Assessment of HMC Parameter Updates for Piping Zone Boundary Detection

Estimation of the extent of piping or internal erosion in an earthen embankment/ levee can provide valuable information for the serviceability of such geotechnical structures. Working in a Bayesian framework, different parameter updates are discussed for the update of domain geometry, using a statistically efficient gradient based Markov Chain Monte Carlo (MCMC) algorithm called Hamiltonian Monte Carlo (HMC). Although, the hydraulic conductivity spatial random field is generally uncertain, it is assumed to be known exactly in this study (for simplicity), and the updates presented only consider uncertainty in domain boundary. The parameter updates are discussed for volume integral method based discretization of domains e.g. finite element method. The main challenge in such shape detection problems is to ensure a high quality of mesh as the boundary is updated. As such, the parameter update methods can be classified into two groups, one without remeshing and the other with remeshing. Additionally, in HMC, parameter updates have to be designed in a manner that finite element nodal coordinate functions are differentiable w.r.t the parameters. For the class of updates that do not involve remeshing, methods that maintain mesh quality even under large distortions, such as the mesh moving method are considered. The reversibility aspects of such updates and their implications on the inverse analysis are highlighted. A method to compute nodal coordinate gradients w.r.t parameters, in the remeshing case is also discussed. Finally, the merits and demerits of the two classes of methods are highlighted and an extension of the remeshing class to trans-dimensional parameter updates is briefly introduced.

Keywords: Inverse problems, HMC, boundary detection, piping zone, mesh moving method, remeshing.