Titel: Quasi-Monte Carlo for Bayesian inverse problems governed by PDEs

Abstract:
We study the application of quasi-Monte Carlo (QMC) methods for Bayesian inverse problems governed by PDEs. For the parameterization of the unknown quantities, we consider a model recently studied by Chernov and Le [1,2] as well as Harbrecht, Schmidlin, and Schwab [3] in which the input random field is assumed to belong to a Gevrey class. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Specifically, we consider the application of QMC for Bayesian shape inversion [4] and electrical impedance tomography [5] using the techniques developed in [6].  

References:
[1] A. Chernov and T. Le. Analytic and Gevrey class regularity for parametric elliptic eigenvalue problems and applications. SIAM J. Numer. Anal., 62(4):1874-1900, 2024.
[2] A. Chernov and T. Le. Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification. Comput. Math. Appl., 164:116-130, 2024.
[3] H. Harbrecht, M. Schmidlin, and Ch. Schwab. The Gevrey class implicit mapping theorem with applications to UQ of semilinear elliptic PDEs. Math. Models Methods Appl. Sci., 34(5):881-917, 2024.
[4] A. Djurdjevac, V. Kaarnioja, M. Orteu, and C. Schillings. Quasi-Monte Carlo for Bayesian shape inversion governed by the Poisson problem subject to Gevrey regular domain deformations. Preprint 2025, arXiv:2502.14661 [math.NA].
[5] L. Bazahica, V. Kaarnioja, and L. Roininen. Uncertainty quantification for electrical impedance tomography using quasi-Monte Carlo methods. Preprint 2024, arXiv:2411.11538 [math.NA].
[6] V. Kaarnioja and C. Schillings. Quasi-Monte Carlo for Bayesian design of experiment problems governed by parametric PDEs. Preprint 2024, arXiv:2405.03529 [math.NA].