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Bayesian inference calibration of the modulus of elasticity

J. Dick, Q. T. Le Gia, K. Mustapha

TL;DR

This work tackles the Bayesian calibration of the modulus of elasticity by treating the Young's modulus as a random field modeled through a Karhunen-Loève expansion $E(x,y) = E_0(x) + \sum_{j=1}^{\infty} y_j \psi_j(x)$ and solving the resulting stochastic linear elasticity problem with finite elements. The forward map delivers a displacement field $u$ as a function of the random parameters, and the posterior expectations of quantities of interest are computed by a higher-order quasi-Monte Carlo scheme applied to a truncated, discretized surrogate. The authors derive error bounds that separate truncation, discretization, and quadrature contributions, and demonstrate the approach on a unit square with synthetic noisy data, showing convergence of the QoI estimates $Z'_N/Z$ as the QMC dimension and sampling budget increase. Overall, the paper provides a scalable, rigorous framework for uncertainty quantification in elasticity calibration using advanced Bayesian inversion and high-order QMC techniques.

Abstract

This work uses the Bayesian inference technique to infer the Young modulus from the stochastic linear elasticity equation. The Young modulus is modeled by a finite Karhunen Loéve expansion, while the solution to the linear elasticity equation is approximated by the finite element method. The high-dimensional integral involving the posterior density and the quantity of interest is approximated by a higher-order quasi-Monte Carlo method.

Bayesian inference calibration of the modulus of elasticity

TL;DR

This work tackles the Bayesian calibration of the modulus of elasticity by treating the Young's modulus as a random field modeled through a Karhunen-Loève expansion and solving the resulting stochastic linear elasticity problem with finite elements. The forward map delivers a displacement field as a function of the random parameters, and the posterior expectations of quantities of interest are computed by a higher-order quasi-Monte Carlo scheme applied to a truncated, discretized surrogate. The authors derive error bounds that separate truncation, discretization, and quadrature contributions, and demonstrate the approach on a unit square with synthetic noisy data, showing convergence of the QoI estimates as the QMC dimension and sampling budget increase. Overall, the paper provides a scalable, rigorous framework for uncertainty quantification in elasticity calibration using advanced Bayesian inversion and high-order QMC techniques.

Abstract

This work uses the Bayesian inference technique to infer the Young modulus from the stochastic linear elasticity equation. The Young modulus is modeled by a finite Karhunen Loéve expansion, while the solution to the linear elasticity equation is approximated by the finite element method. The high-dimensional integral involving the posterior density and the quantity of interest is approximated by a higher-order quasi-Monte Carlo method.

Paper Structure

This paper contains 6 sections, 5 theorems, 35 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The forward problem
  3. Higher order QMC method
  4. Bayesian inversion
  5. Numerical experiments
  6. Acknowledgements

Key Result

Theorem 2.1

Assume that ass A11 is satisfied for some $0<p<1$, and $\chi:{\bf V} \to \mathbb{R}$ is a bounded linear functional, ($|\chi({\bf w})|\le \|\chi\|_{{\bf V}^*}\|{\bf w}\|_{\bf V}$ for all ${\bf w} \in {\bf V}$). Then, for every ${\bf f} \in {\bf V}^*$, ${\boldsymbol{y}}\in U$, and $s \in \mathbb{N}$,

Figures (1)

  • Figure 1: Un-normalised posterior density with $s=2$

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • proof