Sparse Grid Stochastic Collocation Research Articles

Evaluating two sparse grid surrogates and two adaptation criteria for groundwater Bayesian uncertainty quantification

Summary Sparse grid (SG) stochastic collocation methods have been recently used to build accurate but cheap-to-run surrogates for groundwater models to reduce the computational burden of Bayesian uncertainty analysis. The surrogates can be built for either a log-likelihood function or state variables such as hydraulic head and solute concentration. Using a synthetic groundwater flow model, this study evaluates the log-likelihood and head surrogates in terms of the computational cost of building them, the accuracy of the surrogates, and the accuracy of the distributions of model parameters and predictions obtained using the surrogates. The head surrogates outperform the log-likelihood surrogates for the following four reasons: (1) the shape of the head response surface is smoother than that of the log-likelihood response surface in parameter space, (2) the head variation is smaller than the log-likelihood variation in parameter space, (3) the interpolation error of the head surrogates does not accumulate to be larger than the interpolation error of the log-likelihood surrogates, and (4) the model simulations needed for building one head surrogate can be recycled for building others. For both log-likelihood and head surrogates, adaptive sparse grids are built using two indicators: absolute error and relative error. The adaptive head surrogates are insensitive to the error indicators, because the ratio between the two indicators is hydraulic head, which has small variation in the parameter space. The adaptive log-likelihood surrogates based on the relative error indicators outperform those based on the absolute error indicators, because adaptation based on the relative error indicator puts more sparse-grid nodes in the areas in the parameter space where the log-likelihood is high. While our numerical study suggests building state-variable surrogates and using the relative error indicator for building log-likelihood surrogates, selecting appropriate type of surrogates and error indicators depends on the shapes of response surfaces. The shapes should be approximated and examined before building sparse grid surrogates.

Uncertainty quantification in numerical simulation of particle-laden flows

Numerical models can help to push forward the knowledge about complex dynamic physical systems. Modern approaches employ detailed mathematical models, taking into consideration inherent uncertainties on input parameters (phenomenological parameters or boundary and initial conditions, among others). Particle-laden flows are complex physical systems found in nature, generated due to the (possible small) spatial variation on the fluid density promoted by the carried particles. They are one of the main mechanisms responsible for the deposition of sediments on the seabed. A detailed understanding of particle-laden flows, often referred to as turbidity currents, helps geologists to understand the mechanisms that give rise to reservoirs, strategic in oil exploration. Uncertainty quantification (UQ) provides a rational framework to assist in this task, by combining sophisticated computational models with a probabilistic perspective in order to deepen the knowledge about the physics of the problem and to access the reliability of the results obtained with numerical simulations. This work presents a stochastic analysis of sediment deposition resulting from a turbidity current considering uncertainties on the initial sediment concentrations and particles settling velocities. The statistical moments of the deposition mapping, like other important features of the currents, are approximated by a Sparse Grid Stochastic Collocation method that employ a parallel flow solver for the solution of the deterministic problems associated to the grid points. The whole procedure is supported and steered by a scientific workflow management engine designed for high performance computer applications.

Distributed Control of the Stochastic Burgers Equation with Random Input Data

AbstractWe discuss a control problem involving a stochastic Burgers equation with a random diffusion coefficient. Numerical schemes are developed, involving the finite element method for the spatial discretisation and the sparse grid stochastic collocation method in the random parameter space. We also use these schemes to compute closed-loop suboptimal state feedback control. Several numerical experiments are performed, to demonstrate the efficiency and plausibility of our approximation methods for the stochastic Burgers equation and the related control problem.

