Scale-invariant Monte Carlo and multilevel Monte Carlo estimation of mean and variance: An application to simulation of linear elastic bone tissue

We propose novel scale-invariant error estimators for the Monte Carlo and multilevel Monte Carlo estimation of mean and variance. For any linear transformation of the distribution of the quantity of interest, the computation cost across fidelity levels is optimized using a normalized error estimate, which is not only fully dimensionless but also remains robust to variations in the characteristics of the distribution. We demonstrate the effectiveness of the algorithms through application to a mechanical simulation of linear elastic bone tissue, where material uncertainty incorporating both heterogeneity and random anisotropy is considered in the constitutive law.