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Transport in Porous Media

, Volume 126, Issue 1, pp 23–38 | Cite as

Efficient Uncertainty Quantification for Unconfined Flow in Heterogeneous Media with the Sparse Polynomial Chaos Expansion

  • Jin Meng
  • Heng LiEmail author
Article

Abstract

In this study, we explore an efficient stochastic approach for uncertainty quantification of unconfined groundwater flow in heterogeneous media, where a sparse polynomial chaos expansion (PCE) surrogate model is constructed with the aid of the feature selection method. The feature selection method is introduced to construct a sparse PCE surrogate model with a reduced number of basis functions, which is accomplished by the least absolute shrinkage and selection operator-modified least angle regression and cross-validation. The training samples are enriched sequentially with the quasi-optimal samples until the results are satisfactory. In this study, we test the performance of the sparse PCE method for unconfined flow with the presence of random hydraulic conductivity and recharge, as well as pumping well. Numerical experiments reveal that, even with large spatial variability and high random dimensionality, the sparse PCE approach is able to accurately estimate the flow statistics with greatly reduced computational efforts compared to Monte Carlo simulations.

Keywords

Uncertainty quantification Unconfined flow Sparse PCE Feature selection Quasi-optimal sampling 

1 Introduction

In realistic subsurface problems, description of flow and transport in porous media often involves some degree of uncertainty, due to the strong heterogeneity and incomplete knowledge of the formations. In order to quantify the uncertainties, stochastic partial differential equations need to be solved with some stochastic approaches. The most common one is the Monte Carlo (MC) method, where the model inputs are randomly sampled and the statistical properties of the outputs can be obtained from the simulated realizations. However in the MC, a large number of realizations are usually required to reduce the sampling errors for obtaining high-order moments and the probability density function (PDF), which leads to unaffordable computational efforts (Ballio and Guadagnini 2004). Besides, perturbation method and moment equation method (Zhang 2001; Zhang and Lu 2004) are other stochastic approaches to uncertainty quantification. Due to the need of manipulating the governing flow equations, the solving procedures could be complicated and cumbersome.

As an efficient alternative, the polynomial chaos expansion (PCE) method has drawn more and more attentions over the past decades (Ghanem 1998; Xiu and Karniadakis 2002a, b; Oladyshkin et al. 2012; Dai et al. 2014). In the PCE method, the random output is represented by a series of orthogonal polynomial basis functions (Ghanem and Spanos 2003). The orthogonal polynomials are associated with the distribution of the input random variables. For instance, Hermite, Laguerre, and Jacobi polynomials are optimal for Gaussian, Gamma, and Beta random variables, respectively, and the generalized PCE is developed for arbitrary variables (Xiu and Karniadakis 2002a, b). If the random inputs are correlated or characterized by random field, parameterization methods such as Karhunen–Loève (K–L) expansion can be used to transform them to independent random variables (Li and Zhang 2007, 2013; Li 2014). Once the PCE coefficients are evaluated, the polynomial surrogate model can be used for uncertainty quantification (UQ) and global sensitivity analysis (GSA) (Oladyshkin et al. 2012; Dai et al. 2014).

The PCE method can be classified as intrusive and non-intrusive approaches. The intrusive approach requires manipulation of governing equations. The best known is the stochastic finite element method, which utilizes the Galerkin scheme and requires solving a set of coupled equations (Maître et al. 2002; Ghanem and Spanos 2003). In contrast, the non-intrusive approach allows using the existing deterministic simulator, which can be treated as a black box. In this category, the probabilistic collocation method (PCM) and stochastic collocation method (SCM) are widely used for UQ (Webster et al. 1996; Li and Zhang 2007; Babuška et al. 2007; Chang and Zhang 2009). In these methods, a set of decoupled equations are derived on some collocation points, which are associated with Lagrange interpolation or Galerkin integral (Li and Zhang 2007; Chang and Zhang 2009). The PCE coefficients can be obtained by directly solving the equation set or by regression (Blatman and Sudret 2010, 2011). These approaches are found to be accurate and computationally efficient for problems with relatively low random dimensionality (Li and Zhang 2007; Chang and Zhang 2009; Liao and Zhang 2015).

Stochastic modeling of flow in heterogeneous porous media has been widely studied (e.g., Polmann et al. 1991; Zhang 2001; Li and Zhang 2007; Nezhad et al. 2011). For unconfined groundwater flow, strong spatial nonstationarity exists due to the presence of nonlinearity, recharge/discharge, and injection/pumping wells. Larger random dimensionality and higher degree of the PCE are required to better describe such complex process, where the number of basis functions will explode (Xiu and Hesthaven 2005). Shi et al. (2009) and Li et al. (2009) utilized the PCM for unconfined and unsaturated flow problems but they are restricted to relatively low random dimensionality. To overcome this “curse of dimension” problem, some dimension reduction methods need to be applied for UQ. Some researchers introduced the idea of compressive sensing to guarantee sparsity with the L1-norm restriction (Doostan and Owhadi 2011; Yan et al. 2012; Hampton and Doostan 2015). Fajraoui et al. (2012) reduced the dimensions by retaining terms that make significant contributions to the variance of the model. Blatman and Sudret (2010, 2011) made use of the stepwise regression and least angle regression (LAR) to construct the PCE model with a reduced number of basis functions. Elsheikh et al. (2014) further applied LAR for Bayesian inference of subsurface flow models. In fact, both the stepwise regression and LAR are common algorithms in the field of feature selection (Tibshirani 1996; Guyon and Elisseeff 2003). Feature selection was first proposed in the area of statistical learning for removing redundant and irrelevant features. A mature system for feature selection has been established, where extensive algorithms and successful applications have been published in the last two decades (Tibshirani 1996; Efron et al. 2004; Hastie et al. 2009). Meng and Li (2017) proposed the sparse PCE for UQ of confined groundwater flow, where the least absolute shrinkage and selection operator (LASSO) modified least angle regression (LASSO-LAR) algorithm is used for feature selection.