Uncertainty and global sensitivity analysis of neutron survival and extinction probabilities using polynomial chaos

Generalised Polynomial Chaos (GPC) in conjunction with sparse grid stochastic collocation and High Dimensional Model Representation (HDMR) is used to perform uncertainty and global sensitivity analysis for the neutron chain survival and extinction probabilities, with and without an intrinsic random source. Starting with a lumped backward Master equation formulation, uncertainty is introduced by allowing the factorial moments of the fission multiplicity distribution, the neutron lifetime, and strength of the intrinsic source to be independent and uniformly distributed random variables. A multidimensional Legendre chaos representation of the random survival and extinction probabilities is used to achieve optimal numerical convergence in the stochastic dimension and the relative variance contributions from each random parameter are then quantified using High Dimensional Model Representation (HDMR).The underlying deterministic results of the model are found to closely match analytical benchmarks and, once uncertainty is introduced, the GPC results match both Monte Carlo simulations and analytical results for polynomial order greater than two. The GPC method is found to require significantly less computational time to achieve a given accuracy on the survival and extinction probabilities than the Monte Carlo method. It is found that, the probabilities are most sensitive to χi for lower i and have a significant sensitivity to χ2 in all cases. A chain’s survival probability is moderately sensitive to the neutron lifetime early in simulation. In the subcritical case this sensitivity increases as the simulation continues whilst it decreases in the supercritical case. The extinction probability is sensitive to the source strength.

Parametric uncertainty and global sensitivity analysis in a model of the carotid bifurcation: Identification and ranking of most sensitive model parameters

In computational cardiovascular models, parameters are one of major sources of uncertainty, which make the models unreliable and less predictive. In order to achieve predictive models that allow the investigation of the cardiovascular diseases, sensitivity analysis (SA) can be used to quantify and reduce the uncertainty in outputs (pressure and flow) caused by input (electrical and structural) model parameters.In the current study, three variance based global sensitivity analysis (GSA) methods; Sobol, FAST and a sparse grid stochastic collocation technique based on the Smolyak algorithm were applied on a lumped parameter model of carotid bifurcation. Sensitivity analysis was carried out to identify and rank most sensitive parameters as well as to fix less sensitive parameters at their nominal values (factor fixing). In this context, network location and temporal dependent sensitivities were also discussed to identify optimal measurement locations in carotid bifurcation and optimal temporal regions for each parameter in the pressure and flow waves, respectively.Results show that, for both pressure and flow, flow resistance (R), diameter (d) and length of the vessel (l) are sensitive within right common carotid (RCC), right internal carotid (RIC) and right external carotid (REC) arteries, while compliance of the vessels (C) and blood inertia (L) are sensitive only at RCC. Moreover, Young’s modulus (E) and wall thickness (h) exhibit less sensitivities on pressure and flow at all locations of carotid bifurcation. Results of network location and temporal variabilities revealed that most of sensitivity was found in common time regions i.e. early systole, peak systole and end systole.

Uncertainty quantification in virtual surgery hemodynamics predictions for single ventricle palliation.

The adoption of simulation tools to predict surgical outcomes is increasingly leading to questions about the variability of these predictions in the presence of uncertainty associated with the input clinical data. In the present study, we propose a methodology for full propagation of uncertainty from clinical data to model results that, unlike deterministic simulation, enables estimation of the confidence associated with model predictions. We illustrate this problem in a virtual stage II single ventricle palliation surgery example. First, probability density functions (PDFs) of right pulmonary artery (PA) flow split ratio and average pulmonary pressure are determined from clinical measurements, complemented by literature data. Starting from a zero-dimensional semi-empirical approximation, Bayesian parameter estimation is used to find the distributions of boundary conditions that produce the expected PA flow split and average pressure PDFs as pre-operative model results. To reduce computational cost, this inverse problem is solved using a Kriging approximant. Second, uncertainties in the boundary conditions are propagated to simulation predictions. Sparse grid stochastic collocation is employed to statistically characterize model predictions of post-operative hemodynamics in models with and without PA stenosis. The results quantify the statistical variability in virtual surgery predictions, allowing for placement of confidence intervals on simulation outputs.

