# Adaptive surrogate modeling for response surface approximations with application to bayesian inference

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## Abstract

Parameter estimation for complex models using Bayesian inference is usually a very costly process as it requires a large number of solves of the forward problem. We show here how the construction of adaptive surrogate models using a posteriori error estimates for quantities of interest can significantly reduce the computational cost in problems of statistical inference. As surrogate models provide only approximations of the true solutions of the forward problem, it is nevertheless necessary to control these errors in order to construct an accurate reduced model with respect to the observables utilized in the identification of the model parameters. Effectiveness of the proposed approach is demonstrated on a numerical example dealing with the Spalart–Allmaras model for the simulation of turbulent channel flows. In particular, we illustrate how Bayesian model selection using the adapted surrogate model in place of solving the coupled nonlinear equations leads to the same quality of results while requiring fewer nonlinear PDE solves.

### Keywords

Goal-oriented error estimation Adjoint problem Turbulence modeling Spalart–Allmaras model Parameter identification## Background

A general issue in parameter estimation using Bayesian inference is that one has to sample the forward model a very large number of times in order to obtain accurate posterior distributions of the parameters. In the case of complex models involving many parameters, the process often results in computational costs that far exceed computer resources currently available. Kennedy and O’Hagan [1] suggested to consider emulators of response surfaces in order to reduce the number of forward simulations and showed that it could lead to considerable computational savings. A variety of reduced models to estimate the likelihood function have since then been proposed in the literature [2, 3, 4, 5]. However, reduced models only provide approximations of the true solution and the accuracy of the response needs to be verified in order to obtain meaningful results.

We propose in this work to develop a goal-oriented error estimation procedure to adaptively construct reduced models, also referred to as surrogate models, of the Reynolds averaged Navier–Stokes (RANS) models for the simulation of turbulent flows. Goal-oriented error estimation was initially designed as an adjoint-based method to estimate discretization errors in finite element solutions of boundary-value problems with respect to quantities of interest. The method was later extended to the estimation and control of modeling error when a fine scale model is replaced by a coarse-scale model [6, 7]. The goal-oriented error estimation framework will be used here not only to assess the accuracy of the surrogate models but also to guide the adaptive process for improving the representation of the true response provided by the reduced model.

We illustrate the methodology on examples dealing with the simulation of turbulent channel flows. Turbulence will be modeled here by the Reynolds averaged Navier–Stokes (RANS) equations, supplemented by the Spalart–Allmaras model for the description of the Reynolds stress. This closure model involves several parameters that need to be calibrated in order to be useful. Values of the parameters have been proposed in [8, 9, 10]. As the turbulence community is well aware that these parameters may include some level of uncertainty, Bayesian inference has recently been used to quantify uncertainties in simulations of turbulence (see e.g. [10, 11]). In the present study, our objective will be to reproduce some of the numerical examples described in [10] in order to demonstrate that one can confidently use reduced models rather than the full models to estimate the parameters of the Spalart–Allmaras model.

The content of the paper is as follows: we describe in "Methods" the model problem, namely the Reynolds averaged Navier–Stokes equations and the Spalart–Allmaras model for the Reynolds stress, and derive the weak formulation of the deterministic problem. We also recall, briefly, some concepts of probability theory and present the parameterized reduced model. We then review the basic principles for goal-oriented error estimation and adaptivity methodology for the construction of the reduced model as presented in [12, 13]. Lastly, we describe the concepts of Bayesian inference and model selection. In "Results and discussion", we provide numerical results of a Bayesian model selection study, where the full model is replaced with the adapted surrogate model, before providing some concluding remarks.

## Methods

### Model problem

*T*) and \(\varvec{u}'\) is the fluctuation about the mean. Substituting this decomposition into the Navier–Stokes equations and taking the average over (0,

*T*), we obtain the so-called RANS equations,

*P*, \(\rho\), and \(\nu\), denote the mean pressure, the density, and the kinematic viscosity, respectively. In order to be able to solve the equations, one usually considers a closure model for the Reynolds stress tensor \(r_{ij}:=\overline{ u'_i u'_j }\) (scaled here by the constant density \(\rho\)) based on the eddy viscosity assumption. That is, the Reynolds stress is expressed as the viscosity term,

Many closure models have been proposed based on the eddy viscosity assumption, see e.g. [8, 9, 14, 17, 18, 19], some of which have been examined in the study by Oliver and Moser [10]. Here we will focus on one of the most commonly used models, the eddy viscosity transport model of Spalart and Allmaras [9]. The form considered in this work, as well as in [10], has been modified to avoid negative values of turbulent production and to ignore the transition to turbulence from a laminar initial condition; see [8, 20] for more details on the modified form of the model.

