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:我们开发了新的多层次蒙特卡洛(MLMC)方法,以估计一个随机系数的蒸馏对流操作员最小的顶值期望值。 MLMC方法基于一个数量元素的序列(FE)的离异性,它基于一个越来越精细的模层。对于分解的、代数的变异性蛋白质,我们使用雷利商(RQ)迭代(RQ)和隐性地重新启动Arnoldi(IRA),对每个案例的成本进行分析。通过研究每个级别的差异和将经典的FE误差与沙变设置相适应,我们能够将我们MLMC测算器的总误差捆绑起来,并提供一个复杂分析。正如所预期的那样,我们的MLMC测算器的复杂程度高于普通的蒙特卡洛。为了进一步提高MLMC的效率,我们利用了所有MSeshes的等级,并使用星际缩缩略图的近值作为精度的起始值。在精细的精度中,我们用FIKSL法的稳定性分析方法,我们运用了一种新的方法,让我们的变变变变的变的变的变法。</s>