Stochastic analysis of unsaturated flow with probabilistic collocation method

Uncertainty in ocean environmental parameters is often the dominant source of uncertainty in an underwater acoustic field calculation. In addition, the relationship between environmental parameters and the resulting field prediction may be highly nonlinear, leading to the explicit determination of the resulting field uncertainty uneasy. The direct Monte Carlo (MC) sampling method which is conventionally used to compute the field uncertainty is computationally prohibitive. To overcome the drawbacks of the existing methods, this paper proposed a Probabilistic Collocation Method (PCM) based method to propagate the uncertainty efficiently. The basic concept of the method is to represent the random acoustic field by Polynomial Chaos Expansions (PCE) and to calculate the coefficients in the PCE using PCM. Statistical properties can then be obtained from the PCE easily. Computer simulations show that PCM can propagate the uncertainty efficiently when the distribution of the result acoustic is unimodal.

Improving parameter estimation with an efficient sequential probabilistic collocation-based optimal design method

Development of clustered polynomial chaos expansion model for stochastic hydrological prediction

Uncertainty, Sensitivity and Geostatistical Studies of Flow and Contaminant Transport in Heterogeneous Unsaturated Zone by Feng Pan Dr. Zhongbo Yu, Examination Committee Chair Associate Professor of Hydrology University of Nevada Las Vegas Dr. Jianting Zhu, Examination Committee Co-Chair Associate Professor of Hydrologic Science Desert Research Institute The objectives of this study are: (1) to develop a methodology of estimating probability density functions (PDFs) of unsaturated hydraulic parameters when field samples are sparse, (2) to evaluate the predictive uncertainties in flow and contaminant transport due to parameter uncertainties in the layerand local-scale heterogeneities of hydraulic parameters in unsaturated zone (UZ), (3) to investigate the contributions of the parameter uncertainties to the flow and transport uncertainties, and (4) to estimate the spatial correlation structures of hydraulic parameters by incorporating prior information and site measurements. At layer scale, the uncertainty assessment of flow and contaminant transport in UZ entails PDFs of the hydraulic parameters. A non-conventional maximum likelihood (ML) approach is used in this study to estimate the PDFs of water retention parameters (e.g., van Genuchten a and ri) for situations common in field scale applications where core samples are sparse and prior PDFs of the parameters are unknown. This study also investigates the effects of the uncertainties in the water retention parameters on the predictive uncertainties in flow and transport in UZ. By comparing the predictive uncertainties with and without incorporating the random water retention parameters, it is found that the random water retention parameters have limited effects on the mean predictions of the state variables including percolation flux, normalized cumulative mass arrival, and contaminant travel time. However, incorporating the uncertainties in the water retention parameters significantly increases the magnitude and spatial extent of predictive uncertainties of the state variables. The layer-scale uncertainty is specific to hydrogeologic layers, while the localscale heterogeneity refers to the spatial variation of hydraulic properties within a layer. The local-scale heterogeneity is important in predicting flow path, velocity, and travel time of contaminants, but it is often neglected in modeling practices. This study incorporates the local-scale heterogeneity and examines its relative effects to the layerscale uncertainty on flow and transport uncertainties in UZ. Results illustrate that localscale heterogeneity significantly increases predictive uncertainties in the percolation fluxes and contaminant plumes, whereas the mean predictions are only slightly affected by the local-scale heterogeneity. Layer-scale uncertainty is more important than localscale heterogeneity for simulating overall contaminant travel time, suggesting that it would be more cost-effective to reduce the layer-scale parameter uncertainty in order to reduce predictive uncertainty in contaminant transport.

The in situ soil properties exhibit natural spatial variability due to various factors and are difficult to be estimated based on limited test data. In this paper, a probabilistic back analysis method is adopted to characterize the spatial variability of soil parameters based on the responses of displacement. The Karhunen-Loeve (K-L) expansion method is used to discretize the random field of soil spatial variability. A finite number of basic random variables from the truncated K-L expansion are the random variables to be back estimated using the Markov Chain Monte Carlo (MCMC) method. To improve computational efficiency, a surrogate model based on the polynomial chaos expansion (PCE) is established to substitute the finite element model of slope stability analysis. A hypothetical example with a slope subject to surcharge load is presented to illustrate the accuracy and efficiency of the method. The spatially varied elastic moduli are estimated using the responses of displacement. It is found that the back analysis based on horizontal displacements is more accurate than that based on vertical displacements. When the correlation length decreases, the accuracy of the estimated field is reduced. The vertical correlation length has a greater effect on the estimation of soil spatial variability than the horizontal correlation length. When the coefficient of variation increases, the accuracy of the estimated field is reduced.

Modern large-scale infrastructure systems are typically complicated in nature and require extensive simulations to evaluate their performance. The Probabilistic Collocation Method (PCM) is developed to effectively simulate system performance under uncertainty. In this paper, we extend the formal analysis of the single-variable PCM to the multivariate case, where the parameters may or may not be independent. Specifically, we provide conditions that permit the multivariate PCM to precisely predict the mean of the original system output. We also explore additional capabilities of the multivariate PCM, in terms of cross-statistics prediction, relation to the minimum mean-square estimator, and computational feasibility for large dimensional data. At the end of the paper, we demonstrate the application of the multivariate PCM in air traffic management.

