Stochastic Collocation Research Articles

Impact of Hydrogeological Uncertainty on Estimation of Environmental Risks Posed by Hydrocarbon Transportation Networks

AbstractUbiquitous hydrogeological uncertainty undermines the veracity of quantitative predictions of soil and groundwater contamination due to accidental hydrocarbon spills from onshore pipelines. Such predictions, therefore, must be accompanied by quantification of predictive uncertainty, especially when they are used for environmental risk assessment. We quantify the impact of parametric uncertainty on quantitative forecasting of temporal evolution of two key risk indices, volumes of unsaturated and saturated soil contaminated by a surface spill of light nonaqueous‐phase liquids. This is accomplished by treating the relevant uncertain parameters as random variables and deploying two alternative probabilistic models to estimate their effect on predictive uncertainty. A physics‐based model is solved with a stochastic collocation method and is supplemented by a global sensitivity analysis. A second model represents the quantities of interest as polynomials of random inputs and has a virtually negligible computational cost, which enables one to explore any number of risk‐related contamination scenarios. For a typical oil‐spill scenario, our method can be used to identify key flow and transport parameters affecting the risk indices, to elucidate texture‐dependent behavior of different soils, and to evaluate, with a degree of confidence specified by the decision‐maker, the extent of contamination and the correspondent remediation costs.

Model form uncertainty quantification in turbulent combustion simulations: Peer models

Turbulent combustion simulations invoke a number of component models for chemical kinetics, turbulence, flame structure, etc., each of which has an error associated with its structural form and contributes to overall uncertainties in simulation results. These model form errors arise from the necessity of making assumptions in deriving a model. Conventional approaches to estimating model form errors rely on an ad hoc additive error that is then calibrated against experimental or computational data. These approaches inherently neglect any a priori knowledge of physics in developing the model error estimate. Instead, in this work, an inherently physics-based approach to estimating model form error is developed based on the notion of “peer” models. In the generic approach, the error in a candidate model is determined by taking the difference between it and an equally plausible alternative “peer” model with a different set of assumptions. The generic approach is applied in this work to the modeling of the subfilter mixture fraction dissipation rate, which is typically modeled as the ratio of the subfilter mixture fraction variance and a time scale. The typical time scale approximation invokes a turbulent time scale, and a “peer” model is proposed in which a chemical time scale is invoked to estimate the model form error. Using stochastic collocation, the subfilter mixture fraction dissipation rate model form error as well as the uncertainty in a model parameter are propagated through LES calculations of the Sandia D Flame utilizing the steady flamelet model. The results indicate that the mixture fraction, temperature, and carbon monoxide uncertainties increase with downstream distance due to an increase in the relative subfilter mixture fraction variance and increased sensitivity to the time scale approximations, which diverge in magnitude with downstream distance. Uncertainties in these quantities arising from the model form error are shown to be more significant than uncertainties arising from the model constant uncertainty. For the temperature, uncertainties due to chemical kinetic rate uncertainty are shown to be slightly smaller than uncertainties due to the subfilter mixture fraction dissipation rate model error; for the carbon monoxide mass fraction, uncertainties due to chemical kinetic rate uncertainty are twice as large as uncertainties due to the subfilter mixture fraction dissipation rate error since carbon monoxide is more kinetically-controlled than the temperature.

Sparse Recovery via ℓq-Minimization for Polynomial Chaos Expansions

AbstractIn this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.

Stochastic Solutions to Rough Surface Scattering Using the Finite Element Method

We propose stochastic solutions to the scattering of electromagnetic waves from penetrable randomly rough surfaces using the vector-based finite-element method (FEM). The random nature of the surface requires the computation to be performed on multiple surface instances. We propose a mesh deformation scheme, which allows the use of a single FEM mesh for computing ensemble averaged quantities. This scheme is used to perform Monte Carlo (MC) iterations, which is much faster than the conventional techniques where a different mesh is utilized for each surface instance. This scheme also allows for MC-free formulations of the problem; in the first kind, we expand the solution in a stochastic basis using the concept of generalized polynomial chaos obtaining ensemble averaged quantities. This results in a larger set of equations that need to be solved just once. In the second kind —the stochastic collocation method—we run the deterministic solver at certain specified points in the random domain corresponding to the rough surface description. We compare these results with those obtained by MC iterations, and outline the computational costs and convergence behavior. We find that stochastic methods are ideal for surfaces with large correlation lengths. For other parameters, the MC approach is preferable.