This present paper is an extension of our earlier work for confined flow (Meng and Li 2017) to unconfined flow problems, where higher random dimensionality is involved. In the process of constructing the sparse PCE, the LASSO-LAR is applied for feature selection. Different factors are studied in this paper, e.g., coupling of random hydraulic conductivity and recharge, as well as pumping well. In contrast to confined flow, quantifying uncertainty for unconfined flow with these factors is of more difficulties. As a result, the S-value constrained quasi-optimal sampling method is adopted to improve the optimality of the samples in constructing the sparse PCE, in contrast to Meng and Li (2017), where only Latin hypercube sampling was used. In all the studied cases, comparisons are made between the sparse PCE and MC in terms of both computational accuracy and efficiency. The numerical results show that the sparse PCE is accurate and efficient for UQ of unconfined flow.

2 Methodology

2.1 Governing Equations

We consider the unconfined groundwater flow under Dupuit assumption (Bear 1972):
$$\begin{aligned} S_y \frac{\partial h(\mathbf{x},t)}{\partial t}+\nabla \times \mathbf{q}(\mathbf{x},t)= & {} R(\mathbf{x},t)+Q(\mathbf{x},t), \end{aligned}$$
(1)
$$\begin{aligned} \mathbf{q}(\mathbf{x},t)= & {} -K_s (\mathbf{x})h(\mathbf{x},t)\nabla h(\mathbf{x},t), \end{aligned}$$
(2)
subject to the initial and boundary conditions:
$$\begin{aligned} h(\mathbf{x},0)= & {} h_0 (\mathbf{x}),\quad \mathbf{x}\in D, \end{aligned}$$
(3)
$$\begin{aligned} h(\mathbf{x},t)= & {} g_\mathrm{D} (\mathbf{x},t),\quad \mathbf{x}\in \varGamma _\mathrm{D}, \end{aligned}$$
(4)
$$\begin{aligned} \mathbf{q}(\mathbf{x},t)\cdot \mathbf{n}(x)= & {} g_N (\mathbf{x},t),\quad \mathbf{x}\in \varGamma _N, \end{aligned}$$
(5)
where \(h(\mathbf{x},t)\) is hydraulic head; \(S_y \) is specific yield; \(K_s (\mathbf{x})\) is hydraulic conductivity; \(R(\mathbf{x},t)\) is areal recharge/discharge (e.g., rainfall or evaporation); \(Q(\mathbf{x},t)\) is point source/sink term (e.g., injection/pumping well); \(h_0 (\mathbf{x})\) is initial head in domain \(D\in {\mathbb {R}}^{d}; g_\mathrm{D} (\mathbf{x},t)\) is prescribed head on Dirichlet boundary segments \(\varGamma _\mathrm{D} ; g_N (\mathbf{x},t)\) is prescribed flux across Neumann boundary segments \(\varGamma _N \); and \(\mathbf{n}(x)=(n_1 ,\ldots ,n_\mathrm{d} )^{T}\) is an outward unit vector normal to the boundary \(\varGamma =\varGamma _\mathrm{D} \cup \varGamma _N \). In this study, \(\ln [K_s (\mathbf{x},\theta )]\) and \(R(\mathbf{x},t)\) are assumed to be second-order stationary Gaussian random fields.

