Sampling efficiency in Monte Carlo based uncertainty propagation strategies: Application in seawater intrusion simulations

Efficiency enhancement of optimized Latin hypercube sampling strategies: Application to Monte Carlo uncertainty analysis and meta-modeling

Cost-effective and efficient sampling methods are of main concern in many social, biological and environmental studies. In this article, an efficient sampling scheme, named manipulation-based ranked set sampling (MBRSS) scheme is introduced with its properties for estimating population mean and median. The MBRSS is a mixture of simple random sampling (SRS), ranked set sampling (RSS) and median ranked set sampling (MRSS) schemes and is applicable in the situation when ordinary RSS cannot be conducted. It is shown that the proposed scheme provides unbiased mean estimator provided underlying distribution is symmetric. For asymmetric distributions, a weighted mean is proposed, where optimal weights are computed using Shannon's entropy. Monte Carlo simulation is used to ascertain effectiveness of the proposed mean and median estimators in the presence of outliers. We also compared the efficiency of MBRSS and truncation-based ranked set sampling (TBRSS) scheme with respect to SRS under the situation of perfect and imperfect ranking i.e error in rankings with respect to variable of interest. It is observed, on the basis of theoretical and numerical studies that MBRSS is more efficient than SRS. Further, a real data set is used to illustrate the proposed MBRSS scheme.

Progressive Latin Hypercube Sampling: An efficient approach for robust sampling-based analysis of environmental models

An efficient soil moisture sampling scheme for the improvement of remotely sensed soil moisture validation over an agricultural field

A surrogate model based groundwater optimization model was developed to solve the non-aqueous phase liquids (NAPLs) contaminated groundwater remediation optimization problem. To illustrate the impact of sampling method improvement to the surrogate model performance improvement, aiming at a nitrobenzene contaminated groundwater remediation problem, optimal Latin hypercube sampling (OLHS) method was introduced to sample data in the input variables feasible region, and a radial basis function artificial neural network was used to construct a surrogate model. Considering the surrogate model's uncertainty, a chance-constrained programming (CCP) model was constructed, and it was solved by genetic algorithm. The results showed the following, for the problem considered in this study. (1) Compared with the Latin hypercube sampling (LHS) method, the OLHS method improves the space-filling degree of sample points considerably. (2) The effects of the two sampling methods on surrogate model performance were analyzed through comparison of goodness of fit, residual and uncertainty. The results indicated that the OLHS-based surrogate model performed better than the LHS-based surrogate model. (3) The optimal remediation strategies at 99%, 95%, 90%, 85%, 80% and 50% confidence levels were obtained, which showed that the remediation cost increased with the confidence level. This work would be helpful for increasing surrogate model performance and lowering the risk of a groundwater remediation strategy.

To Sobol or not to Sobol? The effects of sampling schemes in systems biology applications

Simulation uncertainty determination of single flank rolling tests using monte carlo simulation and skin model shapes for zero defect manufacturing of micro gears