*d*is the distance to the nearest wall and parameter \(c_{w1}\) is defined as:

*S*is the magnitude of the vorticity, and

Standard parameter values for the Spalart–Allmaras turbulence model [10]

Parameter | Nominal value |
---|---|

\(c_{b1}\) | 0.1355 |

\(\sigma _\text {SA}\) | 2/3 |

\(c_{b2}\) | 0.622 |

\(\kappa\) | 0.41 |

\(c_{w2}\) | 0.3 |

\(c_{w3}\) | 2 |

\(c_{v1}\) | 7.1 |

\(c_{v2}\) | 0.7 |

\(c_{v3}\) | 0.9 |

We suppose that our primary goal is the prediction of the centerline velocity in a fully-developed incompressible channel flow at \(Re_\tau =5000\). The turbulence is assumed non-homogeneous in the *y*-direction (wall normal direction) but homogeneous in the *x*-direction, reducing the complexity of the RANS equations significantly. Except for the mean pressure gradient in the *x*-direction, derivatives of statistical variables are all assumed to vanish with respect to *x* and *t*; thus, \(U_2 = 0\), and \(U_1 = U(y)\) is only a function of *y*, and the gradient of *P* can be shown to be constant [14]. To simplify the presentation, we set \(1/\rho \ \partial _x P = 1\) and control the dynamics of the flow purely through the Reynolds number.

*H*represents the half height of the channel. Combining the simplified form of the RANS equations with the Spalart–Allmaras turbulence model, the strong form of the equations now reads:

*U*; Eq. (11) is the transport equation for the Spalart–Allmaras working variable \(\tilde{\nu }\). The equations are supplemented with the boundary conditions:

*D*. Let \(\mathcal V^{h} \subset \mathcal V\) be the finite element subspace consisting of piecewise linear functions, on a suitable partition of

*D*with maximal element diameter

*h*. The finite element approximation of (13) is given by,

Problem (18) represents the model problem we shall consider throughout this work. However, since the RANS turbulence model parameters are uncertain, these equations can be parameterized by random variables; we discuss the characterizations of the uncertain parameters in the following section.

### Uncertainty characterization and reduced model

As previously discussed, the parameters of the Spalart–Allmaras turbulence model (3) and (5) are usually assumed constant with the values provided in Table 1. In this work, we suppose that a subset of these parameters are in fact unknown, or random. Therefore, the boundary-value problem can be viewed as parameterized by parameters of the Spalart–Allmaras turbulence model.

We briefly review some relevant concepts of probability theory and the use of polynomial expansions, commonly referred to as generalized polynomial chaos, to represent the effects of uncertainty in the model response.

Let \(\{\Theta , \Sigma , P\}\) be a probability space, where \(\Theta\) is the sample space of random events, \(\Sigma\) is a \(\sigma\)-algebra, and *P* is the probability measure on \(\Sigma\), meaning \(P(\Theta )=1\). A random variable on the probability space is defined as a *P*-measurable function of \(\Theta\). Let \(\Xi \subset \mathbb R^n\). We use the notation \(\varvec{\xi }:\Theta \rightarrow \Xi\) to denote a random variable and denote by \(p_\xi\) the associated probability density function.

A number of methods for computing the coefficients \(U^{h}_i\) of the expansion have been developed in the uncertainty quantification literature and generally fall into two categories: intrusive and non-intrusive. Non-intrusive approaches attempt to estimate the coefficients of a generalized polynomial chaos expansion by solving the deterministic problem at a set of realizations of \(\varvec{\xi }\). As a result, existing simulation codes can usually be used directly. In contrast, intrusive approaches, such as those based on Galerkin methods, solve a system of equations for the entire set of expansion coefficients; typically this requires the use of specially designed solvers. We shall focus on non-intrusive approaches here since they allow for the existing turbulence simulation codes to be used with minimal modification. Moreover, the set of independent parameter values can usually be run in parallel, making non-intrusive approaches more efficient.

As discussed previously, non-intrusive approaches aim to compute the coefficients of the expansion based on independent realizations of the deterministic solution \(U^h(y,\varvec{\xi }).\) One can use sampling-based methods, such as Monte Carlo or Latin hypercube sampling, to compute projections onto the polynomial bases \(\Phi\). In high-dimensional parameter spaces sampling may be preferred, since convergence is based on the number of samples and not the dimension of the space. In our case, the dimension will remain relatively low so that we can rely on direct numerical integration using quadrature techniques. Even in a high number of dimensions authors have proposed the use of sparse representations to make the process more efficient [27, 28, 29, 30, 31, 32, 33].

