This study is devoted to the construction of a multifidelity Monte Carlo (MFMC) method for the uncertainty quantification of a nonlocal, non-mass-conserving Cahn-Hilliard model for phase transitions with an obstacle potential. We are interested in the estimation of the expected value of an output of interest (OoI) that depends on the solution of the nonlocal Cahn-Hilliard model. As opposed to its local counterpart, the nonlocal model captures sharp interfaces without the need for significant mesh refinement. However, the computational cost of the nonlocal Cahn-Hilliard model is higher than that of its local counterpart with similar mesh refinement, inhibiting its use for outer-loop applications such as uncertainty quantification. The MFMC method augments the desired high-fidelity, high-cost OoI with a set of lower-fidelity, lower-cost OoIs to alleviate the computational burden associated with nonlocality. Most of the computational budget is allocated to sampling the cheap surrogate models to achieve speedup, whereas the high-fidelity model is sparsely sampled to maintain accuracy. For the non-mass-conserving nonlocal Cahn-Hilliard model, the use of the MFMC method results in, for a given computational budget, about one-order-of-magnitude reduction in the mean-squared error of the expected value of the OoI relative to that of the Monte Carlo method.

翻译：本研究致力于构建一种多保真度蒙特卡洛方法，用于带有障碍势的非局部、非质量守恒Cahn-Hilliard相变模型的不确定性量化。我们关注于估计依赖于非局部Cahn-Hilliard模型解的输出兴趣值的期望值。与其局部对应模型不同，非局部模型无需显著网格细化即可捕捉尖锐界面。然而，在类似网格细化条件下，非局部Cahn-Hilliard模型的计算成本高于局部对应模型，这限制了其在不确定性量化等外环应用中的使用。MFMC方法通过将所需的高保真度、高成本OoI与一组低保真度、低成本OoI相结合，以缓解非局部性带来的计算负担。大部分计算预算分配用于采样廉价的替代模型以实现加速，而高保真度模型则通过稀疏采样来保持精度。对于非质量守恒非局部Cahn-Hilliard模型，在给定计算预算下，使用MFMC方法可使OoI期望值的均方误差相较于蒙特卡洛方法降低约一个数量级。