## 1. Introduction

[2] Groundwater models are vital tools for predicting the effects of future anthropogenic and/or natural occurrences in the subsurface environment. Model predictions are inherently uncertain due to epistemic and aleatory uncertainties in data and model parameters and structures; uncertainty quantification in groundwater modeling is indispensable, and many methods of uncertainty quantification have been developed to facilitate science-informed decision making in water resource management (see recent review articles of *Matott et al*. [2009] and *Tartakovsky* [2013], and references therein). While this study is only for quantification of parametric uncertainty, the results can be used directly for quantification of model uncertainty, because quantifying parametric uncertainty is the basis of quantifying model structure uncertainty in the popular multimodel analysis methods [*Neuman*, 2003; *Ye et al*., 2004; *Poeter and Hill*, 2007; *Refsgaard et al*., 2012; *Neuman et al*., 2012; *Lu et al*., 2012a]. The Bayesian method is one of the most widely utilized approaches for quantifying parametric uncertainty [*Kitanidis*, 1986; *Box and Tiao*, 1992; *Ezzedine et al*., 1999; *Beck and Au*, 2002; *Marshall et al*., 2005; *Marzouk et al*., 2007; *Ma and Zabaras*, 2009; *Marzouk and Xiu*, 2009; *Allaire and Willcox*, 2010; *Renard*, 2011; *Zeng et al*., 2012; *Kitanidis*, 2012; *Lu et al*., 2012b; *Shi et al*., 2012], wherein model parameters and predictions are modeled as random variables. The Bayesian methods are well connected with and complementary to other methods of uncertainty quantification [e.g., *Woodbury*, 2011; *Nott et al*., 2012]. They are flexible and can be applied to different models to incorporate multiple types of data and prior information [e.g., *Woodbury*, 2007; *Rubin et al*., 2010; *Chen et al*., 2012]. The outputs of Bayesian methods are probability density functions of quantities of interest that can be directly used for uncertainty quantification, risk assessment, and decision making.

[3] In the Bayesian inference framework, this paper presents a computationally efficient method developed using an adaptive sparse-grid high-order stochastic collocation (aSG-hSC) method to reduce the computational cost of Bayesian computation, which is always a burden for practical Bayesian applications especially to computationally demanding models with a large number of parameters. When estimating the posterior probability density function (PPDF) in Bayesian inference, except in special cases in which analytical expressions of the PPDF can be derived [*Woodbury and Ulrych*, 2000; *Hou and Rubin*, 2005], the PPDF is usually estimated numerically using sampling techniques. One of the most popular and robust sampling techniques is the Markov Chain Monte Carlo (MCMC) method [*Marshall et al*., 2005; *Gamerman and Lopes*, 2006; *Vrugt et al*., 2008, 2009; *Keating et al*., 2010; *Liu et al*., 2010]. However, MCMC methods are in general computationally expensive, because a large number of model executions are needed to estimate the PPDF and sample from it. Many MCMC algorithms have been developed to improve computational efficiency, such as delayed rejection and adaptive Metropolis sampling [*Haario et al*., 2006] and differential evolution adaptive Metropolis (DREAM) sampling [*Vrugt et al*., 2008, 2009], by reducing the needed number of model executions. The number of model executions is of primary interest, because computational cost of solving the models dominates over that of other MCMC calculations that are simple algebraic operations. However, even with these advanced methods, the number of model executions is still often in the order of magnitude of tens of thousands or even hundreds of thousands. As a result, applications of MCMC approaches are prohibitive for computationally demanding models such as those of groundwater reactive transport, one solution of which may take tens of minutes and even hours [*Zhang et al*., 2012].

[4] In this study, the problem of high computational cost of MCMC simulations is resolved by incorporating sparse-grid methods into MCMC operation to develop sparse-grid-based MCMC algorithms. The sparse-grid methods in a broader sense are one of surrogate methods that have been used to improve computational efficiency in water resources research [*Razavi et al*., 2012]. The key idea of sparse-grid methods is to place a grid in the parameter space with sparse parameter samples (as opposed to a full tensor-product grid). Then the forward model is solved only for the sparse parameter samples to save computational cost. More specifically speaking, the method used in this study is a stochastic collocation method at sparse grids, also known as the sparse-grid stochastic collocation method [*Nobile et al*., 2008a, 2008b]. Another popular collocation method is the probabilistic collocation method that uses the finite-dimensional polynomial chaos expansion [*Marzouk et al*., 2007; *Li and Zhang*, 2007; *Shi et al*., 2009]. A comprehensive comparison of the two stochastic collocation methods can be found in the study of *Chang and Zhang* [2009] in terms of their accuracy and efficiency. While such a comparison is of high significance to the selection of an appropriate method for different applications, it is beyond the scope of this study. The sparse-grid methods have been demonstrated to be efficient and effective for dealing with high-dimensional interpolation and integration, and they have been used recently in groundwater uncertainty quantification. In the studies of, e.g., *Shi and Yang* [2009], *Lin and Tartakovsky* [2009, 2010], and *Lin et al*. [2010], the sparse-grid methods were used to estimate the mean and covariance of groundwater state variables such as hydraulic head and solute concentrations. In these studies, parameter distributions were assumed known, and Bayesian inference was not conducted. Bayesian inference using the sparse-grid method was conducted in the study of *Ma and Zabaras* [2009] and *Zeng et al*. [2012], in which surrogate of geophysical models was built and then used to evaluate parameter distributions using observations of state variables.