Uncertainty is endemic in geospatial data due to the imperfect means of recording, processing, and representing spatial information. Propagating geospatial model inputs inherent uncertainty to uncertainty in model predictions is a critical requirement in each model's impact assessment and risk-conscious policy decision-making. It is still extremely difficult, however, to perform in practice uncertainty analysis of model outputs, particularly in complex spatially distributed environmental models, partially due to computational constraints.In the field of groundwater hydrology, the "stochastic revolution" has produced an enormous number of theoretical publications and greatly influenced our perspective on uncertainty and heterogeneity; it has had relatively little impact, however, on practical modeling. Monte Carlo simulation using simple random (SR) sampling from a multivariate distribution is one of the most widely used family of methods for uncertainty propagation in hydrogeological flow and transport model predictions, the other being analytical propagation.Real-life hydrogeological problems however, consist of complex and non-linear three dimensional groundwater models with millions of nodes and irregular boundary conditions. The number of Monte-Carlo runs required in these cases, depends on the number of uncertain parameters and on the relative accuracy required for the distribution of model predictions. In the context of sensitivity studies, inverse modelling or Monte-Carlo analyses, the ensuing computational burden is usually overwhelming and computationally impractical. These tough computational constrains have to be relaxed and removed before meaningful stochastic groundwater modeling applications are possible.A computationally efficient alternative to classical Monte Carlo simulation based on SR sampling is Latin hypercube (LH) sampling, a form of stratified random sampling. The latter yields a more representative distribution of model outputs (in terms of smaller sampling variability of their statistics) for the same number of input simulated realizations. The ability to generate unbiased LH realizations becomes critical in a spatial context, where random variables are geo-referenced and exhibit spatial correlation, to ensure unbiased outputs of complex models. On this regard, this dissertation offers a detailed analysis of LH sampling and compares it with SR sampling in a hydrogeological context. Additionally, two alternative stratified sampling methods, here named stratified likelihood (SL) sampling and minimum energy (ME) sampling, are examined (proposed in a spatial context) and their efficiency is further compared to SR and LH in a hydrogeological context; also accounting for the uncertainty related to the particular model at hand via a two step sampling method. All three stratified sampling methods (accounting for model sensitivity in the second case study) were found in this work to be more efficient than simple random sampling.Additionally, this thesis proposes a novel method for the expansion of the application domain of LH sampling to very large regular grids which is the common case in environmental (hydrogeological or not) models. More specifically, a novel combination of Stein's Latin Hypercube sampling with a Monte Carlo simulation method applicable over high discretization domains is proposed, and its performance is further validated in 2D and 3D hydrogeological problems of flow and transport in a mid-heterogeneous porous media, both consisting of about $1$ million nodes. Last, an additional novel extension of the proposed LH sampling on large grids is adopted for conditional high discretized problems. In this case too, the performance of the proposed approach is evaluated in a 3D hydrogeological model of flow and transport. Results indicate that both extensions (conditional and not) of LH sampling on large grids facilitate efficient uncertainty propagation with fewer model runs due to more representative model inputs. Overall, it could be argued that all the proposed methodological approaches could reduce the time and computer resources required to perform uncertainty analysis in hydrogeological flow and transport problems. Additionally, since it is the first time that stratified sampling is performed over high discretization domains, it could be argued that the proposed extensions of LH sampling on large grids could be considered a milestone for future uncertainty analysis efforts. Moreover, all the proposed stratified methods could contribute to a wider application of uncertainty analysis endeavors in a Monte Carlo framework for any spatially distributed impact assessment study.

An efficient sampling scheme: Updated Latin Hypercube Sampling

The application of Latin Hypercube and Monte Carlo (MC) sampling techniques for ground motion selection purposes is investigated. Latin Hypercube Sampling (LHS) works by first stratifying a probability distribution domain into multiple equally spaced and non-overlapping stripes and then by permutationally drawing samples from those stripes. To examine the efficiency of these two distinct sampling methods, a set of conditional multivariate distributions was fit to an intensity measure vector based on a single, two, or average of more-than-two (average) conditioning intensity measure. LHS was then utilized for sampling purposes from the conditional multivariate distributions, which in turn demonstrated superiority over MC given the same number of realization samples. Accordingly, it was utilized as an underlying peace of a broader ground motion selection framework to facilitite the selection of a number of ground motion suites based on different methods of conditioning. Using the selected suites, response history and subsequent damage/loss analyses were conducted on a generic 4-story non-ductile reinforced concrete building. The outcomes of these latter studies demonstrated that the ground motion suite selected based on an average-intensity-measure conditioning criterion performed better than those selected through single- and two-intensity-measure conditioning criteria.