*h*, the level of discretization of the physical domain, as well as

*N*, the maximum order of polynomials used in the expansion.

In an effort to produce more accurate surrogate models goal-oriented error estimation techniques, common to the finite element community, have recently been extended to address problems with uncertainty [27, 35, 36]. The following section outlines the adaptive approach developed in [12, 13].

### Goal-oriented adaptive surrogate modeling

In this section, we provide details of the error estimation and adaptive procedure for boundary-value problems parameterized by uncertainty that was developed in [12, 13]. We review the extension of goal-oriented error estimation to the case of uncertainty as proposed in previous works [12, 13, 34, 35, 36, 37, 38]. In addition we suggest an adaptive procedure based on the contribution of higher-order expansion terms to the error in the quantity of interest.

#### Goal-oriented error estimation

Since its introduction in the 1990s [39, 40, 41, 42], goal-oriented error estimation has grown in popularity in the finite element community. Here we only provide a brief outline of the approach and refer the interested reader to more extensive descriptions and reviews of the methodology [42, 43, 44, 45].

Equation (20) is in fact a deterministic problem and the standard goal-oriented error estimation framework can be applied for a fixed \(\varvec{\xi }\); the effect of variability in \(\varvec{\xi }\) can then be handled using the expansion techniques of the previous section as we will show below.

*U*over the channel cross-section,

*U*depends on the value of \(\varvec{\xi }\) and thus so does \(\mathcal Q\). Since the velocity profile is expected to reach the maximum at the center of the channel, this quantity of interest is expected to be more sensitive to the centerline velocity.

*D*as \(V^h\). Thus, a computable error estimate for the error in the quantity of interest at a specified value of \(\varvec{\xi }\) is provided by

**Theorem 2.1**

*Assume*\(B_{\varvec{\xi }}\)

*to be continuously differentiable, for any*\(\varvec{\xi }\in \Xi\)

*, in a subset of*\(\mathcal V\)

*that contains*\((U,\tilde{\nu })\), \((U^h,\tilde{\nu }^h)\), \((U^{h,N},\tilde{\nu }^{h,N})\).

*Let*\((\hat{z}_U^{+,N},\hat{z}_{\tilde{\nu }}^{+,N})\)

*be an approximation of the adjoint solution according to*(31).

*The error estimate*(32)

*for*(30)

*satisfies the bound*,

The extension of the goal-oriented error estimation framework to boundary-value problems with uncertain data simply means that uncertainty can be considered in complex physical problems using a reduced-order surrogate model since its accuracy can be assessed in terms of the quantity of interest. Even more valuable, much as in the case of deterministic goal-oriented error estimation, estimates can be used to drive adaptivity to improve the approximations further, as we discuss in the next section.

#### Adapting the surrogate model

One major disadvantage of using quadrature-based sampling strategies for surrogate construction is that the number of points required for a fixed expansion order grows exponentially with parameter dimension; this is commonly referred to as the *curse of dimensionality*.

A number of different approaches have been proposed to minimize the effect of the curse of dimensionality. For example, one may choose to alter the quadrature formula used in calculating the expansion coefficients; in the case a tensor product quadrature formula is employed, authors have suggested sparse quadrature grids that can reduce the number of evaluations necessary for the same level of accuracy [3, 27, 31, 32, 33, 51]. Alternatively, or in a combined manner, higher-order information can be used to identify the more influential parameters. Instead of increasing the expansion order uniformly (isotropic refinement), one may then choose to improve the surrogate model by adding basis functions that only correspond to the most influential parameters (anisotropic refinement).

While our proposed error estimates can be used with a sparse quadrature formula, we will restrict our discussion to full tensor product quadrature for simplicity. Also the advantage of using an anisotropic refinement strategy is more easily observed in this setting. In contrast to existing techniques for using higher-order information to drive anisotropic surrogate refinement, which are often based on heuristic measures, our approach identifies the components associated with the error in the quantity of interest.

