Robustness of Maximin Space-filling Designs

In this report, we first have a review of the maximin space-filling design methods that is often applied and discussed in the literature (for example, Müller (2007)). Then we will discuss the robustness of the maximin space-filling design against model misspecification via numerical simulation. For this purpose, we will generate spatial data sets on a n x n grid and design points are selected from the n2 locations. The predictions at the unsampled locations are made based on the observations at these design points. Then the mean of the squared prediction errors are estimated as a measure of the robustness of the designs against possible model misspecification. Surprisingly, according to the simulation results, we find that the maximin space-filling designs may be robust against possible model misspecification in the sense that the mean of the squared prediction error does not increase significantly when the model is misspecified. Although the results were obtained based on simple models, this result is very inspiring. It will guide further numerical and theoretical studies which will be done as future work.

MP-CE Method for Space-Filling Design in Constrained Space with Multiple Types of Factors

Space-filling design selects points uniformly in the experimental space, bringing considerable flexibility to the complex-model-based and model-free data analysis. At present, space-filling designs mostly focus on regular spaces and continuous factors, with a lack of studies into the discrete factors and the constraints among factors. Most of the existing experimental design methods for qualitative factors are not applicable for discrete factors, since they ignore the potential order or spatial distance between discrete factors. This paper proposes a space-filling method, called maximum projection coordinate-exchange (MP-CE), taking into account both the diversity of factor types and the complexity of factor constraints. Specifically, the maximum projection criterion and distance criterion are introduced to capture the “bad” coordinates, and the coordinate-exchange and the optimization of experimental design are realized by solving one-dimensional constrained optimization problem. Meanwhile, by adding iterative perturbations to the traditional coordinate exchange process, the adjacent areas of the local optimal solution are explored and the optimum performances of the current optimal solution are retained, while the shortcomings of random restart are effectively avoided. Experiments in the regular space and constraint space, as well as experimental design for the terminal interception effectiveness of a missile defense system, show that the MP-CE method significantly outperforms existing popular space-filling design methods in terms of space-projection properties, while yielding comparable or superior space-filling properties.

Management of Uncertainty in Reservoir Modelling for an Onshore Carbonate Reservoir in the Middle East: A Case Study

As part of a reservoir modelling study for an onshore oil field in the Middle East, our study implemented a workflow with the objective to evaluate the impact of uncertainty on the long-term development scenario. The presence of several geological uncertainties characterized the field: many faults with uncertainty in juxtaposition and conductivity, lateral distribution of permeability in high permeability layers, and uncertainty on the rock typing. A deterministic geological model was available. There were also many dynamic uncertainties. The workflow started with an identification of uncertain variables, both from the static and the dynamic point of view, through an integrated team approach supported by a previous reservoir synthesis (Major Field Review). Subsequently, a screening analysis allowed identifying the relative impact of uncertain variables. After selecting the uncertainties with the largest impact on recovery, use of an experimental design methodology with a space-filling design resulted in alternative history matches. Statistical analysis of forecasts yielded probability density functions and low and high estimates of ultimate recovery. Forty-five uncertain variables, including both static and dynamic uncertainties, characterized the production profiles. Screening allowed reducing these to 11 main uncertain variables. A Wootton, Sergent, Phan-Tan-Luu (WSP) space-filling design yielded 162 simulation runs. Only five out of these corresponded to acceptable history matches. This number being statistically insignificant, a reexamination of the uncertainty ranges followed by a narrowing, allowed obtaining 45 history matches (out of 198 runs). The obtained spread in the cumulative oil production was narrow, with a slightly skewed distribution around the base case (closer to P90 than to P10). The study resulted in an estimation of final uncertainty in reserves that is smaller than the typical uncertainty found in post-mortem analysis of oil field development projects. Other reservoir studies in the company and in literature, employing a similar workflow, yielded outcomes with a similar bias. To tackle this issue, as a way forward we suggest history matching of multiple geological scenarios, either with multiple deterministic cases (min, base, max) or with an ensemble history matching loop including structural model generation, in-filling, and dynamic parameter uncertainty.

Fast and simple dataset selection for machine learning

The task of data reduction is discussed and a novel selection approach which allows to control the optimal point distribution of the selected data subset is proposed. The proposed approach utilizes the estimation of probability density functions (pdfs). Due to its structure, the new method is capable of selecting a subset either by approximating the pdf of the original dataset or by approximating an arbitrary, desired target pdf. The new strategy evaluates the estimated pdfs solely on the selected data points, resulting in a simple and efficient algorithm with low computational and memory demand. The performance of the new approach is investigated for two different scenarios. For representative subset selection of a dataset, the new approach is compared to a recently proposed, more complex method and shows comparable results. For the demonstration of the capability of matching a target pdf, a uniform distribution is chosen as an example. Here the new method is compared to strategies for space-filling design of experiments and shows convincing results.

Evaluation of zero-dimensional stochastic reactor modelling for a Diesel engine application

Prediction of engine-out emissions with high fidelity from in-cylinder combustion simulations is still a significant challenge early in the engine development process. This article contributes to this fast evolving body of knowledge by focusing on the evaluation of NO x emission prediction capability of a probability density function–based stochastic reactor engine models for a Diesel engine. The research implements a systematic approach to the study of the stochastic reactor engine model performance, underpinned by a detailed space-filling design of experiments (DoE)-based sensitivity analysis of both external and internal parameters, evaluating their effects on the accuracy in matching physical measurements of both in-cylinder conditions and NO x output. The approach proposed in this article introduces an automatic stochastic reactor engine model calibration methodology across the engine operating envelope, based on a multi-objective optimization approach. This aims to exploit opportunities for internal stochastic reactor engine model parameters tuning to achieve good overall modelling performance as a trade-off between physical in-cylinder measurements accuracy and the output NO x emission predictions error. The results from the case study provide a valuable insight into the effectiveness of the stochastic reactor engine model, showing good capability for NO x emissions prediction and trends, while pointing out the critical sensitivity to the external input parameters and modelling conditions.