2.2 Karhunen–Loève (K–L) Expansion

Let \(Y(\mathbf{x},\theta )\) be a random field, where \(\mathbf{x}\in D, \theta \in \Theta \) (a probability space). One may write \(Y(\mathbf{x},\theta )=\bar{{Y}}(\mathbf{x})+{Y}'(\mathbf{x},\theta )\), where \(\bar{{Y}}(\mathbf{x})\) and \({Y}'(\mathbf{x},\theta )\) are the mean and fluctuation, respectively. Since the covariance of the random field \(C_Y (\mathbf{x},{\mathbf{x}}')=\left\langle {{Y}'(\mathbf{x},\theta ){Y}'({\mathbf{x}}',\theta )} \right\rangle \) is bounded, symmetric and positive-definite, it may be decomposed as (Ghanem and Spanos 2003):
$$\begin{aligned} C_Y (\mathbf{x},{\mathbf{x}}')=\sum _{i=1}^\infty {\lambda _i f_i (\mathbf{x})} f_i ({\mathbf{x}}'), \end{aligned}$$
(6)
where \(\lambda _i \) and \(f_i (\mathbf{x})\) are eigenvalue and eigenfunction, respectively. The eigenpairs can be solved from the following Fredholm equation of the second type:
$$\begin{aligned} \int _D {C_Y (\mathbf{x},{\mathbf{x}}')f(\mathbf{x})\text {d}{} \mathbf{x}=\lambda f({\mathbf{x}}')}. \end{aligned}$$
(7)
Then the random field can be expressed by K–L expansion as:
$$\begin{aligned} Y(\mathbf{x},\theta )=\bar{{Y}}(\mathbf{x})+\sum _{i=1}^\infty {\sqrt{\lambda _i}f_i (\mathbf{x})} \xi _i (\theta ). \end{aligned}$$
(8)
where \(\xi _i (\theta )\) are orthogonal random variables with zero mean and unit variance. For Gaussian random field, \(\xi _i (\theta )\) are independent, standard Gaussian distributed and the K–L approximation is optimal with mean square convergence (Ghanem and Spanos 2003). For non-Gaussian field, kernel principle component analysis (KPCA) can be applied for parameterization (Li and Zhang 2013).
The eigenpairs are solved numerically (Eq. 7) in most cases; however, analytical or semi-analytical solutions exist under certain conditions. For a 2D Gaussian random field, assume the covariance function between \(\mathbf{x}(x_1 ,y_1 )\) and \({\mathbf{x}}'(x_2 ,y_2 )\) to be exponential and separable \(C_Y (\mathbf{x},{\mathbf{x}}')=\sigma _Y^2 \exp (-\left| {x_1 -x_2 } \right| /\eta _x -\left| {y_1 -y_2 } \right| /\eta _y )\), where \(\sigma _Y^2 \) and \(\eta \) are the variance and correlation length, respectively. In this setting, the eigenpairs in each direction (e.g., x direction) can be expressed analytically (Zhang and Lu 2004):
$$\begin{aligned} \lambda _j^{(x)} =\frac{2\eta _x \sigma _Y^2 }{\eta _x^2 (\omega _j^{(x)} )^{2}+1},\quad f_j^{(x)} (x)=\frac{\eta _x \omega _j^{(x)} \cos (\omega _j^{(x)} x)+\sin (\omega _j^{(x)} x)}{\sqrt{[(\eta _x \omega _j^{(x)} )^{2}+1]L_x /2+\eta _x}}, \end{aligned}$$
(9)
where \(\omega _j^{(x)} \) are positive roots of the characteristic equation:
$$\begin{aligned} (\eta _x^2 \omega ^{2}-1)\sin (\omega L_x )=2\eta _x \omega \cos (\omega L_x). \end{aligned}$$
(10)
Then the 2D eigenpairs are expressed as:
$$\begin{aligned} \lambda _i =\frac{1}{\sigma _Y^2 }\lambda _j^{(x)} \lambda _k^{(y)} ,\quad f_i (\mathbf{x})=f_i (x,y)=f_j^{(x)} (x)f_k^{(y)} (y). \end{aligned}$$
(11)
In real applications, the truncated K–L expansion is applied for computation. The number of truncated terms is associated with the decay rate of \(\lambda _i \) (Zhang and Lu 2004). The dimensionality of the random field is determined by these retained random variables \({\varvec{\xi }} (\theta ) = (\xi _1 (\theta ),\ldots ,\xi _n (\theta ))^{T}\).

2.3 Polynomial Chaos Expansion (PCE)

Since the covariance function of random output is yet to be found, the polynomial chaos expansion (PCE) is utilized to represent the random structure. With the PCE technique, the random output is decomposed into a series of orthogonal polynomials in terms of the independent random variables \({\varvec{\xi }} (\theta )\) (Xiu and Karniadakis 2002a, b):
$$\begin{aligned} h(\theta )=\sum _{{\varvec{\upalpha }} \in {\mathbb {N}}^{n}} {a_{\varvec{\upalpha }} \phi _{\varvec{\upalpha }} ({\varvec{\xi }} (\theta ))}, \end{aligned}$$
(12)
where \(a_{\varvec{\upalpha }}\) are unknown PCE coefficients and \(\phi _\upalpha \) are multivariate orthogonal polynomial basis functions associated with the joint probability density function (PDF). Due to the independence of \({\varvec{\xi }} (\theta )\), the multivariate polynomial can be constructed by tensor product of the univariate polynomials in each dimension:
$$\begin{aligned} \phi _{\varvec{\upalpha }} ({\varvec{\xi }})=\prod _{i=1}^n {\varphi _{\alpha _i }^{(i)} (\xi _i )} , \end{aligned}$$
(13)
where \(\alpha _i \) indicates the degree of univariate polynomial \(\varphi _{\alpha _i}^{(i)}\) in the i-th dimension. If \(\xi _i \) obey a certain basic distribution, e.g., Gaussian, Gamma or Beta distribution, \(\varphi _{\alpha _i }^{(i)} \) is Hermite, Laguerre or Jacobi polynomials, respectively (Xiu and Karniadakis 2002a, b). For arbitrary distribution, \(\varphi _{\alpha _i }^{(i)} \) is derived by generalized PCE technique (Xiu and Karniadakis 2002a, b; Li et al. 2011).
For practical application, the total degree of polynomials is restricted to be no more than d, so that a finite number of terms are retained in PCE. The truncated PCE is:
$$\begin{aligned} h_\mathrm{d} (\theta )=\sum _{{\varvec{\upalpha }} \in \Lambda } {a_{\varvec{\upalpha }} \phi _{\varvec{\upalpha }} ({\varvec{\xi }} (\theta ))} , \end{aligned}$$
(14)
where \(\Lambda \triangleq \{{\varvec{\upalpha }} \in {\mathbb {N}}^{n}:\left| {\varvec{\upalpha }} \right| =\sum _{i=1}^n {\alpha _i} \le d\}\) is a finite set of \({\varvec{\upalpha }}\). There are \(P=(n+d)!/n!d!\) basis functions in the truncated PCE and the coefficients of these P terms have to be determined. Once the coefficients are evaluated, then the mean and variance of the random output can be written as:
$$\begin{aligned} \mu _h =a_{\alpha =0},\quad \sigma _h^2 =\sum _{{\varvec{\upalpha }} \in \{\Lambda \backslash 0\}} {a_{\varvec{\upalpha }}^2 }. \end{aligned}$$
(15)

