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 obtain the eigenvalue of the convection-diffusion operator by following a continuous path starting from the pure diffusion operator. 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：我们发展了新的多层蒙特卡罗（MLMC）方法，用于估计随机系数对流扩散算子最小特征值的期望。MLMC方法基于在逐渐加密的网格层级上对特征值问题进行有限元（FE）离散化序列。对于离散后的代数特征值问题，我们分别使用Rayleigh商（RQ）迭代和隐式重启Arnoldi（IRA）方法，并分析了每种情况下的计算成本。通过研究各层级的方差，并将经典有限元误差界适应到随机环境，我们能够界定MLMC估计器的总误差，并进行复杂度分析。正如预期，MLMC估计器的复杂度界优于普通蒙特卡罗方法。为进一步提升MLMC效率，我们利用网格层级结构，将粗糙层级的近似解作为精细层级特征值求解器的初始值。为改善对流主导问题中MLMC方法的稳定性，我们采用两种补充策略：首先，考虑离散特征值问题的流线迎风Petrov-Galerkin格式，这使得我们能够在比标准有限元更粗糙的网格上启动MLMC方法；其次，应用同伦方法，通过追踪从纯扩散算子出发的连续路径来获取对流扩散算子的特征值。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的可行性。这些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MLMC方法相较于普通蒙特卡罗的优越性。