Robust calibration of macroscopic traffic simulation models using stochastic collocation

The predictions of a well-calibrated traffic simulation model are much more valid if made for various conditions. Variation in traffic can arise due to many factors such as time of day, work zones and weather. Calibration of traffic simulation models for traffic conditions requires larger datasets to capture the stochasticity in traffic conditions. In this study we use datasets spanning large time periods to incorporate variability in traffic flow, speed for various time periods. However, large data poses a challenge in terms of computational effort. With the increase in number of stochastic factors, the numerical methods suffer from the curse of dimensionality. In this study, we propose a novel methodology to address the computational complexity due to the need for the calibration of simulation models under highly stochastic traffic conditions. This methodology is based on sparse grid stochastic collocation, which, treats each stochastic factor as a different dimension and uses a limited number of points where simulation and calibration are performed. A computationally efficient interpolant is constructed to generate the full distribution of the simulated flow output. We use real-world examples to calibrate for different times of day and conditions and show that this methodology is much more efficient that the traditional Monte Carlo-type sampling. We validate the model using a hold out dataset and also show the drawback of using limited data for the calibration of a macroscopic simulation model. We also discuss the drawbacks of the predictive ability of a single calibrated model for all the conditions.

A Sparse Grid Stochastic Collocation Method for Elliptic Interface Problems with Random Input

In this paper, numerical solutions of elliptic partial differential equations with both random input and interfaces are considered. The random coefficients are piecewise smooth in the physical spac...

Uncertain eigenvalue analysis by the sparse grid stochastic collocation method

In this paper, the eigenvalue problem with multiple uncertain parameters is analyzed by the sparse grid stochastic collocation method. This method provides an interpolation approach to approximate eigenvalues and eigenvectors’ functional dependencies on uncertain parameters. This method repetitively evaluates the deterministic solutions at the pre-selected nodal set to construct a high-dimensional interpolation formula of the result. Taking advantage of the smoothness of the solution in the uncertain space, the sparse grid collocation method can achieve a high order accuracy with a small nodal set. Compared with other sampling based methods, this method converges fast with the increase of the number of points. Some numerical examples with different dimensions are presented to demonstrate the accuracy and efficiency of the sparse grid stochastic collocation method.

Robust optimal shape design for an elliptic PDE with uncertainty in its input data

We consider a shape optimization problem for an elliptic partial differential equation with uncertainty in its input data. The design variable enters the lower-order term of the state equation and is modeled through the characteristic function of a measurable subset of the spatial domain. As usual, a measure constraint is imposed on the design variable. In order to compute a robust optimal shape, the objective function involves a weighted sum of both the mean and the variance of the compliance. Since the optimization problem is not convex, a full relaxation of it is first obtained. The relaxed problem is then solved numerically by using a gradient-based optimization algorithm. To this end, the adjoint method is used to compute the continuous gradient of the cost function. Since the variance enters the cost function, the underlying adjoint equation is non-local in the probabilistic space. Both the direct and adjoint equations are solved numerically by using a sparse grid stochastic collocation method. Three numerical experiments in 2D illustrate the theoretical results and show the computational issues which arise when uncertainty is quantified through random fields.

Robust Calibration of Macroscopic Traffic Simulation Models Using Stochastic Collocation

The predictions of a well-calibrated traffic simulation model are much more valid if made for various conditions. Variation in traffic can arise due to many factors such as time of day, work zones, weather, etc. Calibration of traffic simulation models for traffic conditions requires larger datasets to capture the stochasticity in traffic conditions. In this study we use datasets spanning large time periods to incorporate variability in traffic flow, speed for various time periods. However, large data poses a challenge in terms of computational effort. With the increase in number of stochastic factors, the numerical methods suffer from the curse of dimensionality. In this study, we propose a novel methodology to address the computational complexity due to the need for the calibration of simulation models under highly stochastic traffic conditions. This methodology is based on sparse grid stochastic collocation, which, treats each stochastic factor as a different dimension and uses a limited number of points where simulation and calibration are performed. A computationally efficient interpolant is constructed to generate the full distribution of the simulated flow output. We use real-world examples to calibrate for different times of day and conditions and show that this methodology is much more efficient that the traditional Monte Carlo-type sampling. We validate the model using a hold out dataset and also show the drawback of using limited data for the calibration of a macroscopic simulation model. We also discuss the drawbacks of using a single calibrated model for all the data.

