连续级别蒙特卡洛方法与样本自适应模型层次结构
Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies
AI总结：
本文提出连续级别蒙特卡洛方法，用于PDE应用中根据样本自适应调整模型层次。通过目标导向的有限元误差估计器对数作为新级别参数，证明了无偏性和复杂度定理，并通过数值实验验证了样本自适应细化策略的优势。
摘要：
本文提出将多级蒙特卡洛方法推广到连续级别参数的设置中，即连续级别蒙特卡洛（CLMC）估计器，为PDE应用提供自然框架以根据每个样本调整模型层次。CLMC估计器在目标量足够快速收敛时可无偏估计其期望值。CLMC的实际实现基于在有限分辨率上插值目标量的实际评估。我们使用目标导向的有限元误差估计器的对数作为新级别参数。我们证明了CLMC的无偏性，并证明其复杂度定理，显示CLMC与MLMC具有相同的复杂度速率。最后，我们通过在标准PDE测试问题上成功测试CLMC来支持我们的理论结果。数值实验显示，样本自适应细化策略在样本层面比均匀细化有明显优势。
英文摘要：
In this paper, we present a generalisation of the Multilevel Monte Carlo (MLMC) method to a setting where the level parameter is a continuous variable. This Continuous Level Monte Carlo (CLMC) estimator provides a natural framework in PDE applications to adapt the model hierarchy to each sample. In addition, it can be made unbiased with respect to the expected value of the true quantity of interest provided the quantity of interest converges sufficiently fast. The practical implementation of the CLMC estimator is based on interpolating actual evaluations of the quantity of interest at a finite number of resolutions. As our new level parameter, we use the logarithm of a goal-oriented finite element error estimator for the accuracy of the quantity of interest. We prove the unbiasedness, as well as a complexity theorem that shows the same rate of complexity for CLMC as for MLMC. Finally, we provide some numerical evidence to support our theoretical results, by successfully testing CLMC on a standard PDE test problem. The numerical experiments demonstrate clear gains for sample-wise adaptive refinement strategies over uniform refinements.
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