*N*, the residual is not restricted to be of order

*N*. The higher-order coefficients in (34) capture additional information about the error in the quantity of interest. The relative magnitudes of the coefficients in \(\mathcal {E}\) provide a weighting of the most important parameter directions or, more specifically, which basis functions should be added to the solution expansions. To precisely define the refinement strategy, we need a generalization of the set of multi-indices; let

### Bayesian inference and model selection

To this point we have restricted the discussion to the construction of a reduced model. Often the full model is yet still in need of validation against experimental observations. A reduced model can certainly make validation studies more computationally feasible, the more pressing question is whether using the surrogate model in place of the full model will lead to the same conclusions on the model’s validity.

Bayesian inference requires a large number of model simulations; replacing the full simulation with an accurate surrogate model may lead to considerable computational savings. We first provide an overview of the general Bayesian methodology and then discuss how it can be used in a model selection procedure.

Bayes’ theorem is a fundamental result of probability. Relatively recently, it has been adapted toward parameter identification for complex mathematical models. The advantage of Bayesian inference for model calibration is that it provides for a distribution of probable parameter values instead of the one best fitting parameter value obtained from traditional optimization procedures. Bayesian parameter identification can be interpreted as an update of the degree of belief in the parameters.

We rely here on a Bayesian model selection procedure that compares a set of models to decide which uncertainty description most likely matches the data. We will only consider the methodology to select the most probable model among a class of uncertainty models, but one can just as well employ the technique to decide between models governing the physical response of a system, such as different RANS closure models, or a combination of physical and uncertainty models [10].

*C*is a normalization constant similar to that in (37), \(p(\varvec{q}|M_i,\mathcal M)\) acts like a likelihood for model \(M_i\), and \(p(M_i|\mathcal M)\) is the prior model plausibility. The likelihood of the model is given by the evidence, which is simply the normalization constant discussed in the previous section, conditioned on model \(M_i\),

We have explicitly left \(\mathcal M\) in the conditional distributions above since the whole process is strictly conditional on the original set of models. In other words, only the models included in the set \(\mathcal M\) are evaluated, thus any conclusions or observations resulting from the quantitative analysis are limited by the quality of the models under consideration; the process can not be used to identify a *truth* model unless it is present in \(\mathcal M\).

## Results and discussion

In this section we present two sets of results. First, we illustrate the adaptive algorithm for the construction of a reduced model for the mean velocity of a turbulent flow in a channel. Secondly, we demonstrate that, as a result of the error estimation and adaptation, the surrogate model can be used in place of the full model for a Bayesian model selection procedure, providing a more computationally efficient method for performing uncertainty quantification in complex simulations.

### Adapted surrogate model for turbulence model problem

We apply here the adaptive surrogate refinement procedure to the model problem of RANS turbulence modeling for incompressible flows. We will consider the physical discretization as fixed and focus on the adaptive construction of a surrogate model for the Spalart–Allmaras turbulence model with six uncertain parameters: \(\kappa\), \(c_{b1}\), \(\sigma _\text {SA}\), \(c_{b2}\), \(c_{v1}\), \(c_{w2}\). To evaluate the surrogate model, we will use both the error estimates reviewed in "Goal-oriented error estimation" as well as simulations of the quantity of interest provided by the full model to further establish the accuracy of our error estimates.

The prior distribution for all uncertain parameters will be taken to be uniform. The exact descriptions will be based on the nominal values presented in Table 1 with a range from 50 to 150 corresponding value; for example \(\kappa \sim \mathcal U(0.205,0.615)\).

*p*-refinement, the anisotropic refinement of the surrogate model leads to significant improvement of the error for an equal number of forward model evaluations, roughly two orders of magnitude reduction. The progression of the expansion order is shown in Table 2. We observe that the initial refinements are associated with the \(\kappa\) and \(c_{v1}\) parameters, demonstrating that their values have the greatest influence on the quantity of interest. Following initial refinement of \(\kappa\) and \(c_{v1}\), we also see a continued increase in the expansion order for \(\kappa\), which we would expect to see since it has a significant impact on the flow velocity away from the wall where the velocity is higher and thus contributes more significantly to the quantity of interest. Refinements are suggested for all model parameters, though \(c_{b2}\) is only modeled linearly, suggesting that the gradient of the working variable \(\tilde{\nu }\) does not have a notable impact on the average velocity.