Uncertain eigenvalue analysis by sparse grid stochastic collocation method

In this paper, the eigenvalue problem with multiple uncertain parameters is analyzed by the sparse grid stochastic collocation method. This method provides an interpolation approach to approximate eigenvalues and eigenvectors’ functional dependencies on uncertain parameters. This method repetitively evaluates the deterministic solutions at the pre-selected nodal set to construct a highdimensional interpolation formula of the result. Taking advantage of the smoothness of the solution in the uncertain space, the sparse grid collocation method can achieve a high order accuracy with a small nodal set. Compared with other sampling based method, this method converges fast with the increase of the number of points. Some numerical examples with different dimensions are presented to demonstrate the accuracy and efficiency of the sparse grid stochastic collocation method.

A sparse grid stochastic collocation method for structural reliability analysis

This paper develops a sparse grid stochastic collocation method for the reliability analysis of structures with uncertain parameters and loads. The method consists of two standard techniques in uncertainty quantification: the moment-based Gauss transformation and Smolyak-type sparse grid quadrature rule. Unlike the first-order reliability method (FORM) or second-order reliability method (SORM), the developed method does not need the evaluation of the first- or second-order partial derivatives of the limit state function considered and, moreover, does not suffer from the problem of multiple design points. In addition, the developed method is suitable for all problems whose deterministic solutions can be found and usually needs much fewer function evaluations than the Monte Carlo simulation method. Numerical examples demonstrate that the developed method is exact enough for evaluating the mean values, standard deviations, skewness and kurtosis of the limit state functions and small probabilities of failure as low as 10−4. Even for probabilities of failure as low as 10−5, the quality of approximation obtained by the method is also acceptable.

Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation

We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja-sparse grid approach to several high-dimensional and problems and demonstrate that Leja sequences are often superior to more standard sparse grid constructions (e.g. Clenshaw-Curtis), at least for interpolatory metrics.

An efficient framework for optimization and parameter sensitivity analysis in arterial growth and remodeling computations

Computational models for vascular growth and remodeling (G&R) are used to predict the long-term response of vessels to changes in pressure, flow, and other mechanical loading conditions. Accurate predictions of these responses are essential for understanding numerous disease processes. Such models require reliable inputs of numerous parameters, including material properties and growth rates, which are often experimentally derived, and inherently uncertain. While earlier methods have used a brute force approach, systematic uncertainty quantification in G&R models promises to provide much better information. In this work, we introduce an efficient framework for uncertainty quantification and optimal parameter selection, and illustrate it via several examples. First, an adaptive sparse grid stochastic collocation scheme is implemented in an established G&R solver to quantify parameter sensitivities, and near-linear scaling with the number of parameters is demonstrated. This non-intrusive and parallelizable algorithm is compared with standard sampling algorithms such as Monte-Carlo. Second, we determine optimal arterial wall material properties by applying robust optimization. We couple the G&R simulator with an adaptive sparse grid collocation approach and a derivative-free optimization algorithm. We show that an artery can achieve optimal homeostatic conditions over a range of alterations in pressure and flow; robustness of the solution is enforced by including uncertainty in loading conditions in the objective function. We then show that homeostatic intramural and wall shear stress is maintained for a wide range of material properties, though the time it takes to achieve this state varies. We also show that the intramural stress is robust and lies within 5% of its mean value for realistic variability of the material parameters. We observe that prestretch of elastin and collagen are most critical to maintaining homeostasis, while values of the material properties are most critical in determining response time. Finally, we outline several challenges to the G&R community for future work. We suggest that these tools provide the first systematic and efficient framework to quantify uncertainties and optimally identify G&R model parameters.

Reduced Basis Collocation Methods for Partial Differential Equations with Random Coefficients

The sparse grid stochastic collocation method is a new method for solving partial differential equations with random coefficients. However, when the probability space has high dimensionality, the number of points required for accurate collocation solutions can be large, and it may be costly to construct the solution. We show that this process can be made more efficient by combining collocation with reduced basis methods, in which a greedy algorithm is used to identify a reduced problem to which the collocation method can be applied. Because the reduced model is much smaller, costs are reduced significantly. We demonstrate with numerical experiments that this is achieved with essentially no loss of accuracy.