A Two-Level Sparse Grid Collocation Method for Semilinear Stochastic Elliptic Equation

Abstract In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯 H ⁢ ( D ) {\mathcal{T}_{H}(D)} with small scale of sparse collocation points η ⁢ ( L , N ) {\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} using large scale of sparse collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} (where η ⁢ ( L , N ) , η ⁢ ( ℓ , N ) {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L , ℓ {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h ≈ H 3 , η ⁢ ( ℓ , N ) ≈ ( η ⁢ ( L , N ) ) 3 {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}} , the novel two-level discretization method achieves the same convergence accuracy in norm ∥ ⋅ ∥ ℒ ρ 2 ⁢ ( Γ ) ⊗ ℒ 2 ⁢ ( D ) {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ( ℒ ρ 2 ⁢ ( Γ ) {\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ 2 {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} and large scale collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} in random spaces.

Practical application of collocation method in power flow study of South Australia grid

In this paper stochastic collocation method is proposed to solve probabilistic power flow (PPF) model of South Australia (SA). And this model is based upon historical acquisition of power system data of SA. In SA, numbers of wind farms are installed and the variability of wind speed brings more uncertainties into the power system. The traditional deterministic power flow (DPF) computation does not take into account the probabilistic nature of power system uncertainties, so PPF computation is imperative. However, as a commonly used PPF simulation method, Monte Carlo simulation (MCS) has a very high computation cost. Hence, in this paper sparse grid interpolation (SGI) is presented to accomplish PPF analysis with striking high computation efficiency. Meanwhile, instead of using theoretical wind power generation model, probabilistic collocation method (PCM) is used to construct realistic relationship between wind speed and wind power. In addition, fuzzy logic optimization is applied to PCM to improve the accuracy of the model output. The paper concludes with presentation of an aggregated DC power flow model of SA to compare the computation efficiency of the SGI and MC.

Impact of parametric uncertainty on estimation of the energy deposition into an irradiated brain tumor

Predictions of the total energy deposited into a brain tumor through X-ray irradiation are notoriously error-prone. We investigate how this predictive uncertainty is affected by uncertainty in both the location of the region occupied by a dose-enhancing iodinated contrast agent and the agent's concentration. This is done within the probabilistic framework in which these uncertain parameters are modeled as random variables. We employ the stochastic collocation (SC) method to estimate statistical moments of the deposited energy in terms of statistical moments of the random inputs, and the global sensitivity analysis (GSA) to quantify the relative importance of uncertainty in these parameters on the overall predictive uncertainty. A nonlinear radiation–diffusion equation dramatically magnifies the coefficient of variation of the uncertain parameters, yielding a large coefficient of variation for the predicted energy deposition. This demonstrates that accurate prediction of the energy deposition requires a proper treatment of even small parametric uncertainty. Our analysis also reveals that SC outperforms standard Monte Carlo, but its relative efficiency decreases as the number of uncertain parameters increases from one to three. A robust GSA ameliorates this problem by reducing this number.

Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli–Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.

Quantifying uncertainty in the chemical master equation.

We describe a novel approach to quantifying the uncertainty inherent in the chemical kinetic master equation with stochastic coefficients. A stochastic collocation method is coupled to an analytical expansion of the master equation to analyze the effects of both extrinsic and intrinsic noise. The method consists of an analytical moment-closure method resulting in a large set of differential equations with stochastic coefficients that are in turn solved via a Smolyak sparse grid collocation method. We discuss the error of the method relative to the dimension of the model and clarify which methods are most suitable for the problem. We apply the method to two typical problems arising in chemical kinetics with time-independent extrinsic noise. Additionally, we show agreement with classical Monte Carlo simulations and calculate the variance over time as the sum of two expectations. The method presented here has better convergence properties for low to moderate dimensions than standard Monte Carlo methods and is therefore a superior alternative in this regime.

