Bayesian inference calibration of the modulus of elasticity

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 Karhunen–Loève expansion, while the solutionto 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. References A. Alexanderian. Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: A review. Inverse Problems 37, 043001 (2021). doi: 10.1088/1361-6420/abe10c J. Beck, B. M. Dia, L. F. R. Espath, Q. Long, and R. Tempone. Fast Bayesian experimental design: Laplace-based importance sampling for the expected information gain. Comput. Meth. Appl. Mech. Eng. 334 (2018), pp. 523–553. doi: 10.1016/j.cma.2018.01.053 M. Clarke, J. Dick, Q. T. Le Gia, K. Mustapha, and T. Tran. Quasi-Monte Carlo sparse grid Galerkin finite element methods for linear elasticity equations with uncertainties. Int. J. Uncert. Quant. 15 (2025), pp. 77–94. doi: 10.1615/Int.J.UncertaintyQuantification.2025056524 M. Dashti and A. M. Stuart. The Bayesian approach to inverse problems. Handbook of Uncertainty Quantification. Springer, 2017. doi: 10.1007/978-3-319-12385-1_7 J. Dick. Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46 (2008), pp. 1519–1553. doi: 10.1137/060666639 J. Dick, R. N. Gantner, Q. T. Le Gia, and C. Schwab. Higher order quasi-Monte Carlo integration for Bayesian PDE inversion. Comput. Math. Appl. 77.1 (2019), pp. 144–172. doi: 10.1016/j.camwa.2018.09.019 J. Dick, F. Y. Kuo, Q. T. Le Gia, D. Nuyens, and C. Schwab. Higher order QMC Petrov–Galerkin discretization for affine parametric operator equations with random field inputs. SIAM J. Numer. Anal. 52 (2014), pp. 2676–2702. doi: 10.1137/130943984 S. Eberle-Blick and N. Hyvönen. Bayesian experimental design for linear elasticity. Inv. Prob. Imag. 18.6 (2024), pp. 1294–1319. doi: 10.3934/ipi.2024015 R. N. Gantner and C. Schwab. Computational higher order quasi-Monte Carlo integration. Monte Carlo and Quasi-Monte Carlo Methods. Ed. by R. Cools and D. Nuyens. Vol. 163. Springer Proc. Math. Stat. Springer, 2016, pp. 271–288. doi: 10.1007/978-3-319-33507-0_12 T. Goda and J. Dick. Construction of interlaced scrambled polynomial lattice rules of arbitrary high order. Found. Comput. Math. 15 (2015), pp. 1245–1278. doi: 10.1007/s10208-014-9226-8