A stochastic dimension reduction multiscale finite element method for groundwater flow problems in heterogeneous random porous media

Summary In this paper we present a stochastic dimension reduction multiscale finite element method for solving groundwater flow problems in heterogeneous random porous media. The stochastic conductivity is parameterized in a stochastic space using a truncated Karhunen–Loeve expansion with finite random variables. This often results in high stochastic dimensionality due to small correlation length and large variance of the inherent covariance function. In order to treat the high-dimensional stochastic problems efficiently, a truncated high-dimensional model representation (HDMR) technique is proposed that decomposes the high-dimensional stochastic problem into a moderate-dimensional stochastic problem and a few one-dimensional stochastic problems. The derived low-dimensional stochastic problems are solved by incorporating a sparse grid stochastic collocation method (SGSCM) with the proposed HDMR. This leads to a set of uncoupled deterministic problems at the sparse grid collocation points. To capture the small-scale heterogeneities in physical space, a multiscale finite element method (MsFEM) is utilized to solve these deterministic problems. Since the SGSCM and MsFEM are implemented in the different spaces, the integrated approach inherits the advantages from both of the SGSCM and the MsFEM, in which each low-dimensional stochastic problem is decoupled each other at small number of stochastic collocation points and is solved on a coarse spatial mesh. To demonstrate the efficiency and accuracy of the proposed stochastic MsFEM with the HDMR technique, a few numerical experiments are carried out for the transient water flow problems in heterogeneous random media with different correlation lengths and various degrees of spatial variabilities.

A Stochastic Finite Element Heterogeneous Multiscale Method for Seepage Field in Heterogeneous Ground

The finite element heterogeneous multiscale method (FEHM) combined with stochastic collocation method (SCM) called SHMFE is applied to studying the seepage field of naturally heterogeneous multiscale subsurface formations. Kinds of stochastic finite element (SFEM) are mainly computational techniques for the class of problems. But those methods do not report the multiscale nature of the properties of subsurface formations. When the random permeability field is heterogeneous in fine scale comparing to study domain, the simulation by the classic SFEM is not a trivial task. The SHMFE can efficiently solve the problems. In the method, Karhunen-Loµeve (KL) decomposition is used to represent the log hydraulic conductivity Y = lnKεin fine scale. The SCM which couples the generalized polynomial chaos is used to make the problem determined, and then the FEHM method is used to solve it. Sparse grid stochastic collocation method is used when KL expansion has many random variables. The numerical examples demonstrate that the SHMFE approach can efficiently simulate the flow in naturally multiscale heterogeneous subsurface formations with relatively lower computational cost comparing with the SFEM methods.

Effects of Approximate Deconvolution Models on the Solution of the Stochastic Navier-Stokes Equations

The direct numerical simulation of Navier-Stokes equations in the turbulent regime is not computationally feasible either in the deterministic or (especially) in the stochastic case. Therefore, turbulent modeling must be employed. We consider the family of approximate deconvolution models (ADM) for the simulation of the turbulent stochastic Navier-Stokes equations (NSE). For moderate values of the Reynolds number, we investigate the effect stochastic forcing (through the boundary conditions) has on the accuracy of solutions of the ADM equations compared to direct numerical simulations. Although the existence, uniqueness and verifiability of the ADM solutions has already been proven in the deterministic setting, the analyticity of a solution of the stochastic NSE is difficult to prove. Hence, we approach the problem from the computational point of view. A Smolyak-type sparse grid stochastic collocation method is employed for the approximation of first two statistical moments of the solution − the expected value and variance. We show that for different test problems, the modeling error in the stochastic case is the same as predicted for the deterministic setting. Although the ADMs are arguably only applicable for certain boundary conditions (zero or periodic), we test the model on a problem with a boundary layer and recirculation region and demonstrate that the model correctly predicts the solution of the stochastic NSE with the noise in the boundary data.

A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.