On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations

Polynomial chaos-based methods have been extensively applied in electrical and other engineering problems for the stochastic simulation of systems with uncertain parameters. Most of the implementations are based on either the intrusive stochastic Galerkin method or on non-intrusive collocation approaches, of which a very common example is the pseudo-spectral method based on Gaussian quadrature rules. This paper shows that, for the important class of linear differential algebraic equations, the latter can be cast as an approximate factorization of the stochastic Galerkin approach, thus generalizing recent discussions in literature in this regard. Consistently with this literature, we show that the factorization turns out to be exact for first-order random inputs, and hence the two methods coincide under this assumption. Further, the presented results also generalize recent work in the field of electrical circuit simulation, in which a similar decomposition was derived ad hoc, via error minimization, for the case of Hermite chaos. We demonstrate that the factorization stems from the general properties of orthogonal polynomials and the error introduced by the approximation—or in other terms, the error of the stochastic collocation method in comparison with the stochastic Galerkin method—is carefully quantified and assessed. An illustrative example concerning the stochastic analysis of an RLC circuit is used to illustrate the main findings of this paper. In addition, a more complex and real-life example allows emphasizing the generality of the achieved results.

Analysis of the variability of food texture properties: Application to the fracturability of dry pet food.

Food texture is affected by their material property and geometrical structure. Material properties are difficult to estimate when the size and shape of the samples cannot be standardized, for example, for snack or pet food. In addition, the intrinsic variability of material properties due to the recipe and the process cannot be neglected. But this cannot be easily distinguished from the variability of geometrical parameters using conventional methods. An innovative approach using stochastic finite element method is developed and is able to exhibit the major parameters influencing the variability of the response to mechanical testing. This methodology may be helpful to improve acceptance sampling control procedures, for example, by combining texture and size measurements when necessary, or by better determining the most robust response variables for decision-making.

Inverse uncertainty quantification of TRACE physical model parameters using sparse gird stochastic collocation surrogate model

Within the BEPU (Best Estimate plus Uncertainty) methodology uncertainties must be quantified in order to prove that the investigated design remains within acceptance criteria. For best-estimate system thermal-hydraulics codes like TRACE and RELAP5, significant uncertainties come from the closure laws which are used to describe transfer terms in the balance equations. The accuracy and uncertainty information of these correlations are usually unknown to the code users, which results in the user simply ignoring or describing them using expert opinion or personal judgment during uncertainty and sensitivity analysis. The purpose of this paper is to replace such ad-hoc expert judgment of the uncertainty information of TRACE physical model parameters with inverse Uncertainty Quantification (UQ) based on OECD/NRC BWR Full-size Fine-Mesh Bundle Tests (BFBT) benchmark steady-state void fraction data. Inverse UQ seeks statistical descriptions of the physical model random input parameters that are consistent with the experimental data. Inverse UQ always captures the uncertainty of its estimates rather than merely determining point estimates of the best-fit input parameters. Bayesian analysis is used to establish the inverse UQ problems based on experimental data, with systematic and rigorously derived surrogate models based on Sparse Gird Stochastic Collocation (SGSC). Global sensitivity analysis including Sobol' indices and correlation coefficients are used to identify the important TRACE input parameters. Several adaptive Markov Chain Monte Carlo (MCMC) sampling techniques are investigated and implemented to explore the posterior probability density functions. This research solves the problem of lack of uncertainty information for TRACE physical model parameters for the closure relations. The quantified uncertainties are necessary for future uncertainty and sensitivity study of TRACE code in nuclear reactor system design and safety analysis.

Big-Data Tensor Recovery for High-Dimensional Uncertainty Quantification of Process Variations

Fabrication process variations are a major source of yield degradation in the nanoscale design of integrated circuits (ICs), microelectromechanical systems (MEMSs), and photonic circuits. Stochastic spectral methods are a promising technique to quantify the uncertainties caused by process variations. Despite their superior efficiency over Monte Carlo for many design cases, stochastic spectral methods suffer from the curse of dimensionality, i.e., their computational cost grows very fast as the number of random parameters increases. In order to solve this challenging problem, this paper presents a high-dimensional uncertainty quantification algorithm from a big data perspective. Specifically, we show that the huge number of (e.g., $1.5 \times 10^{27}$ ) simulation samples in standard stochastic collocation can be reduced to a very small one (e.g., 500) by exploiting some hidden structures of a high-dimensional data array. This idea is formulated as a tensor recovery problem with sparse and low-rank constraints, and it is solved with an alternating minimization approach. The numerical results show that our approach can efficiently simulate some IC, MEMS, and photonic problems with over 50 independent random parameters, whereas the traditional algorithm can only deal with a small number of random parameters.