Expansion orders for parameters in adaptive surrogate of Spalart–Allmaras turbulence model

Iteration | \(\kappa\) | \(c_{b1}\) | \(\sigma _\text {SA}\) | \(c_{b2}\) | \(c_{v1}\) | \(c_{w2}\) |
---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 1 | 0 | 0 | 0 | 0 | 0 |

3 | 1 | 0 | 0 | 0 | 1 | 0 |

4 | 2 | 0 | 0 | 0 | 1 | 0 |

5 | 3 | 0 | 0 | 0 | 1 | 0 |

6 | 3 | 0 | 1 | 0 | 1 | 0 |

7 | 3 | 1 | 1 | 0 | 1 | 0 |

8 | 3 | 1 | 1 | 0 | 1 | 1 |

9 | 4 | 1 | 1 | 0 | 1 | 1 |

10 | 4 | 2 | 1 | 0 | 1 | 1 |

11 | 4 | 2 | 2 | 0 | 1 | 1 |

12 | 4 | 2 | 2 | 0 | 2 | 1 |

13 | 5 | 2 | 2 | 0 | 2 | 1 |

14 | 5 | 2 | 2 | 1 | 2 | 1 |

15 | 5 | 2 | 3 | 1 | 2 | 1 |

16 | 5 | 3 | 3 | 1 | 2 | 1 |

17 | 5 | 3 | 3 | 1 | 2 | 2 |

18 | 6 | 3 | 3 | 1 | 2 | 2 |

### Bayesian model selection

As a basis for comparison we use the work of Oliver and Moser [10], where Bayesian methods were used to evaluate the validity of a number of turbulence models and discrepancies. To simplify the presentation here, we restrict our investigation to the Spalart–Allmaras model for eddy viscosity discussed previously and consider four different uncertainty models: independent homogeneous, correlated homogeneous, correlated inhomogeneous, and an additive Reynolds stress error model.

Specifically, we use the same calibration data as Oliver and Moser [10], which was obtained from direct numerical simulations by Jiménez et al. [55, 56]. Mean velocity measurements were taken at \(Re_\tau =944\) and \(Re_\tau =2003\). The uncertainty in the observations from the direct simulation is the result of calculating the sample mean rather than the true mean. The authors of [57] provide an estimate of the variance in the error, however the covariance between data points in the profile is not provided. To minimize the impact of the correlation of sampling errors between measurement points, Oliver and Moser [10] downsampled the data and considered points that are farther apart; we will do the same and assume the sampling errors to be independent. Since the simulation of the channel flow is dependent on the Reynolds number \(Re_\tau\), we will construct two surrogate models, one for each flow scenario represented in the calibration data. For all error models we will use the same surrogate construction of the approximate forward models.

As we did in the examination of the forward model, we consider six uncertain parameters. This set of parameters is naturally augmented with the calibration parameters for the uncertainty models considered. For both Reynolds numbers we will use the final expansion order in Table 2, \(\mathcal N=(6,3,3,1,2,2)\), which yields error estimates \(\eta _{944}=1.388788\times 10^{-2}\) and \(\eta _{2003}=1.878746\times 10^{-2}\).

#### Independent homogeneous covariance

#### Correlated homogeneous covariance

*l*are additional calibration parameters. While undoubtedly more reasonable than independent errors, a homogeneous correlation length still seems improbable; generally the accuracy of the turbulence models near the wall differs considerably from that in the region far from the wall.

#### Correlated inhomogeneous covariance

*l*(

*z*) is given by,

### Reynolds stress uncertainty model

Finally, we introduce an uncertainty model based on the Reynolds stress. While more complex, and thus more difficult to implement in practice, a Reynolds stress uncertainty model is appealing since it directly targets the source of error, the approximation of the Reynolds stress tensor.

#### Numerical results

Some of the above models are obviously deficient while others may be overly complex for the present analysis; all models are now evaluated based on their agreement with the calibration data. With the model set \(\mathcal M\) comprised of the four models proposed in this section, we are prepared to move forward with the application of the Bayesian model comparison procedure with the adaptive surrogate model for the turbulent channel flow problem.

The same uniform distributions used to define the parameter ranges in "Adapted surrogate model for turbulence model problem" are carried over here as the prior distributions for each parameter. The results of this section were obtained using the QUESO library of algorithms for statistical inverse problems [59]; more specifically, we used the multi-level sampling functionality.

Evidences computed for the four uncertainty models (\(\log (E)\) is reported in the table)

Surrogate | Full model | |
---|---|---|

Independent homogeneous | −1.457 | 8.862 |

Correlated homogeneous | 1.963 | 8.045 |

Correlated inhomogeneous | 164.9 | 164.0 |

Reynolds stress | 164.8 | 169.0 |

Number of nonlinear PDE solves computed in the MCMC process for each uncertainty model

Independent homogeneous | 7776 |

Correlated homogeneous | 8453 |

Correlated inhomogeneous | 8084 |

Reynolds stress | 7595 |

Total | 31,908 |

Our results suggest that for uncertainty quantification studies of this nature, requiring many simulations of a relatively complex forward model, the reduced model described here leads to the same conclusion as using the full model to perform the same analysis. In fact, some work has been done to show that, in limited cases with mostly Gaussian assumptions, the error in the surrogate model can be used to prove a bound on the error in the posterior distributions obtained through Bayesian inference [2, 3, 4, 5]. However, further effort is needed to extend these results to more general cases.