2.4 Coefficients Evaluation

The PCE coefficients can be evaluated with the help of Galerkin method (Ghanem 1998), where a set of coupled equations have to be solved. As an efficient alternative, the collocation methods are more widely used today (Babuška et al. 2007; Li and Zhang 2007). In these methods, a set of determined and decoupled equations are derived on some collocation points \(\{{\varvec{\xi }} (\theta _1 ),\ldots ,{\varvec{\xi }} (\theta _N )\}\):
$$\begin{aligned} \sum _{j=1}^P {a_j \phi _j ({\varvec{\xi }} (\theta _i ))}\,=\, h(\theta _i ),\quad i=1,\ldots ,N. \end{aligned}$$
(16)
We rewrite the equation set in the form of matrix:
$$\begin{aligned} {\varvec{\Phi }} \mathbf{a}=\mathbf{h}, \end{aligned}$$
(17)
where \(\mathbf{a}=(a_1 ,\ldots ,a_P )^{T}\) is the vector of coefficients, \(\mathbf{h}=(h(\theta _1 ),\ldots ,h(\theta _N ))^{T}\) is the vector of model realizations, and \({\varvec{\Phi }}_{N\times P} \) is a Vandermonde-like matrix of basis functions whose elements are \({\varvec{\Phi }}_{ij} =\phi _j ({\varvec{\xi }} (\theta _i )), i=1,\ldots ,N, j=1,\ldots ,P\). When \(N>P\), the coefficients can be directly solved by ordinary least square (OLS) regression:
$$\begin{aligned} \mathbf{a}^{OLS}= & {} \arg \min \left\{ {(\mathbf{h}- {\varvec{\Phi }} \mathbf{a})^{T}(\mathbf{h}- {\varvec{\Phi }} \mathbf{a})} \right\} \nonumber \\= & {} ({\varvec{\Phi }}^{T}{\varvec{\Phi }})^{-1}{\varvec{\Phi }}^{T}\mathbf{h}. \end{aligned}$$
(18)
However in many cases, there exist a large amount of random inputs and higher-degree polynomials, which leads to an enormous number of basis functions. Since each realization represents a call to the simulator, the number of realizations one can afford is usually comparably small to the unknown coefficients, i.e., \(N\ll P\). In this setting, the equation set is strongly ill-conditioned and the coefficients evaluated by OLS are inconvincible. In fact among all the basis functions, only a part of them significantly affect the random process and those insignificant or redundant terms can be ignored. Therefore, a sparse PCE surrogate model can be constructed with a reduced number of basis functions. More specifically, the feature selection method is applied in this study to select those significant terms.

2.5 Feature Selection

The feature selection methods originate from statistical learning area and have been widely used for model constructions in many fields (Guyon and Elisseeff 2003; Saeys et al. 2007). The redundant or irrelevant features are discarded to improve prediction accuracy and provide a better understanding for datasets. A number of algorithms have been proposed for feature selection, e.g., Fisher Score, sequential forward selection, regularization model, etc. (Dash and Liu 1997; Guyon and Elisseeff 2003). Some of them have been applied for UQ problems in previous works (Blatman and Sudret 2010, 2011; Elsheikh et al. 2014). In this study, we utilize the LASSO-LAR algorithm, to construct the sparse PCE surrogate model, which is highly efficient for basis reduction (Meng and Li 2017).

The least absolute shrinkage and selection operator (LASSO) shrinks the regression coefficients by imposing a penalty on their size. The LASSO estimate is defined by:
$$\begin{aligned} \mathbf{a}^\mathrm{LASSO}=\arg \min \left\{ {\left\| {\mathbf{h}- {\varvec{\Phi }} \mathbf{a}} \right\| _2 } \right\} ,\hbox { s.t. }\left\| \mathbf{a} \right\| _1 \le t, \end{aligned}$$
(19)
where \(\left\| {\mathbf{h}- {\varvec{\Phi }} \mathbf{a}} \right\| _2 =\sum \nolimits _{i=1}^N {(h_i -\sum \nolimits _{j=1}^P {a_j {\varvec{\Phi }}_{ij} } )^{2}}\) is the loss function between model targets and estimations, and \(\left\| \mathbf{a} \right\| _1 =\sum _{j=1}^P {\left| {a_j } \right| } \le t\) is the L1-norm penalty term. Though the size of nonzero coefficients is defined by the L0-norm \(\left\| \mathbf{a} \right\| _0 =\# \left\{ {k:a_k \ne 0} \right\} \), the optimal solution is difficult to find with it. Since L1-norm is the optimal convex approximation to L0-norm (Tibshirani 1996), by making t sufficiently small, some of the coefficients will be exactly zero and the correlated features will be discarded.
The LASSO coefficients can be solved by several optimization algorithms, e.g., the (block) coordinate descent, proximal method, etc. (Hastie et al. 2009). When a large number of features exist, the least angle regression (LAR) method is especially efficient where the entire solution path of LASSO is provided. The so-called LASSO-modified LAR (LASSO-LAR) algorithm is briefly revisited as follows (Efron et al. 2004):

LASSO-LAR algorithm:

1.

