Surrogate models are commonly used in Bayesian approaches such as Markov Chain Monte Carlo (MCMC) to avoid repetitive CPU‐demanding model evaluations. However, the approximation error of a surrogate may lead to biased estimation of the posterior distribution. This bias can be corrected by constructing a very accurate surrogate or implementing MCMC in a two‐stage manner. Since the two‐stage MCMC requires extra original model evaluations after surrogate evaluations, the computational cost is still high. If the information of measurement is incorporated, a locally accurate surrogate can be adaptively constructed with low computational cost. Based on this idea, we integrate Gaussian process (GP) and MCMC to adaptively construct locally accurate surrogates for Bayesian experimental design in groundwater contaminant source identification problems. Moreover, the uncertainty estimate of GP approximation error is incorporated in the Bayesian formula to avoid over‐confident estimation of the posterior distribution. The proposed approach is tested with a numerical case study. Without sacrificing the estimation accuracy, the new approach achieves about 200 times of speed‐up compared to our previous work which implemented MCMC in a two‐stage manner.

In this study, an efficient full Bayesian approach is developed for the optimal sampling well location design and source parameters identification of groundwater contaminants. An information measure, i.e., the relative entropy, is employed to quantify the information gain from concentration measurements in identifying unknown parameters. In this approach, the sampling locations that give the maximum expected relative entropy are selected as the optimal design. After the sampling locations are determined, a Bayesian approach based on Markov Chain Monte Carlo (MCMC) is used to estimate unknown parameters. In both the design and estimation, the contaminant transport equation is required to be solved many times to evaluate the likelihood. To reduce the computational burden, an interpolation method based on the adaptive sparse grid is utilized to construct a surrogate for the contaminant transport equation. The approximated likelihood can be evaluated directly from the surrogate, which greatly accelerates the design and estimation process. The accuracy and efficiency of our approach are demonstrated through numerical case studies. It is shown that the methods can be used to assist in both single sampling location and monitoring network design for contaminant source identifications in groundwater.

Bayesian analysis is widely used in science and engineering for real‐time forecasting, decision making, and to help unravel the processes that explain the observed data. These data are some deterministic and/or stochastic transformations of the underlying parameters. A key task is then to summarize the posterior distribution of these parameters. When models become too difficult to analyze analytically, Monte Carlo methods can be used to approximate the target distribution. Of these, Markov chain Monte Carlo (MCMC) methods are particularly powerful. Such methods generate a random walk through the parameter space and, under strict conditions of reversibility and ergodicity, will successively visit solutions with frequency proportional to the underlying target density. This requires a proposal distribution that generates candidate solutions starting from an arbitrary initial state. The speed of the sampled chains converging to the target distribution deteriorates rapidly, however, with increasing parameter dimensionality. In this paper, we introduce a new proposal distribution that enhances significantly the efficiency of MCMC simulation for highly parameterized models. This proposal distribution exploits the cross covariance of model parameters, measurements, and model outputs and generates candidate states much alike the analysis step in the Kalman filter. We embed the Kalman‐inspired proposal distribution in the DiffeRential Evolution Adaptive Metropolis algorithm during burn‐in and present several numerical experiments with complex, high‐dimensional, or multimodal target distributions. Results demonstrate that this new proposal distribution can greatly improve simulation efficiency of MCMC. Specifically, we observe a speedup on the order of 10–30 times for groundwater models with more than 100 parameters.