## Conclusion

We have examined the application of goal-oriented error estimation to the adaptivity of surrogate models for boundary-value problems with uncertainty. In contrast to existing anisotropic refinement strategies, a new refinement algorithm was proposed that uses higher-order information from the goal-oriented error estimate to identify the most influential parameters and adapt the surrogate model accordingly.

Based on our approach, an accurate surrogate model need to be constructed for the Spalart–Allmaras turbulence model and the solution of the RANS equations in a fully-developed channel. The reduced model was then used in a Bayesian model calibration study in place of the full simulation. Posterior distributions for the parameters showed excellent agreement with those obtained using the original forward model. The results demonstrate that the newly developed methodology can be a valuable resource to computational scientists in assessing complex physical systems using Bayesian techniques where a large number of model simulations are required. In our case, the quantity of interest and calibration observables were both defined in terms of velocities; an interesting question is how to alter the approach taken here if the observable measurements are rather different from the quantity of interest for which we would like to use the model. One would thus be required to construct surrogate models with respect to both the quantity of interest and the observable calibration data. This will be the subject of future work.

## Notes

### Authors' contributions

CB carried out the derivation of the error estimator and adaptive method for the construction of surrogate models, the implementation of the algorithms and numerical studies, and the drafting of the manuscript. SP participated in the development of the theoretical results on the error estimator and adaptive method, in the analysis of the numerical results, and contributed to the drafting of the manuscript. Both authors read and approved the final manuscript.

### Acknowledgements

SP is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He is also a participant of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. CB acknowledges the support by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615]. The authors are also grateful to Todd Oliver from the Institute for Computational Engineering and Sciences at The University of Texas at Austin for useful discussions on the Spalart–Allmaras model and the calibration of the model parameters using Bayesian inference.

