Multifidelity Methods for Uncertainty Quantification of a Nonlocal Model for Phase Changes in Materials

TL;DR

This paper develops a multifidelity Monte Carlo method to efficiently quantify uncertainty in a nonlocal Cahn-Hilliard model for phase transitions, significantly reducing computational costs while maintaining accuracy.

Contribution

It introduces a novel multifidelity approach tailored for nonlocal phase transition models, enabling efficient uncertainty quantification with reduced computational expense.

Findings

Achieves about tenfold reduction in mean-squared error compared to standard Monte Carlo.

Effectively balances high- and low-fidelity model sampling to optimize computational resources.

Demonstrates the method's applicability to complex nonlocal phase change models.

Abstract

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.