Bayesian inference with subset simulation in varying dimensions applied to the Karhunen-Loeve expansion

Uncertainties associated with spatially varying parameters are modeled through random fields discretized into a finite number of random variables. Standard discretization methods, such as the Karhunen-Loeve expansion, use series representations for which the truncation order is specified a priori. However, when data is used to update random fields through Bayesian inference, a different truncation order might be necessary to adequately represent the posterior random field. This is an inference problem that not only requires the determination of the often high-dimensional set of coefficients, but also their dimension. In this article, we develop a sequential algorithm to handle such inference settings and propose a penalizing prior distribution for the dimension parameter. The method is a variable-dimensional extension of BUS (Bayesian Updating with Structural reliability methods), combined with subset simulation (SuS). The key idea is to replace the standard Markov Chain Monte Carlo (MCMC) algorithm within SuS by a trans-dimensional MCMC sampler that is able to populate the discrete-continuous parameter space. To address this task, we consider two types of MCMC algorithms that operate in a fixed-dimensional saturated space. The performance of the proposed method with both MCMC variants is assessed numerically for two examples: a 1D cantilever beam with spatially varying flexibility and a 2D groundwater flow problem with uncertain permeability field.