### Compliance with ethical guidelines

**Competing interests** The authors declare that they have no competing interests.

### References

- 1.Kennedy MC, O’Hagan A. Bayesian calibration of computer models. J Royal Statist Soc Series B. 2001;63(3):425–64.MATHMathSciNetCrossRefGoogle Scholar
- 2.Li J, Marzouk Y. Adaptive construction of surrogates for the Bayesian solution of inverse problems. SIAM J Sci Comput. 2014;36:1164–86.MathSciNetGoogle Scholar
- 3.Ma X, Zabaras N. An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method. Inverse Probl. 2009;25(3):035013.MathSciNetCrossRefGoogle Scholar
- 4.Marzouk Y, Najm H, Rahn L. Stochastic spectral methods for efficient Bayesian solution of inverse problems. J Comput Phys. 2007;224:560–86.MATHMathSciNetCrossRefGoogle Scholar
- 5.Marzouk Y, Xiu D. A stochastic collocation approach to bayesian inference in inverse problems. Commun Comput Phys. 2009;6(4):826–47.MathSciNetCrossRefGoogle Scholar
- 6.Oden JT, Prudhomme S. Estimation of modeling error in computational mechanics. J Comput Phys. 2002;182:496–515.MATHMathSciNetCrossRefGoogle Scholar
- 7.Oden JT, Prudhomme S, Romkes A, Bauman PT. Multiscale modeling of physical phenomena: adaptive control of models. SIAM J Sci Comput. 2006;28(6):2359.MATHMathSciNetCrossRefGoogle Scholar
- 8.Allmaras SR, Johnson FT, Spalart PR. Modifications and clarifications for the implementation of the Spalart–Allmaras turbulence model. In: Seventh International Conference on Computational Fluid Dynamics (ICCDF7), Big Island, Hawaii, USA, 9–13 July 2012.Google Scholar
- 9.Spalart PR, Allmaras SR. A one-equation turbulence model for aerodynamic flows. Recherche Aérospatiale. 1994;1:5–21.Google Scholar
- 10.Oliver TA, Moser RD. Bayesian uncertainty quantification applied to RANS turbulence models. J Phys Conf Ser. 2011;318(4):042032.CrossRefGoogle Scholar
- 11.Cheung SH, Oliver TA, Prudencio EE, Prudhomme S, Moser RD. Bayesian uncertainty analysis with applications to turbulence modeling. Reliab Eng Syst Saf. 2011;96(9):1137–49.CrossRefGoogle Scholar
- 12.Bryant CM, Prudhomme S, Wildey T. A posteriori error control for partial differential equations with random data. ICES Report 13-08, 2013.Google Scholar
- 13.Bryant CM, Prudhomme S, Wildey T. Error decomposition and adaptivity for response surface approximations from PDEs with parametric uncertainty. SIAM/ASA J. Uncert Quant. 2015.
**(accepted)**.Google Scholar - 14.Durbin PA, Petterson Reif BA. Statistical theory and modeling for turbulent flows. New York: Wiley; 2001.MATHGoogle Scholar
- 15.Moser RD. Turbulence (lecture notes). 2013.Google Scholar
- 16.Pope SB. Turbulent flows. New York: Cambridge University Press; 2000.MATHCrossRefGoogle Scholar
- 17.Baldwin B, Lomax H. Thin-layer approximation and algebraic model for separated turbulent flows. AIAA Paper 257, 1978.Google Scholar
- 18.Chien KY. Predictions of channel and boundary-layer flows with a low-Reynolds-number two-equation model of turbulence. AIAA J. 1982;20(1):33–8.MATHMathSciNetCrossRefGoogle Scholar
- 19.Durbin PA. Separated flow computations with the \(k-\epsilon -v^{2}\) model. AIAA J. 1995;33:659–64.CrossRefGoogle Scholar
- 20.Oliver TA, Darmofal DL. Impact of turbulence model irregularity on high-order discretizations. AIAA Paper 953, 2009.Google Scholar
- 21.Ghanem R, Red-Horse J. Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach. Phys D Nonlinear Phenom. 1999;133:137–44.MATHMathSciNetCrossRefGoogle Scholar
- 22.Ghanem R, Spanos P. Stochastic finite elements: a spectral approach. New York: Springer; 2002.Google Scholar
- 23.Xiu D, Karniadakis G. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput. 2002;24:619–44.MATHMathSciNetCrossRefGoogle Scholar
- 24.Wan X, Karniadakis GE. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J Comput Phys. 2005;209(2):617–42.MATHMathSciNetCrossRefGoogle Scholar
- 25.Wan X, Karniadakis G. Solving elliptic problems with non-Gaussian spatially-dependent random coefficients. Comput Methods Appl Mech Eng. 2009;198(21–26):1985–95.MATHMathSciNetCrossRefGoogle Scholar
- 26.Wan X, Karniadakis G. Error control in multi-element generalized polynomial chaos method for elliptic problems with random coefficients. Commun Computat Phys. 2009;5(2–4):793–820.MathSciNetGoogle Scholar
- 27.Constantine PG, Eldred MS, Phipps ET. Sparse pseudospectral approximation method. Comput Methods Appl Mech Eng. 2012;229–232:1–12.MathSciNetCrossRefGoogle Scholar
- 28.Babuška I, Nobile F, Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal. 2007;45(3):1005–34.MATHMathSciNetCrossRefGoogle Scholar
- 29.Ganis B, Klie H, Wheeler MF, Wildey T, Yotov I, Zhang D. Stochastic collocation and mixed finite elements for flow in porous media. Comput Methods Appl Mech Eng. 2008;197(43–44):3547–59.MATHMathSciNetCrossRefGoogle Scholar
- 30.Gerstner T, Griebel M. Dimension-adaptive tensor-product quadrature. Computing. 2003;71(1):65–87.MATHMathSciNetCrossRefGoogle Scholar
