Preconditioned Crank‐Nicolson Markov Chain Monte Carlo Coupled With Parallel Tempering: An Efficient Method for Bayesian Inversion of Multi‐Gaussian Log‐Hydraulic Conductivity Fields

Abstract

Geostatistical inversion with quantified uncertainty for nonlinear problems requires techniques for providing conditional realizations of the random field of interest. Many first‐order second‐moment methods are being developed in this field, yet almost impossible to critically test them against high‐accuracy reference solutions in high‐dimensional and nonlinear problems. Our goal is to provide a high‐accuracy reference solution algorithm. Preconditioned Crank‐Nicolson Markov chain Monte Carlo (pCN‐MCMC) has been proven to be more efficient in the inversion of multi‐Gaussian random fields than traditional MCMC methods; however, it still has to take a long chain to converge to the stationary target distribution. Parallel tempering aims to sample by communicating between multiple parallel Markov chains at different temperatures. In this paper, we develop a new algorithm called pCN‐PT. It combines the parallel tempering technique with pCN‐MCMC to make the sampling more efficient, and hence converge to a stationary distribution faster. To demonstrate the high‐accuracy reference character, we test the accuracy and efficiency of pCN‐PT for estimating a multi‐Gaussian log‐hydraulic conductivity field with a relative high variance in three different problems: (1) in a high‐dimensional, linear problem; (2) in a high‐dimensional, nonlinear problem and with only few measurements; and (3) in a high‐dimensional, nonlinear problem with sufficient measurements. This allows testing against (1) analytical solutions (kriging), (2) rejection sampling, and (3) pCN‐MCMC in multiple, independent runs, respectively. The results demonstrate that pCN‐PT is an asymptotically exact conditional sampler and is more efficient than pCN‐MCMC in geostatistical inversion problems.

Bayesian Inversion of Multi‐Gaussian Log‐Conductivity Fields With Uncertain Hyperparameters: An Extension of Preconditioned Crank‐Nicolson Markov Chain Monte Carlo With Parallel Tempering

“…The angle of anisotropy is fixed as in Table2. This extends the test cases already used byXu et al (2020) toward uncertain hyperparameters, while using the same reference field.…”

“…We consider fully saturated, steady‐state groundwater flow as test case, which is an extension of the test case of Xu et al. (2020). The flow equation can be written as $$$\nabla \cdot (K\nabla H)\hspace*{.5em}+S=0,$$$ where $$\nabla \cdot $$ denotes the divergence operator, $$K$$ is the hydraulic conductivity (m/day), $$\nabla $$ is the Nabla operator, $$H$$ denotes hydraulic head (m), and $$S$$ denotes the source/sink term as volumetric injection flow rate per unit volume of aquifer (1/day).…”

Efficient Discretization‐Independent Bayesian Inversion of High‐Dimensional Multi‐Gaussian Priors Using a Hybrid MCMC

“…We consider an artificial steady state groundwater flow in a confined aquifer test case as proposed in Xu et al. (2020). It has a size of with a constant depth of 50 m and is discretized into cells as shown in Figure 2.…”

“…Here, however, we use a typical benchmarking setup of Xu et al. (2020) with hydraulic head or concentration measurements, because it ensures intercomparability of results. However, our method is not restricted to this choice of data.…”

“…A variety of different measurement types are used for geostatistical Bayesian inversion (e.g., Butera & Soffia, 2017;Ezzedine & Rubin, 1996;Gutjahr et al, 1994;Zimmerman et al, 1998) and optimal choices can be made (Nowak et al, 2010). Here, however, we use a typical benchmarking setup of Xu et al (2020) with hydraulic head or concentration measurements, because it ensures intercomparability of results. However, our method is not restricted to this choice of data.…”