In preparation, each column vector of features in the training dataset is normalized and the target values are centered, i.e., \(\frac{1}{N}\sum \nolimits _{i=1}^N {{\varvec{\Phi }} _{ij} } =0, \frac{1}{N}\sum \nolimits _{i=1}^N {{\varvec{\Phi }}_{ij}^2 } =1, j=1,\ldots ,P\) and \(\frac{1}{N}\sum \nolimits _{i=1}^N {h_i} =0\). Initialize with the residual \(\mathbf{r}=\mathbf{h}\) and set \(a_1 ,\ldots ,a_P=0\)

2.

Find the column vector \(\Phi _j \) most correlated with \(\mathbf{r}\)

3.

Move \(a_j \) from 0 toward its least square coefficient \(\left\langle {\Phi _j ,\mathbf{r}} \right\rangle \), until some other competitor \(\Phi _k \) has as much correlation with the current residual as does \(\Phi _j \)

4.

Move \(\{a_j ,a_k \}\) in the direction defined by their joint least square coefficient of the current residual on \((\Phi _j ,\Phi _k )\), until some other competitor \(\Phi _l \) has as much correlation with the current residual

5.

If a nonzero coefficient hits zero, drop its feature from the active set of features and recompute the current joint least square direction

6.

Continue in this way until \(\min (N-1,P)\) features have been entered

The coefficients derived by LASSO-LAR are used to determine the direction of the solution path, which is not optimal for model construction. When the active feature subset \({\varvec{\Phi }} _{{{\mathcal {A}}}_k } \) is selected after k steps of LASSO-LAR, the hybrid LAR coefficients are generated for higher accuracy, which is defined as:
$$\begin{aligned} \mathbf{a}_{{{\mathcal {A}}}_k } =({\varvec{\Phi }}_{{{\mathcal {A}}}_k }^T {{\varvec{\Phi }}} _{{{\mathcal {A}}}_k })^{-1}{\varvec{\Phi }}_{{{\mathcal {A}}}_k }^T \mathbf{h}. \end{aligned}$$
(20)
The models constructed by each feature subset should be assessed to find the best one. To provide a quantitative criterion and avoid overfitting, the cross-validation error is utilized for assessment. In cross-validation, a testing dataset is set aside from the training dataset iteratively to evaluate the goodness of the constructed model (Hastie et al. 2009). Since the model prediction is a linear transformation of the target:
$$\begin{aligned} \hat{{\mathbf{h}}}={\varvec{\Phi }}_{{\mathcal {A}}_k } \mathbf{a}_{{{\mathcal {A}}}_k } ={\varvec{\Phi }}_{{\mathcal {A}}_k } ({\varvec{\Phi }} _{{{\mathcal {A}}}_k}^T {\varvec{\Phi }} _{{{\mathcal {A}}}_k})^{-1}{\varvec{\Phi }} _{{{\mathcal {A}}}_k }^T \mathbf{h}=\mathbf{Sh}, \end{aligned}$$
(21)
the generalized cross-validation (GCV) error can be derived to approximate the leave-one-out cross-validation error (Golub et al. 1979). The GCV error is defined as:
$$\begin{aligned} \mathrm{Err}_\mathrm{GCV} =\frac{1}{N}\sum _{i=1}^N {\left[ {\frac{h_i -\hat{{h}}_i }{1-\hbox {trace}(\mathbf{S})/N}} \right] } ^{2}, \end{aligned}$$
(22)
where \(\hbox {trace}(\mathbf{S})\) is the sum of diagonal elements of \(\mathbf{S}\) and N is the number of samples. When \(\mathrm{Err}_\mathrm{GCV} \) reaches a certain stopping criterion, the surrogate model constructed by current feature subset is adopted for further uncertainty analysis.

2.6 Sequential Quasi-Optimal Sampling

The surrogate model is also highly influenced by the sampling method to generate the sampling (collocation) points. With higher-quality samples, the predictions can be more accurate. Space-filling samples are commonly used for experimental design, such as Latin hypercube samples and quasi Monte Carlo samples. In the SCM and PCM, specific interpolation points and Gaussian integral points are directly utilized as training dataset (Li and Zhang 2007; Chang and Zhang 2009). Besides, some new methods have been proposed in recent years such as coherence-optimal sampling (Hampton and Doostan 2015), subsampling with QR column pivoting (Seshadri et al. 2016), etc. Since in the sparse PCE, the basis functions are selected from a large number of candidates, we have no idea how many samples are sufficiently enough for coefficients evaluation beforehand. Therefore, we may start from a conservatively small size of samples at first and gradually enrich the experiment design to improve the surrogate model. A sequential sampling method, the quasi-optimal sampling with S-value constraint (Shin and Xiu 2016), is adopted in this study for this purpose.

