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

Sharana Kumar Shivanand, Bojana Rosić

TL;DR

The paper introduces scale-invariant NMSE estimators for MC and MLMC to estimate mean and variance of a QoI under linear transformations, and develops unbiased, h-statistic-based normalizers to ensure full invariance to scale and distributional changes. It derives MLMC variants for mean and variance with NMSE-based error control and optimal sampling across levels. The methods are applied to a linear elastic bone-tissue model with matrix-valued random elasticity capturing heterogeneity and random anisotropy, demonstrating significant computational savings of MLMC over MC while maintaining accuracy. The work provides a robust framework for uncertainty quantification in solid-mechanics problems where scale and distributional characteristics vary across simulations.

Abstract

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 variation in 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.

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

TL;DR

The paper introduces scale-invariant NMSE estimators for MC and MLMC to estimate mean and variance of a QoI under linear transformations, and develops unbiased, h-statistic-based normalizers to ensure full invariance to scale and distributional changes. It derives MLMC variants for mean and variance with NMSE-based error control and optimal sampling across levels. The methods are applied to a linear elastic bone-tissue model with matrix-valued random elasticity capturing heterogeneity and random anisotropy, demonstrating significant computational savings of MLMC over MC while maintaining accuracy. The work provides a robust framework for uncertainty quantification in solid-mechanics problems where scale and distributional characteristics vary across simulations.

Abstract

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 variation in 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.

Paper Structure

This paper contains 21 sections, 4 theorems, 89 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Problem description
  3. Monte Carlo method
  4. Monte Carlo estimation of mean
  5. Monte Carlo estimation of variance
  6. Multilevel Monte Carlo method
  7. Multilevel Monte Carlo estimation of mean
  8. Multilevel Monte Carlo estimation of variance
  9. Application to linear elasticity: a model problem
  10. Deterministic setting
  11. Stochastic modelling of material uncertainty: a reduced parametric approach
  12. Stochastic setting
  13. Numerical results: 2D human femur
  14. Specifications
  15. Screening test
  16. ...and 6 more sections

Key Result

Proposition 3.1

Let us consider $c_{\alpha}, c_{\gamma}, \alpha$ and $\gamma$ as positive constants, and then one may define the error bounds as follows, where Then, for any $0<\mathring{\epsilon}<e^{-1}$, the Monte Carlo (MC) mean estimator $\widehat{\mu}^{\mathrm{MC}}$ with $N = \mathcal{O}(\mathring{\epsilon}^{-2})$ and $h = \mathcal{O}(\mathring{\epsilon}^{1/\alpha})$ satisfies the normalized mean square err

Figures (8)

  • Figure 1: Geometry and boundary conditions
  • Figure 2: Nested mesh levels of 2D femur bone
  • Figure 3: Decay of eigenvalues across all mesh levels
  • Figure 4: Visualization of random anisotropy $C(\cdot, \omega)$ on mesh $l=0$
  • Figure 5: Visualization of spatial variation of a realization $C_{1, 1}(\boldsymbol x, \cdot)$
  • ...and 3 more figures

Theorems & Definitions (4)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • Proposition 4.2