[5] While the aSG-hSC method presented in this paper is in spirit similar to that of *Ma and Zabaras* [2009] and *Zeng et al*. [2012] in terms of using the sparse-grid method to improve computational efficiency of Bayesian inference, our method tackles a more challenging problem of uncertainty quantification and offers more computationally efficient structures of sparse grids. Different from the previous studies of sparse-grid methods that only quantify uncertainty in flow and advection-dispersion problems, this study conducts uncertainty quantification for groundwater reactive transport models, which are significantly more nonlinear due to nonlinear reactions and coupling between flow, transport, and biogeochemical processes. The nonlinearity causes two challenges to applications of sparse-grid methods. First, if the surrogate systems of the nonlinear models are constructed using linear hierarchical basis functions as in previous groundwater applications, more sparse-grid interpolation points, i.e., more model executions, are needed to obtain the prescribed interpolation accuracy, which plagues the purpose of using sparse-grid methods. The other challenge is that the nonlinearity always leads to extremely complex surface of likelihood function (or its least square equivalent) with a large number of local minima such as those reported in *Matott and Rabideau* [2008] and Shi et al. (Assessment of parametric uncertainty for surface complexation modeling of uranium reactive transport, submitted to *Water Resources Research*, 2013) for nitrogen and uranium reactive transport, respectively. The multiple local minima correspond to multiple modes (significant or insignificant) on the surface of the PPDF. Existing algorithms cannot succeed in capturing all the significant modes or may succeed only with significantly increased computational effort. The two problems caused by nonlinearity are not limited to groundwater reactive transport models but prevalent to all nonlinear models.

[6] The aSG-hSC method is developed to resolve the two challenges above. To resolve the first challenge of efficiently approximating the PPDF involving nonlinear groundwater reactive transport models, the surrogate system with a sparse-grid interpolation is constructed with high-order stochastic collocation (hSC) approach, i.e., utilizing high-order hierarchical polynomial basis with quadratic or cubic polynomials as in *Griebel* [1998] and *Bungartz and Griebel* [2004]. Due to their increased accuracy compared to the linear hierarchical basis, the number of model executions needed for constructing the surrogate system can be greatly reduced. The high-order approach is not a trivial extension from the linear technique [*Zhang et al*., 2010)], and it is the first time that the high-order stochastic collocation method is used not only in groundwater modeling but also in surrogate modeling for Bayesian inference. Furthermore, instead of building the approximate PPDF using isotropic sparse-grid interpolation [*Nobile et al*., 2008a; *Barthelmann et al*., 2000] or dimension-adaptive sparse-grid interpolation [*Nobile et al*., 2008b], a locally adaptive sparse-grid (aSG) interpolation [*Griebel*, 1998] is used. This technique utilizes the hierarchical surplus (discussed in section 3.2) as an error indicator to detect the nonsmooth and/or important regions in the parameter space and adaptively place more points in the regions. This results in further computational gains and guarantees that a user-defined accuracy of the surrogate system is realized.

[7] To resolve the second challenge of reducing the computational cost of constructing the surrogate system for a PPDF with multiple modes, an iterative procedure is developed for the aSG-hSC method to incorporate optimization results into the surrogate construction. Using aSG-hSC together with optimization is considered as a strength, since it can leverage extensive research in the area of optimization. The design of the iterative procedure is based on the following observations. In MCMC-based Bayesian inference, large parameter ranges are always specified in the prior distribution due to lack of information. If multiple modes exist on the PPDF, there are high-probability regions around each significant mode (definition of the high-probability regions is given in section 3 below). Markov chains move toward the high-probability regions and generate random samples by following the Metropolis rule [*Gamerman and Lopes*, 2006]. During this process, a large number of samples are discarded in the burn-in period and rejected due to the Metropolis rule, and model executions corresponding to these samples are wasted. This procedure of sampling can be made more computationally efficient using the adaptive sparse-grid techniques, if the approximate locations of the modes are known from optimization. This motivates the iterative aSG-hSC method. In each iteration, global or local optimization is utilized to detect each significant mode of the PPDF, and the corresponding high-probability region is determined based on optimization results such as Hessian matrix at the found optimum. Subsequently, the high-probability region is incorporated into the prior distribution, and the aSG-hSC method is used to construct surrogate within the high-probability region. This is the key to saving computational cost, because the surrogate is not constructed over a large parameter space where a significant number of sparse grid points are blindly placed in the low-probability regions. However, there is a trade-off between the saved computational cost and that spent on optimization, which is discussed in the numerical examples in section 4. The iteration stops until all significant modes are identified according to a user-specified significance tolerance. It is demonstrated in section 4 that our method can find all the modes whose significance is larger than a user-defined significance tolerance. Note that the aSG-hSC method is independent of MCMC methods, so that it can be used together with any MCMC methods. In addition, because both the aSG-hSC and MCMC methods are model independent, the resulting sparse-grid-based MCMC algorithms can be applied to a wide range of problems.

[8] The rest of the paper is organized as follows. In section 2, the Bayesian framework and the conventional MCMC method used in this study are briefly introduced, followed by the iterative aSG-hSC method of constructing the surrogate system presented in section 3. In section 4, the new approach is applied to reactive transport problems and its effectiveness and efficiency in comparison with the conventional MCMC method is demonstrated.