The quasi-optimal sampling is intrinsically a greedy algorithm. An initial experiment design \({\varvec{\mathcal {X}}}\) is generated at first, normally by a space-filling sampling method. A candidate sample set \({\varvec{\mathcal {X}}}_L \) is also generated where a large number of samples exist and the new samples are selected from it. Then the surrogate model is constructed on current sample set and the PCE matrix is \({\varvec{\Phi }}_{{\mathcal {X}}} \) after feature selection. If the model does not reach the requirement, the experiment design will be enriched. For a certain sample in the candidate set \({\varvec{\xi }}_i \in {\varvec{\mathcal {X}}}_L \), it can be added to the current sample set and the PCE matrix becomes \({\varvec{\Phi }}_{{{\mathcal {X}}}\cup \xi _i } \) of size \(N\times \hbox {card}({\varvec{\Phi }}_{{{\mathcal {X}}}\cup \xi _i } )\). Then the S-value criterion is defined as:
$$\begin{aligned} S_i =\left( {\frac{\sqrt{\det {\varvec{\Phi }}_{{{\mathcal {X}}}\cup \xi _i }^T {\varvec{\Phi }}_{{\mathcal {X}}}\cup \xi _i }}{\prod \nolimits _{j=1}^{\hbox {card}(\Phi _{{\mathcal {X}}}\cup \xi _i )} \left\| {\varvec{\Phi }}_{{{\mathcal {X}}}\cup \xi _i}^{(j)} \right\| }} \right) ^{\frac{1}{\hbox {card}({\varvec{\Phi }} _{{{\mathcal {X}}}\cup \xi _i} )}}. \end{aligned}$$
(23)
where \({\varvec{\Phi }}_{{{\mathcal {X}}}\cup \xi _i }^{(j)} \) is the j-th column of \({\varvec{\Phi }}_{{{\mathcal {X}}}\cup \xi _i}\). The S-value is calculated for each sample in the candidate set. The sample corresponding to the maximum S-value, i.e., \({\varvec{\xi }} _{\max } \), is added to current samples at last. The process is repeated with the updated \({\varvec{\mathcal {X}}}={{\mathcal {X}}}\cup {\varvec{\xi }} _{\max } \) and \({{\mathcal {X}}}_L ={{\mathcal {X}}}_L \backslash {\varvec{\xi }} _{\max } \), until the expected number of new samples are generated. Then the new surrogate model will be constructed and the sample set will be enriched iteratively until satisfactory results are obtained.

In fact the S-value is a criterion representing the differences between the estimations and targets. The computation of S-value requires repetitively calculating the determinant which is an expensive procedure. Such unnecessary cost can be avoided by analytical derivations of the S-value. The readers could refer to Shin and Xiu (2016)) for more details of mathematical proofs and implementations. Comparisons of the quasi-optimal sampling with other sampling methods are also discussed in Fajraoui et al. (2017).

3 Case Studies

3.1 Problem Statement

In this section, we test the performance of the sparse PCE for unconfined groundwater flow in heterogeneous aquifers. Some synthetic numerical experiments are performed and comparisons are made, with consideration of the effects of different spatial variability of hydraulic conductivity, random recharge and the presence of pumping well. The flow domain is a square of size \(L_x =L_y =20\) m, uniformly discretized into \(40\times 40\) square elements. The hydraulic head is prescribed constantly at the boundary \(\varGamma _1 \) (along \(x=0, 0\le y\le 15\) m) and \(\varGamma _2 (\hbox {along } x=20\,\hbox {m}, 5\le y\le 20\,\hbox {m})\) as 7.0 and \(5.0\,\hbox {m}\), respectively. Other borders are defined as no-flow boundaries.

In the following studied cases, the log hydraulic conductivity and recharge are assumed to be second-stationary Gaussian random fields with separable exponential covariance, where the correlation length in each direction is \(\eta _x =\eta _y =5.0\,\hbox {m}\). Each random field can be represented by the truncated K–L expansion with independent variables, which is the source of uncertainty. In each case, we train the surrogate model with a relatively small size of samples initially and gradually enrich the experiment design with the quasi-optimal sampling points. The results are compared with Monte Carlo solutions (with 10,000 simulations), which are regarded as the benchmark. The simulations are accomplished with the existing groundwater flow simulator, MODFLOW. The deterministic solutions from the flow simulator are deemed as reliable.

3.2 Effects of Spatial Variability

In this section, we explore the accuracy and efficiency of the sparse PCE with different spatial variabilities of the random field. For simplicity, a constant recharge over the domain \(R=4\,\hbox {cm/day}\) is given and the random log-conductivity field \(Y=\ln K\) is studied. At first, we set the field of zero mean and unit variance, i.e., \(\left\langle Y \right\rangle =0,\; \sigma _Y^2 {=}1.0\), and the coefficient of variation of the hydraulic conductivity equals to \(Cv_K =\sigma _K /\left\langle K \right\rangle =131\% \). In the MC benchmark, we retain the first 624 terms to represent the field where \(\sum _{i=1}^{624} {\lambda _i } /D\sigma _Y^2 =96\% \) energy is preserved in K–L expansion, which is found to be adequate to reproduce the ensemble statistics of the random field. More energy is unnecessary since the effect of small scale fluctuations in the conductivity field on the flow responses can be neglected (Zhang and Lu 2004). However for constructing the surrogate models, 624 dimensions would lead to enormous computational efforts. Thus for the sparse PCE, we preserve 70% energy and retain 23 terms, with which some unnecessary details are ignored for convenience.

