Efficient evaluation of small failure probability in high‐dimensional groundwater contaminant transport modeling via a two‐stage Monte Carlo method

Abstract

In decision‐making for groundwater management and contamination remediation, it is important to accurately evaluate the probability of the occurrence of a failure event. For small failure probability analysis, a large number of model evaluations are needed in the Monte Carlo (MC) simulation, which is impractical for CPU‐demanding models. One approach to alleviate the computational cost caused by the model evaluations is to construct a computationally inexpensive surrogate model instead. However, using a surrogate approximation can cause an extra error in the failure probability analysis. Moreover, constructing accurate surrogates is challenging for high‐dimensional models, i.e., models containing many uncertain input parameters. To address these issues, we propose an efficient two‐stage MC approach for small failure probability analysis in high‐dimensional groundwater contaminant transport modeling. In the first stage, a low‐dimensional representation of the original high‐dimensional model is sought with Karhunen‐Loève expansion and sliced inverse regression jointly, which allows for the easy construction of a surrogate with polynomial chaos expansion. Then a surrogate‐based MC simulation is implemented. In the second stage, the small number of samples that are close to the failure boundary are re‐evaluated with the original model, which corrects the bias introduced by the surrogate approximation. The proposed approach is tested with a numerical case study and is shown to be 100 times faster than the traditional MC approach in achieving the same level of estimation accuracy.

On the Bayesian calibration of expensive computer models with input dependent parameters

“…We test the proposed method against the 2-D groundwater flow and solute transport model (Zhang et al, 2017). The study addresses the case that, under steady state water flow conditions, some amount of con- Figure 5d.…”

“…In the 2-D groundwater flow and solute transport model (Zhang et al, 2017), it is assumed that the uncertainty only stems from the connectivity field. The log conductivity field Z was modeled as a spatially correlated Gaussian random field with a specific separable exponential correlation form (Zhang et al, 2017).…”

“…We consider, that there is available a sample of 500 points; precisely 50 model evaluations at 10 time steps. The experimental training data were generated such that ξ 0 " 0.5 from the original model of Zhang et al (2017). We wish to recover an estimate for (22), as well as to produce an emulator for the concentration.…”

Inverse Modeling of Hydrologic Systems with Adaptive Multifidelity Markov Chain Monte Carlo Simulations

“…C. Smith, 2014;X. Zeng et al, 2018;Zhang et al, 2017). Inevitably, using the low-fidelity model f L (m) can introduce some bias if no error model is considered (Forrester & Keane, 2009;Razavi et al, 2012).…”

“…A low‐fidelity model could be a data‐driven surrogate based on interpolation or regression, a numerical model that considers fewer processes or has a lower numerical precision (e.g., with a coarser discretization) or is constructed by projecting high‐dimensional variables onto their low‐dimensional subspace, etc. (Asher et al, ; Mo et al, ; Razavi et al, ; R. C. Smith, ; X. Zeng et al, ; Zhang et al, ). Inevitably, using the low‐fidelity model f L ( m ) can introduce some bias if no error model is considered (Forrester & Keane, ; Razavi et al, ).…”