制了这种手段。我们首先证明了我们的想法的背景下，常微分方程，然后提供完整的分析尺蠖蒙特卡罗方法。数值实验验证了我们的理论结果。
We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchworm Monte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.