Sixty Latin hypercube samples are generated as the initial experimental design. The samples are enriched with 20 quasi-optimal points each time until the generalized cross-validation error of the current surrogate model reaches a certain stopping criterion. These quasi-optimal points are selected from 10,000 candidate quasi Monte Carlo points with the aid of S-value constraint. The mean and variance of hydraulic head obtained from the sparse PCE trained by 140 samples are shown in Fig. 1. The mean head can be well estimated owing to the particular setting of the field in our examples; hence, we mainly focus on the variance for comparison. In this case, the PCE up to \(3{\mathrm{rd}}\) degree are provided for model construction and there are \(P=\left( {\begin{array}{l} 23+3 \\ 3 \\ \end{array}} \right) =2600\) basis functions in total. With the traditional PCE, 2600 or more samples are required to estimate the basis coefficients. However with the sparse PCE, around 80 basis functions are selected as the major stochastic features and the statistical moments obtained with 140 samples are close to the benchmark.
Fig. 1

Mean (a) and variance (b) of hydraulic head obtained from the sparse PCE (solid lines) for \(\sigma _Y^2 =1.0\), compared with MC benchmark (dash lines). The borders with bold lines denote no-flow boundaries

It is obvious that the random output is strongly influenced by the spatial variability of the random input. In this case, we examine the validity of sparse PCE with respect to different variance of the random hydraulic conductivity field. A smaller variance \(\sigma _Y^2 =0.5\) is given at first, i.e., \(Cv_K =81\% \) with other conditions unchanged. In this case, the estimated variance trained by just 100 samples agrees well with the benchmark as shown in Fig. 2a. The whole variance field is smoother since the random fluctuation is not that obvious with smaller variability. Therefore, the random structure can be easily represented by fewer basis functions (around 60) with limited samples. In contrast, with larger variability, i.e., \(\sigma _Y^2 =2.0\) and \(Cv_K =253\% \), more than 100 basis are adopted to characterize the details of a more complex field. Figure 2b shows the head variance computed from sparse PCE trained by 200 samples. The hydraulic head varies in a larger scale and more samples are required. The comparison of head variances on the transection \(y=10\,\hbox {m}\) for different spatial variability (i.e., \(\sigma _Y^2 )\) is shown in Fig. 3. The results from the sparse PCE agree well with the MC benchmark in all the cases. It indicates that the method is robust for random fields with various degrees of variability. For even larger variability, more terms in the PCE, especially those polynomials with higher degree will be retained and the computational efforts will be increased accordingly (Meng and Li 2017).
Fig. 2

Head variance obtained from the sparse PCE (solid lines) compared with MC (dash lines) for a\(\sigma _Y^2 =0.5\) and b\(\sigma _Y^2 =2.0\)

Fig. 3

Comparisons of head variance on the transection \(y=10\,\hbox {m}\) for different spatial variability (i.e., \(\sigma _Y^2 )\), obtained with sparse PCE (with 100, 140 and 200 simulations respectively) compared with MC benchmark (with 10,000 simulations)

Here we define the estimated error along the transection y:
$$\begin{aligned} \varepsilon _y =\frac{1}{L_x }\int \limits _y {(\sigma _h^2 (x,y)-\hat{{\sigma }}_h^2 (x,y))^{2}\text {d}x} , \end{aligned}$$
(24)
where \(\sigma _h^2 \) and \(\hat{{\sigma }}_h^2 \) are the reference and estimated variance respectively. With a certain sample size, we run MC simulation 100 times and sparse PCE 50 times to study the error distributions for the case of \(\sigma _Y^2 =1.0\). The convergence of error with increasing sample size is shown in Fig. 4, where the transection \(y=10\,\hbox {m}\) is selected for comparison. The estimated error will gradually converge to zero and the sparse PCE converges at a rapid speed which is significantly faster than MC. Larger spatial variability leads to larger error and slower convergence rate due to the complex random input; however, the sparse PCE is still able to capture the random structure at a fast speed. This is because some major stochastic features have been recognized and more information can be extracted from limited samples.
Fig. 4

Comparisons of estimated error convergence with increasing sample size between the sparse PCE and MC simulation, along the transection \(y=10\,\hbox {m}\) for the case of \(\sigma _Y^2 =1.0\). The error bar shows the mean (±) standard deviation of the estimated error

3.3 Effects of Random Recharge

In this case, random recharge is imposed to the unconfined flow domain, which increases the nonlinearity in the probability space. The log-conductivity is Gaussian with moderate variance, i.e., \(\sigma _Y^2 {=}1.0\) and \(Cv_K =131\% \). We set the spatial recharge with \(\left\langle R \right\rangle =4\,\hbox {cm/day}\) and \(\sigma _R^2 =27.49\,\mathrm{cm}^{2}/\mathrm{day}^{2}\) so that the coefficient of variation also equals to \(Cv_R =\sigma _R /\left\langle R \right\rangle =131\% \). With the K–L expansion, the recharge field is also represented by 23 independent random variables. The conductivity and recharge are assumed to be independent to each other; thus, they are governed by two set of variables \({\varvec{\xi }} _K \) and \({\varvec{\xi }} _R \). Each of them has 23 random dimensions and the problem is involved with 46 random dimensions in total.