Understanding and optimizing complex design problems involves analyzing mathematical models that simulate real-world systems. When such simulations require an enormous amount of time to evaluate a design point, approximation models are created that are simpler and quicker and allow exploration of the design space. Approximation models are also referred to surrogate models or meta-models. Kriging, response surface models, neural networks such as radial basis functions are examples of such approximation methods. The quality of approximations obtained using these methods is dependent upon: (1) the type of approximation technique, (2) the distribution of sample points, namely the sampling strategy, (3) number of sample points available for approximation and (4) the topography of the function being approximated. The quality and performance of an approximation method are measured using prediction errors and time taken to fit respectively. In this work, we present a study of the quality and performance of polynomial regression (general and orthogonal polynomials), Radial basis functions (RBF), Elliptical basis functions (EBF), and Kriging (ordinary and blind) for functions with different topographies. Specifically, the objective of this benchmark is to study the following effects on the accuracy of approximation techniques for comparison: 1. Effect of sampling strategy, 2. Effect of number of points, 3. Effect of function topography. Results indicate that Kriging approximation is best suited for cases where there are few sample points that are uniformly spaced (typically from Optimal Latin Hypercube sampling). Elliptical or Radial Basis Function neural networks are fast, robust and accurate enough to be used for any sampling strategy. For factorial sampling, Chebyshev polynomials have higher accuracy compared to simple polynomial regression.

Automated optimal experimental design strategy for reduced order modeling of aerodynamic flow fields

The generalization of Latin hypercube sampling

In complex engineering systems, approximation models, also called metamodels, are extensively constructed to replace the computationally expensive simulation and analysis codes. With different sample data and metamodeling methods, different metamodels can be constructed to describe the behavior of an engineering system. Then, metamodel uncertainty will arise from selecting the best metamodel from a set of alternative ones. In this study, a method based on Bayes’ theorem is used to quantify this metamodel uncertainty. With some mathematical examples, metamodels are built by six metamodeling methods, i.e., polynomial response surface, locally weighted polynomials (LWP), k-nearest neighbors (KNN), radial basis functions (RBF), multivariate adaptive regression splines (MARS), and kriging methods, and under four sampling methods, i.e., parameter study (PS), Latin hypercube sampling (LHS), optimal LHS and full factorial design (FFD) methods. The uncertainty of metamodels created by different metamodeling methods and under different sampling methods is quantified to demonstrate the process of implementing the method.

We present efficient localization aware sampling and connection strategies for incremental sampling-based stochastic motion planners. For sampling, we introduce a new measure of localization ability of a sample, one that is independent of the path taken to reach the sample and depends only on the sensor measurement at the sample. Using this measure, our sampling strategy puts more samples in regions where sensor data is able to achieve higher uncertainty reduction while maintaining adequate samples in regions where uncertainty reduction is poor. This leads to a less dense roadmap and hence results in significant time savings. We also show that a stochastic planner that uses our sampling strategy is probabilistically complete under some reasonable conditions on parameters. We then present a localization aware efficient connection strategy that uses an uncertainty aware approach in connecting the new sample to the neighbouring nodes, i.e., it uses an uncertainty measure (as opposed to distance) to connect the new sample to a neighboring node so that the new sample is reachable with least uncertainty ("the closest"), and furthermore, connections to other neighbouring nodes are made only if the new path to them (via the new sample) helps to reduce the uncertainty at those nodes. This is in contrast to current incremental stochastic motion planners that simply connect the new sample to all of the neighbouring nodes and therefore, take more search queue iterations to update the paths (i.e., uncertainty propagation). Hence, our efficient connection strategy, in addition to eliminating the inefficient edges that do not contribute to better localization, also reduces the number of search queue iterations. We provide simulation results that show that (a) our localization aware sampling strategy places less samples and find a well-localized path in shorter time with little compromise on the quality of path as compared to existing sampling techniques, (b) our localization aware connection strategy finds a well-localized path in shorter time with no compromise on the quality of path as compared to existing connection techniques, and finally (c) combined use of our sampling and connection strategies further reduces the planner run time.