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Asymptotically-Preserving Large Deviations Principles by Stochastic Symplectic Methods for a Linear Stochastic Oscillator
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2021-01-04 , DOI: 10.1137/19m1306919 Chuchu Chen , Jialing Hong , Diancong Jin , Liying Sun
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2021-01-04 , DOI: 10.1137/19m1306919 Chuchu Chen , Jialing Hong , Diancong Jin , Liying Sun
SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 32-59, January 2021.
It is well known that symplectic methods have been rigorously shown to be superior to nonsymplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the superiority of stochastic symplectic methods by means of the large deviations principle. We propose the concept of asymptotical preservation of numerical methods for large deviations principles associated with the exact solutions of the general stochastic Hamiltonian systems. Considering that the linear stochastic oscillator is one of the typical stochastic Hamiltonian systems, we take it as the test equation in this paper to obtain precise results about the rate functions of large deviations principles for both exact and numerical solutions. Based on the Gärtner--Ellis theorem, we first study the large deviations principles of the mean position and the mean velocity for both the exact solution and its numerical approximations. Then, we prove that stochastic symplectic methods asymptotically preserve these two large deviations principles, but nonsymplectic ones do not. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the “hitting probability" of the mean position and mean velocity of the stochastic oscillator. Finally, numerical experiments are performed to show the superiority of stochastic symplectic methods in computing the large deviations rate functions. To the best of our knowledge, this is the first result about applying the large deviations principle to reveal the superiority of stochastic symplectic methods compared with nonsymplectic ones in the existing literature.
中文翻译：
线性随机振荡器的随机辛方法的渐近保留大偏差原理
SIAM数值分析学报，第59卷，第1期，第32-59页，2021年1月。
众所周知，当应用于确定性汉密尔顿系统时，辛方法已被严格证明优于非辛方法，特别是在长时间计算中。本文试图通过大偏差原理研究随机辛方法的优越性。我们提出了与一般随机哈密顿系统的精确解相关的大偏差原理的数值方法的渐近保存的概念。考虑到线性随机振荡器是典型的随机哈密顿系统，我们将其作为测试方程，以得到关于大偏差原理的速率函数的精确结果，包括精确解和数值解。根据Gärtner-Ellis定理，我们首先研究精确解及其数值逼近的平均位置和平均速度的大偏差原理。然后，我们证明了随机辛方法渐近地保留了这两个大偏差原理，而非渐近方法则没有。这表明随机辛方法能够很好地逼近随机振荡器的平均位置和平均速度的“命中概率”的指数衰减速度，最后进行了数值实验，证明了随机辛方法的优越性。偏差率函数大。据我们所知，
更新日期：2021-01-05
It is well known that symplectic methods have been rigorously shown to be superior to nonsymplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the superiority of stochastic symplectic methods by means of the large deviations principle. We propose the concept of asymptotical preservation of numerical methods for large deviations principles associated with the exact solutions of the general stochastic Hamiltonian systems. Considering that the linear stochastic oscillator is one of the typical stochastic Hamiltonian systems, we take it as the test equation in this paper to obtain precise results about the rate functions of large deviations principles for both exact and numerical solutions. Based on the Gärtner--Ellis theorem, we first study the large deviations principles of the mean position and the mean velocity for both the exact solution and its numerical approximations. Then, we prove that stochastic symplectic methods asymptotically preserve these two large deviations principles, but nonsymplectic ones do not. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the “hitting probability" of the mean position and mean velocity of the stochastic oscillator. Finally, numerical experiments are performed to show the superiority of stochastic symplectic methods in computing the large deviations rate functions. To the best of our knowledge, this is the first result about applying the large deviations principle to reveal the superiority of stochastic symplectic methods compared with nonsymplectic ones in the existing literature.
中文翻译：
线性随机振荡器的随机辛方法的渐近保留大偏差原理
SIAM数值分析学报，第59卷，第1期，第32-59页，2021年1月。
众所周知，当应用于确定性汉密尔顿系统时，辛方法已被严格证明优于非辛方法，特别是在长时间计算中。本文试图通过大偏差原理研究随机辛方法的优越性。我们提出了与一般随机哈密顿系统的精确解相关的大偏差原理的数值方法的渐近保存的概念。考虑到线性随机振荡器是典型的随机哈密顿系统，我们将其作为测试方程，以得到关于大偏差原理的速率函数的精确结果，包括精确解和数值解。根据Gärtner-Ellis定理，我们首先研究精确解及其数值逼近的平均位置和平均速度的大偏差原理。然后，我们证明了随机辛方法渐近地保留了这两个大偏差原理，而非渐近方法则没有。这表明随机辛方法能够很好地逼近随机振荡器的平均位置和平均速度的“命中概率”的指数衰减速度，最后进行了数值实验，证明了随机辛方法的优越性。偏差率函数大。据我们所知，
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