Multi-fidelity stochastic collocation method for computation of statistical moments

We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. The method extends the multi-fidelity approximation method developed in [18,26]. By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. We establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi-fidelity algorithm.

On an efficient multiple time step Monte Carlo simulation of the SABR model

In this paper, we will present a multiple time step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho model. The proposed method is an extension of the one time step Monte Carlo method that we proposed in an accompanying paper Leitao et al. [Appl. Math. Comput. 2017, 293, 461–479], for pricing European options in the context of the model calibration. A highly efficient method results, with many very interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time step Monte Carlo method is especially useful for long-term options and for exotic options.

Performance Robustness of Turbine Squealer Tip Designs Due to Manufacturing and Engine Operation

This paper quantifies the changes in turbine performance due to manufacturing tolerances and profile degradation of the blade-tip region during engine operation. An extensive numerical study was conducted on a modern high-pressure turbine rotor, operating at high Reynolds and high subsonic outlet conditions. The stochastic collocation method was used to investigate the effects of the geometrical variability of a single squealer tip design. The variation of three geometrical characteristics was investigated: tip clearance size, squealer depth, and rim corner radius. Three-dimensional Reynolds-averaged Navier–Stokes simulations were performed to predict the aerodynamic and heat transfer characteristics. The results highlight the influence of the combined geometrical variability on the overtip flow structures, heat transfer signatures, loss development, and downstream flow characteristics. This study measures the robustness of squealer tip designs to changes in its geometry and demonstrates the necessity of anticipating the consequences of these unavoidable tip geometry variabilities in an early development stage of the turbine design optimization.

A Kriging and Stochastic Collocation ensemble for uncertainty quantification in engineering applications

We propose a new surrogate modeling approach by combining two non-intrusive techniques: Kriging and Stochastic Collocation. The proposed method relies on building a sufficiently accurate Stochastic Collocation model which acts as a basis to construct a Kriging model on the residuals, to combine the accuracy and efficiency of Stochastic Collocation methods in describing stochastic quantities with the flexibility and modeling power of Kriging-based approaches. We investigate and compare performance of the proposed approach with state-of-art techniques over benchmark problems and practical engineering examples on various experimental designs.

A novel Monte Carlo approach to hybrid local volatility models

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007, April, 84–89], Tataru and Fisher [Quantitative Development Group, Bloomberg Version 1, 2010], Lipton [Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan [ArXiV preprint math/0604316, 2006], Piterbarg [Risk, 2006, 19, 66–71], Deelstra and Rayée [Appl. Math. Finance, 2012, 1–23], Ren et al. [Risk, 2007, 20, 138–143]. For both model classes a particular (conditional) expectation needs to be evaluated which cannot be extracted from the market and is expensive to compute. We establish accurate and ‘cheap to evaluate’ approximations for the expectations by means of the stochastic collocation method [SIAM J. Numer. Anal., 2007, 45, 1005–1034], [SIAM J. Sci. Comput., 2005, 27, 1118–1139], [Math. Models Methods Appl. Sci., 2012, 22, 1–33], [SIAM J. Numer. Anal., 2008, 46, 2309–2345], [J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context [Available at SSRN 2529691, 2014], [J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.

CLUSTERING-BASED COLLOCATION FOR UNCERTAINTY PROPAGATION WITH MULTIVARIATE DEPENDENT INPUTS

In this article, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation. The key idea is to use cluster centers as the nodes for collocation. In this way, we can extend the use of collocation methods to uncertainty propagation with multivariate, dependent input, in which the output approximation is piecewise constant on the clusters. The approach is particularly useful in situations where the probability distribution of the input is unknown, and only a sample from the input distribution is available. We examine several clustering methods and assess the convergence of collocation based on these methods both theoretically and numerically. We demonstrate good performance of the proposed methods, most notably for the challenging case of nonlinearly dependent inputs in higher dimensions. Numerical tests with input dimension up to 16 are included, using as benchmarks the Genz test functions and a test case from computational fluid dynamics (lid-driven cavity flow).

Multilevel tensor approximation of PDEs with random data

In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels at the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting regularity and additional low-rank structure of the solution. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.