We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection-diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov--Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

翻译：我们提出了新的多层蒙特卡洛（MLMC）方法，用于估计随机对流-扩散算子（具有随机系数）的最小特征值的期望。该MLMC方法基于在一系列逐渐加密网格层次上的特征值问题的有限元（FE）离散化序列。对于离散后的代数特征值问题，我们分别采用瑞利商（RQ）迭代和隐式重启阿诺尔迪（IRA）方法，并对每种情况下的计算成本进行了分析。通过研究各层次方差并将经典有限元误差界推广至随机环境，我们得以界定MLMC估计量的总误差并给出复杂度分析。正如预期，我们的MLMC估计量的复杂度上界优于普通蒙特卡洛方法。为进一步提升MLMC效率，我们利用网格层次结构，将粗网格近似解作为细网格特征值求解器的初始值。针对对流主导问题，我们采用两种额外策略增强MLMC方法的稳定性：首先，考虑离散特征值问题的流线迎风格-勒金公式，这使得我们能够在比标准有限元所能使用的更粗网格上启动MLMC方法；其次，应用同伦方法为每个样本的特征值求解器增加稳定性。最后，我们提出一种多层拟蒙特卡洛方法，该方法在各层次上用拟蒙特卡洛（QMC）规则替代蒙特卡洛。由于QMC更快的收敛速度，这进一步改善了整体复杂度。我们提供了详细的数值结果，比较了不同策略在实际应用场景中MLMC方法的可行性。结果支持了我们的复杂度分析，并进一步证明了在所有情况下该方法均优于普通蒙特卡洛。