The least squares Monte Carlo method is a decision evaluation method that can capture the effect of uncertainty and the value of flexibility of a process. The method is a stochastic approximate dynamic programming approach to decision making. It is based on a forward simulation coupled with a recursive algorithm which produces the near-optimal policy. It relies on the Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir engineering problems because this requires running many reservoir simulations. The objective of this study was to enhance the performance of the least squares Monte Carlo method by improving the sampling method used to generate the technical uncertainties used in obtaining the production profiles. The probabilistic collocation method has been proven to be a robust and efficient uncertainty quantification method. By using the sampling methods of the probabilistic collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision evaluation method. Thus, we introduce the least squares probabilistic collocation method. The decision evaluation considered a number of technical and economic uncertainties. Three reservoir case studies were used: a simple homogeneous model, the PUNQ-S3 model, and a modified portion of the SPE10 model. The results show that using the sampling techniques of the probabilistic collocation method produced relatively accurate responses compared with the original method. Different possible enhancements were discussed in order to practically adapt the least squares probabilistic collocation method to more realistic and complex reservoir models. Furthermore, it is desired to perform the method to evaluate high-dimensional decision scenarios for different chemical enhanced oil recovery processes using real reservoir data.

Uncertainty in surfactant-polymer flooding is an important challenge to the wide scale implementation of this process. Any successful design of this enhanced oil recovery process will necessitate a good understanding of uncertainty. Thus it is essential to have the ability to quantify this uncertainty in an efficient manner. Monte Carlo Simulation is the traditional uncertainty quantification approach that is used for quantifying parametric uncertainty. However, the convergence of Monte Carlo simulation is relatively low requiring a large number of realizations to converge. This study proposes the use of the probabilistic collocation method in parametric uncertainty quantification for surfactant-polymer flooding using four synthetic reservoir models. Four sources of uncertainty were considered: the chemical flood residual oil saturation, surfactant and polymer adsorption and the polymer viscosity multiplier. The output parameter approximated is the recovery factor. The output metrics were the probability density function and the first two moments. These were compared with the results obtained from Monte Carlo simulation over a large number of realizations. Two methods for solving for the coefficients of the output parameter polynomial chaos expansion are compared: Gaussian quadrature and linear regression. The linear regression approach used two types of sampling: Gaussian quadrature nodes and Chebyshev derived nodes. In general, the probabilistic collocation method was applied successfully to quantify the uncertainty in the recovery factor. Applying the method using Gaussian quadrature produced more accurate results compared with using linear regression with quadrature nodes. Applying the method using linear regression with Chebyshev derived sampling also performed relatively well. Possible enhancements to improve the performance of the probabilistic collocation method were discussed. These enhancements include: improved sparse sampling, approximation order independent sampling and using arbitrary random input distribution that could be more representative of reality.

Reliability sensitivity analysis (SA) is a method to identify the relationship between the change in reliability and the change in the characteristics of uncertain variables. Very few methods can be applied for the reliability SA for long-term degeneracy mechanism, especially when the time-dependent limit state function regarding the interested performance is implicit. This paper proposes a new method to compute the reliability sensitivity measures. First, a surrogate model called time-dependent polynomial chaos expansion (PCE) is employed to approximate the uncertain output of the model and the extensive probabilistic collocation method (PCM) and moving least squares (MLS) method-based algorithm is proposed to compute the coefficients of the time-dependent PCE. Then, the explicit time-dependent limit state function can be obtained and the reliability sensitivity measures can be calculated straightforwardly. Finally, this approach is applied to an engineering case and the time-dependent reliability sensitivity measures are obtained for the long-term degeneracy mechanism model.

The characterization of flow in subsurface porous media is associated with high uncertainty. To better quantify the uncertainty of groundwater systems, it is necessary to consider the model uncertainty. Multi-model uncertainty analysis can be performed in the Bayesian model averaging (BMA) framework. However, the BMA analysis via Monte Carlo method is time consuming because it requires many forward model evaluations. A computationally efficient BMA analysis framework is proposed by using the probabilistic collocation method to construct a response surface model, where the log hydraulic conductivity field and hydraulic head are expanded into polynomials through Karhunen–Loeve and polynomial chaos methods. A synthetic test is designed to validate the proposed response surface analysis method. The results show that the posterior model weight and the key statistics in BMA framework can be accurately estimated. The relative errors of mean and total variance in the BMA analysis results are just approximately 0.013% and 1.18%, but the proposed method can be 16 times more computationally efficient than the traditional BMA method.