- 31.Nobile F, Tempone R, Webster CG. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal. 2008;46(5):2411–42.MATHMathSciNetCrossRefGoogle Scholar
- 32.Nobile F, Tempone R, Webster CG. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal. 2008;46(5):2309–45.MATHMathSciNetCrossRefGoogle Scholar
- 33.Todor RA, Schwab C. Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J Numer Anal. 2006;27(2):232–61.MathSciNetCrossRefGoogle Scholar
- 34.Le Maître O, Knio O. Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. New York: Springer; 2010.CrossRefGoogle Scholar
- 35.Butler T, Dawson C, Wildey T. A posteriori error analysis of stochastic spectral methods. SIAM J Sci Comput. 2011;33:1267–91.MATHMathSciNetCrossRefGoogle Scholar
- 36.Butler T, Constantine P, Wildey T. A posteriori error analysis of parameterized linear systems using spectral methods. SIAM J Matrix Anal Appl. 2012;33:195–209.MATHMathSciNetCrossRefGoogle Scholar
- 37.Almeida RC, Oden JT. Solution verification, goal-oriented adaptive methods for stochastic advection-diffusion problems. Comput Methods Appl Mech Eng. 2010;199(37–40):2472–86.MATHMathSciNetCrossRefGoogle Scholar
- 38.Mathelin L, Le Maître O. Dual-based a posteriori error estimate for stochastic finite element methods. Commun Appl Math Comput Sci. 2007;2(1):83–115.MATHMathSciNetCrossRefGoogle Scholar
- 39.Eriksson K, Estep D, Hansbo P, Johnson C. Introduction to adaptive methods for differential equations. Acta Numer. 1995;4:105–158.MathSciNetCrossRefGoogle Scholar
- 40.Eriksson K, Estep D, Hansbo P, Johnson C. Computational differential equations. New York: Cambridge University Press; 1996.MATHGoogle Scholar
- 41.Becker R, Rannacher R. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 2001;10:1–102.MATHMathSciNetCrossRefGoogle Scholar
- 42.Oden JT, Prudhomme S. Goal-oriented error estimation and adaptivity for the finite element method. Comput Math Appl. 2001;41(5):735–56.MATHMathSciNetCrossRefGoogle Scholar
- 43.Ainsworth M, Oden JT. A posteriori error estimation in finite element analysis. New York: Wiley; 2000.MATHCrossRefGoogle Scholar
- 44.Babuška I, Strouboulis T. The finite element method and its reliability. Oxford: Oxford University Press; 2001.Google Scholar
- 45.Verfürth R. A posteriori error estimation techniques for finite element methods. Numerical mathematics and scientific computation. New York: Oxford University Press; 2013.CrossRefGoogle Scholar
- 46.Bryant CM. On goal-oriented error estimation and adaptivity for nonlinear systems with uncertain data and application to flow problems. PhD thesis, The University of Texas at Austin, Austin, TX, 2014.Google Scholar
- 47.van der Zee KG. Goal-adaptive discretization of fluid-structure interaction. PhD thesis, Delft University of Technology, The Netherlands, 2009.Google Scholar
- 48.Prudhomme S, Oden JT. Computable error estimators and adaptive techniques for fluid flow problems. In: Barth T, Deconinck H, editors. Estimation and adaptive discretization methods in computational fluid dynamics, lecture notes in computational science and engineering, vol. 25. Heidelberg: Springer; 2003. p. 207–68.Google Scholar
- 49.Bangerth W, Rannacher R. Adaptive finite element methods for differential equations. Boston: Birkhauser; 2003.MATHCrossRefGoogle Scholar
- 50.Estep D, Larson MG, Williams RD. Estimating the error of numerical solutions of systems of reaction-diffusion equations. Providence: American Mathematical Soc.; 2000.Google Scholar
- 51.Ma X, Zabaras N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J Comput Phys. 2009;228(8):3084–113.MATHMathSciNetCrossRefGoogle Scholar
- 52.Calvetti D, Somersalo E. Introduction to Bayesian scientific computing. New York: Springer; 2007.MATHGoogle Scholar
- 53.Kaipio J, Somersalo E. Statistical and computational inverse problems. New York: Springer; 2005.MATHGoogle Scholar
- 54.Jaynes ET. Probability theory: the logic of science. New York: Cambridge University Press; 2003.CrossRefGoogle Scholar
- 55.Hoyas S, Jiménez J. Scaling of the velocity fluctuations in turbulent channels up to \(Re_\tau = 2003\). Phys Fluids. 2006;18(1):011702.CrossRefGoogle Scholar
- 56.Del Alamo JC, Jiménez J, Zandonade P, Moser RD. Scaling of the energy spectra of turbulent channels. J Fluid Mech. 2004;500:135–44.MATHCrossRefGoogle Scholar
- 57.Hoyas S, Jiménez J. Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys Fluids. 2008;20(10):101511.CrossRefGoogle Scholar
- 58.Rasmussen CE, Williams KI. Gaussian processes for machine learning. Cambridge: MIT Press; 2006.MATHGoogle Scholar
- 59.Prudencio EE, Schulz KW. The parallel C++ statistical library QUESO: quantification of uncertainty for estimation, simulation and optimization. Euro-Par 2011: Parallel Processing Workshops, vol. 7155., Lecture Notes in Computer Science. Heidelberg: Springer; 2012. p. 398–407.Google Scholar

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