The estimated variance of hydraulic head obtained from sparse PCE is shown in Fig. 5. Compared with the case of constant recharge, the hydraulic head varies in a larger scale and more uncertainties are involved. More features and samples are needed for better description of the random process. Since the random dimensions are increased, the surrogate model has to be constructed among \(P=\left( {\begin{array}{l} 46+3 \\ 3 \\ \end{array}} \right) =18,424\) PCE basis functions. But with just 260 samples through the sparse PCE, the variance can be obtained fairly accurately as shown in Fig. 5. The sparse PCE is still highly efficient in this high-dimensional case.
Fig. 5

Comparisons of head variance between sparse PCE (solid lines) and MC benchmark (dash lines), for the case of independent random conductivity and recharge

Fig. 6

Coefficients of hydraulic head at location (5, 5 m) for different terms of PCE, for the case of independent random conductivity and recharge

The coefficient distribution of the basis functions is inspected to study how the surrogate model is constructed at the center of the flow domain. Coefficients for the basis functions derived from \({\varvec{\xi }} _K \) and \({\varvec{\xi }} _R \) are shown in Fig. 6 separately. Although polynomials up to \(3{\mathrm{rd}}\) degree are provided, only a small part of the \(3{\mathrm{rd}}\) degree polynomials are given very limited coefficients. Therefore, only the first 2 degree polynomials are shown in Fig. 6. The \(1^{\mathrm{st}}\) degree polynomials are given larger coefficients for both \({\varvec{\xi }} _K\) and \({\varvec{\xi }}_R \). Since these basis functions strongly affect the random input by the K–L expansion, they are selected as the major stochastic features. Some basis functions are derived by \({\varvec{\xi }}_K \) and \({\varvec{\xi }}_R \) interactively and most of these basis functions are given zero coefficients, indicating that the random field is affected by the conductivity and recharge independently. Thus these interactive polynomials are identified as redundant features and discarded in the selection process. Among all 18,424 basis functions, around 180 terms are presented with nonzero coefficients and others are discarded, and the surrogate model can be constructed with merely 260 samples.

3.4 Effects of Pumping Well

We add a pumping well extracting groundwater at the center of the field to test the performance of the sparse PCE. The unsteady state flow is considered. The field is of random conductivity with \(\sigma _Y^2 =1.0\), constant recharge of \(R=4\,\hbox {cm/day}\) and constant pumping rate of \(Q=8\,\hbox {m}^{3}/\mathrm{day}\). The spares PCE results at the \(10{\mathrm{th}}\) day trained by 200 samples are shown in Fig. 7. Both the mean and variance of the hydraulic head are well estimated, compared with the MC solutions. Due to the presence of the pumping well, a rapid drop of hydraulic head appears near the well which leads to dramatic increase of variance. The transection through the pumping well, i.e., \(y=10\,\hbox {m}\), is selected for better comparison, as shown in Fig. 8. The drawdown of head and the increase of variance are well described with the sparse PCE, while slight deviation of variance is observed at the well.
Fig. 7

Mean head (a) and denary logarithm variance (b) obtained from sparse PCE (solid lines), compared with MC benchmark (dash lines)

Fig. 8

Mean head (a) and variance (b) on the transection \(y=10\,\hbox {m}\) through the pumping well, obtained with the sparse PCE and MC

By randomly running the sparse PCE surrogate model 100,000 times, PDF of the hydraulic head at the pumping well is obtained. The PDFs at the pumping well and two locations nearby are shown in Fig. 9. The estimations agree well with the MC benchmark obtained from 10,000 simulations. However, the PDF at the well has slight deviations. As indicated in Fig. 9b, the hydraulic head at the well varies in a larger range, owing to the drawdown funnel caused by pumping. This greatly increases the nonlinearity of hydraulic head at the pumping well and makes it difficult to quantify uncertainties with polynomials. Fortunately for single phase flow, such nonlinearity is not that obvious and the random structure can be approximately described by the sparse PCE surrogate model.
Fig. 9

PDFs of hydraulic head at different locations: a (4, 5 m), b (5, 5 m) at the pumping well, and c (6, 5 m)

4 Conclusions

In this study, we construct a surrogate sparse PCE model to efficiently quantify uncertainties for unconfined flow. The random process is influenced by several factors, such as the spatial variability of hydraulic conductivity, random recharge/discharge, pumping well, etc. Higher dimensionality is required to represent these uncertainty sources. The computational efforts are alleviated through feature selection, with which the major stochastic features are selected and satisfactory results can be obtained with limited samples. The quasi-optimal samples are added to the experiment design sequentially so that the surrogate model can be gradually improved. The effects of various random factors are studied, and the results are compared with the MC benchmarks. This study leads to the following conclusions:
  1. (1)

    The non-intrusiveness of the sparse PCE method allows one to employ any existing deterministic flow simulators.

     
  2. (2)

    The sparse PCE method is feasible for quantifying uncertainty associated with unconfined flow in random porous media. With relatively small computational efforts, the random process can be accurately and efficiently represented. A much faster convergence rate is observed compared to MC simulation. The number of flow simulations is significantly reduced, which is the most time-consuming step in uncertainty quantification. In comparison, the extra computational efforts caused by the feature selection process and sequential sample generation can be ignored. The method is highly efficient for high-dimensional problems where redundant and irrelevant features are discarded.

     
  3. (3)

    The robustness of the sparse PCE method has been tested with respect to different spatial variability, independent random recharge, and the existence of pumping well. In these numerical cases, different distributions of the statistical moments and PDFs of hydraulic head are well produced. For those complex problems, more basis functions and more samples are required to construct a satisfactory surrogate model. With the sequential quasi-optimal sampling method, one can train the model with a small size of samples at first and gradually enrich the experiment design to improve the results. The method is self-adaptive and the best use of the computational resources can be made.

     

Notes

Acknowledgements

This work is partially funded by the National Science and Technology Major Project of China ( Grant nos. 2017ZX05039-005 and 2016ZX05014-004-006).

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Authors and Affiliations

  1. 1.Department of Energy and Resources Engineering, College of EngineeringPeking UniversityBeijingChina

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