An adaptive sparse-grid high-order stochastic collocation method for Bayesian inference in groundwater reactive transport modeling

Bayesian analysis has become vital to uncertainty quantification in groundwater modeling, but its application has been hindered by the computational cost associated with numerous model executions required by exploring the posterior probability density function (PPDF) of model parameters. This is particularly the case when the PPDF is estimated using Markov Chain Monte Carlo (MCMC) sampling. In this study, a new approach is developed to improve the computational efficiency of Bayesian inference by constructing a surrogate of the PPDF, using an adaptive sparse-grid high-order stochastic collocation (aSG-hSC) method. Unlike previous works using first-order hierarchical basis, this paper utilizes a compactly supported higher-order hierarchical basis to construct the surrogate system, resulting in a significant reduction in the number of required model executions. In addition, using the hierarchical surplus as an error indicator allows locally adaptive refinement of sparse grids in the parameter space, which further improves computational efficiency. To efficiently build the surrogate system for the PPDF with multiple significant modes, optimization techniques are used to identify the modes, for which high-probability regions are defined and components of the aSG-hSC approximation are constructed. After the surrogate is determined, the PPDF can be evaluated by sampling the surrogate system directly without model execution, resulting in improved efficiency of the surrogate-based MCMC compared with conventional MCMC. The developed method is evaluated using two synthetic groundwater reactive transport models. The first example involves coupled linear reactions and demonstrates the accuracy of our high-order hierarchical basis approach in approximating high-dimensional posteriori distribution. The second example is highly nonlinear because of the reactions of uranium surface complexation, and demonstrates how the iterative aSG-hSC method is able to capture multimodal and non-Gaussian features of PPDF caused by model nonlinearity. Both experiments show that aSG-hSC is an effective and efficient tool for Bayesian inference.