This Analysis/Model Report (AMR) documents the development of an expected-case model for unsaturated zone (UZ) flow and transport that will be described in terms of the representativeness of models of the natural system. The expected-case model will provide an evaluation of the effectiveness of the natural barriers, assess the impact of conservatism in the Total System Performance Assessment (TSPA), and support the development of further models and analyses for public confidence building. The present models used in ''Total System Performance Assessment for the Site Recommendation'' (Civilian Radioactive Waste Management System Management and Operating Contractor (CRWMS M&O) 2000 [1532461]) underestimate the natural-barrier performance because of conservative assumptions and parameters and do not adequately address uncertainty and alternative models. The development of an expected case model for the UZ natural barrier addresses issues regarding flow-pattern analysis and modeling that had previously been treated conservatively. This is in line with the Repository Safety Strategy (RSS) philosophy of treating conservatively those aspects of the UZ flow and transport system that are not important for achieving regulatory dose (CRWMS M&O 2000 [153246], Section 1.1.1). The development of an expected case model for the UZ also provides defense-in-depth in areas requiring further analysis of uncertainty and alternative models. In general, the value of the conservative case is to provide a more easily defensible TSPA for behavior of UZ flow and transport processes at Yucca Mountain. This AMR has been prepared in accordance with the ''Technical Work Plan for Unsaturated Zone (UZ) Flow and Transport Process Model Report'' (Bechtel SAIC Company (BSC) 2001 [155051], Section 1.3 - Work Package 4301213UMG). The work scope is to examine the data and current models of flow and transport in the Yucca Mountain UZ to identify models and analyses where conservatism may be reduced and uncertainty may be better characterized either through new analyses or development of new, alternative, models. This AMR supports Site Recommendation (SR) activities by documenting analyses and models reported in ''FY01 Supplemental Science and Performance Analyses, Volume 1: Scientific Bases and Analyses'' (BSC 2001 [155950]). However, except for the model results reported in Section 6.4.2, this AMR does not support TSPA as documented in either ''FY01 Supplemental Science and Performance Analyses, Volume 2: Performance Analyses'' (BSC 2001 [1546591]) or ''Total System Performance Assessment for the Site Recommendation'' (CRWMS M&O 2000 [153246]). Furthermore, this AMR represents scoping studies to aid in determining whether work beyond the scope of the models and analyses used to support TSPA-SR (CRWMS M&O 2000 [153246]) is warranted. If the decision is made to use any of these models and analyses for TSPA for License Application, further verification and validation will be required. The model results reported in Section 6.4.2 are used for TSPA sensitivity studies as documented in ''FY01 Supplemental Science and Performance Analyses, Volume 2: Performance Analyses'' (BSC 2001 [154659], Section 3.2.2.3).

The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications

In this work a design optimization tool to obtain robust optimum designs of trusses under nonlinear conditions is described and implemented. The robustness measures con- sidered here are the expected value and standard deviation of the function involved in the optimization problem. To calculate such quantities, we employ two nonintrusive uncertainty propagation analysis techniques that exploit deterministic computer models: Monte Carlo (MC) method and Probabilistic Collocation Method (PCM). When using these robustness measures combined, the search of optimal design appears as a robust multi-objective opti- mization (RMO) problem. To overcome the time consuming problem inherent in a RMO prob- lem a model reduction technique using the proper orthogonal decomposition (POD) method will be employed to provide fast outputs for nonlinear analysis of trusses. A structural sizing optimization (SSO) algorithm incorporating such procedure in the structural and sensitiv- ity stochastic analyses will be used to obtain efficient optimal trusses design. Optimization studies will be conducted for trusses problems considering different loads level, exploring the material plasticity. Comparisons will be conducted with the SSO approach via traditional FEM and via POD.

This paper performs Robust Design Optimization (RDO) to obtain optimum solutions since some degree of uncertainty in characterizing any real engineering system is inevitable. The robustness measures considered here are the expected value and standard deviation of the function involved in the optimization problem. To calculate such quantities, we employ two nonintrusive uncertainty propagation analysis techniques that exploit deterministic computer models: Monte Carlo (MC) method and Probabilistic Collocation Method (PCM). The uncertainty propagation essentially involves computing the statistical moments of the output. When using these robustness measures combined, the search for optimal design appears as a robust multiobjetive optimization (RMO) problem. Several strategies are implemented to obtain the Pareto front (multiobjective solutions). To overcome the time consuming problem inherent in a RMO problem reduced basis (RB) approximation methodology is added to the optimization system, in the whole optimization process. The integration of all the methodologies described allows the computation of robust design, using a finite element model of 3.900 degrees of freedom, in a practical time (less than a minute).

This paper explores the feasibility of using the Probabilistic Collocation Method (PCM) in power-system studies in the presence of uncertainties. The application of the PCM is illustrated on small disturbance stability studies and voltage stability studies. First, the PCM is used to determine the effects of a supplementary power oscillation damping (POD) controller installed on a VSC-HVDC line in the presences of parameter and operational uncertainties. Second, it is used to calculate the probabilistic distance to voltage collapse in a distribution system based on measurement errors. The PCM is shown to produce accurate statistical results whilst drastically reducing the computational time of the studies in both cases. Additionally, a ranking process based on eigenvalue sensitivity is presented alongside the small disturbance analysis as an appropriate method for reduction of system complexity for extension to large power systems. This technique is validated through the application of the PCM on a large power-system model, yielding accurate probabilistic results.