This paper concerns the estimation of the density function of the solution to a random nonautonomous second-order linear differential equation with analytic data processes. In a recent contribution, we proposed to express the density function as an expectation, and we used a standard Monte Carlo algorithm to approximate the expectation. Although the algorithms worked satisfactorily for most test problems, some numerical challenges emerged for others, due to large statistical errors. In these situations, the convergence of the Monte Carlo simulation slows down severely, and noisy features plague the estimates. In this paper, we focus on computational aspects and propose several variance reduction methods to remedy these issues and speed up the convergence. First, we introduce a pathwise selection of the approximating processes which aims at controlling the variance of the estimator. Second, we propose a hybrid method, combining Monte Carlo and deterministic quadrature rules, to estimate the expectation. Third, we exploit the series expansions of the solutions to design a multilevel Monte Carlo estimator. The proposed methods are implemented and tested on several numerical examples to highlight the theoretical discussions and demonstrate the significant improvements achieved.

中文翻译：

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估计二阶线性微分方程解的密度的方差减少方法和多级蒙特卡洛策略

本文涉及具有解析数据处理的随机非自治二阶线性微分方程解的密度函数估计。在最近的贡献中，我们建议将密度函数表示为期望值，并使用标准的蒙特卡洛算法来近似期望值。尽管该算法对于大多数测试问题均令人满意，但由于较大的统计误差，其他一些数值挑战仍然出现。在这些情况下，蒙特卡洛模拟的收敛速度会严重放慢，并且嘈杂的特征会困扰估计。在本文中，我们着重于计算方面，并提出了几种方差减少方法来补救这些问题并加快收敛速度​​。第一，我们介绍了近似过程的路径选择，旨在控制估计量的方差。其次，我们提出了一种混合方法，将蒙特卡罗和确定性正交规则相结合，以估计期望值。第三，我们利用解决方案的级数展开来设计多层蒙特卡洛估计器。所提出的方法已在几个数值示例上进行了实施和测试，以突出理论讨论并展示所取得的重大改进。