Bayesian inference of random fields represented with the Karhunen–Loève expansion

The integration of data into engineering models involving uncertain and spatially varying parameters is oftentimes key to obtaining accurate predictions. Bayesian inference is effective in achieving such an integration. Uncertainties related to spatially varying parameters are typically represented through random fields discretized into a finite number of random variables. The prior correlation length and variance of the field, as well as the number of terms in the random field discretization, have a considerable impact on the outcome of the Bayesian inference, which has received little attention in the literature. Here, we investigate the implications of different choices in the prior random field model on the solution of Bayesian inverse problems. We employ the Karhunen–Loève expansion for the representation of random fields. We show that a higher-order Karhunen–Loève discretization is required in Bayesian inverse problems as compared to standard prior random field representations, since the updated fields are non-homogeneous. Furthermore, the smoothing effect of the forward operator has a large influence on the posterior solution, particularly if we are analyzing quantities of interest that are sensitive to local random fluctuations of the inverse quantity. This is also reflected in posterior predictions, such as the estimation of rare event probabilities. We illustrate these effects analytically through a 1D cantilever beam with spatially varying flexibility, and numerically using a 2D linear elasticity example where the Young's modulus is spatially variable.