The configurations of instruments fielded on an experiment affect the amount of information captured and the quality of subsequent inference. We investigate the problem of optimizing plasma x-ray radiation detectors in a magneto-inertial fusion experiment at Sandia National Laboratories. It is impossible to directly measure properties such as the temperature of the thermonuclear fusion plasma produced in these experiments because of the extreme environment and destructive nature of the experiment. Among other diagnostics, several detectors are placed with significant standoff from the fusion target to capture the x-rays emitted by the fusion plasma, which can be used to infer some of its properties. To optimize the configuration of these detectors, a high-fidelity model (HFM) is used for simulating outputs and a low-fidelity model (LFM) is used for inference. We develop methods based on A- and L-optimality criteria that are efficient to compute while explicitly accounting for the discrepancy between the HFM and the LFM. The method allows us to find detector configurations that perform similarly to or better than the configuration obtained using an existing sampling-based optimization method while decreasing computational time by a factor of 50. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Computer experiments involving both qualitative and quantitative (QQ) factors have attracted increasing attention. Gaussian process (GP) models have proven effective in this context by choosing specialized covariance functions for QQ factors. In this work, we extend the latent variable-based GP approach, which maps qualitative factors into a continuous latent space, by establishing a general framework to apply standard kernel functions to continuous latent variables. This approach provides a novel perspective for interpreting some existing GP models for QQ factors and introduces new covariance structures in some situations. The ordinal structure can be incorporated naturally and seamlessly in this framework. Furthermore, the Bayesian information criterion and leave-one-out cross-validation are employed for model selection and model averaging. The performance of the proposed method is comprehensively studied on several examples.

Gaussian processes (GPs) are broadly used as surrogate models for expensive computer simulators of complex phenomena. However, a key bottleneck is that its training data are generated from this expensive simulator and thus can be highly limited. A promising solution is to supplement the learning model with boundary information from scientific knowledge. However, despite recent work on boundary-integrated GPs, such models largely cannot accommodate boundary information on irregular (i.e., non-hypercube) domains, and do not provide sample path smoothness control or approximation error analysis, both of which are important for reliable surrogate modeling. We thus propose a novel BdryMatérn GP modeling framework, which can reliably integrate Dirichlet, Neumann and Robin boundaries on an irregular connected domain with a boundary set that is twice-differentiable almost everywhere. Our model leverages a new BdryMatérn covariance kernel derived in path integral form via a stochastic partial differential equation formulation. Similar to the GP with Matérn kernel, we prove that sample paths from the BdryMatérn GP satisfy the desired boundaries with smoothness control on its derivatives. We further present an efficient approximation procedure for the BdryMatérn kernel using finite element modeling with rigorous error analysis. Finally, we demonstrate the effectiveness of the BdryMatérn GP in a suite of numerical experiments on incorporating broad boundaries on irregular domains.

The Expected Improvement (EI) method, proposed by Jones et al. (1998), is a widely-used Bayesian optimization method, which makes use of a fitted Gaussian process model for efficient black-box optimization. However, one key drawback of EI is that it is overly greedy in exploiting the fitted Gaussian process model for optimization, which results in suboptimal solutions even with large sample sizes. To address this, we propose a new hierarchical EI (HEI) framework, which makes use of a hierarchical Gaussian process model. HEI preserves a closed-form acquisition function, and corrects the over-greediness of EI by encouraging exploration of the optimization space. We then introduce hyperparameter estimation methods which allow HEI to mimic a fully Bayesian optimization procedure, while avoiding expensive Markov-chain Monte Carlo sampling steps. We prove the global convergence of HEI over a broad function space, and establish near-minimax convergence rates under certain prior specifications. Numerical experiments show the improvement of HEI over existing Bayesian optimization methods, for synthetic functions and a semiconductor manufacturing optimization problem.

Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific interpretations, are difficult or even impossible to measure directly. Estimating these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations for numerical solutions to PDE through algorithms such as the finite element method, which can be time-consuming, especially for nonlinear PDEs. In this paper, we propose a novel method for the inference of unknown parameters in PDEs, called the PDE-Informed Gaussian Process (PIGP) based parameter inference method. Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that, under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transforms the nonlinear PDE into an equivalent PDE system linear in all derivatives, which our PIGP-based method can handle. The proposed method can be applied to a broad spectrum of nonlinear PDEs. The PIGP-based method can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. Like conventional Bayesian approaches, the method can provide uncertainty quantification for both the unknown parameters and the PDE solution. The PIGP-based method also completely bypasses the numerical solver for PDEs. The proposed method is demonstrated through several application examples from different areas.

In an era where scientific experimentation is often costly, multi-fidelity emulation provides a powerful tool for predictive scientific computing. While there has been notable work on multi-fidelity modeling, existing models do not incorporate an important "conglomerate" property of multi-fidelity simulators, where the accuracies of different simulator components are controlled by different fidelity parameters. Such conglomerate simulators are widely encountered in complex nuclear physics and astrophysics applications. We thus propose a new CONglomerate multi-FIdelity Gaussian process (CONFIG) model, which embeds this conglomerate structure within a novel non-stationary covariance function. We show that the proposed CONFIG model can capture prior knowledge on the numerical convergence of conglomerate simulators, which allows for cost-efficient emulation of multi-fidelity systems. We demonstrate the improved predictive performance of CONFIG over state-of-the-art models in a suite of numerical experiments and two applications, the first for emulation of cantilever beam deflection and the second for emulating the evolution of the quark-gluon plasma, which was theorized to have filled the Universe shortly after the Big Bang.

Computer simulators are widely used for the study of complex systems. In many applications, there are multiple simulators available with different scientific interpretations of the underlying mechanism, and the goal is to identify an optimal simulator based on the observed physical experiments. To achieve the goal, we propose a selection criterion based on leave-one-out cross-validation. This criterion consists of a goodness-of-fit measure and a generalized degrees of freedom penalizing the simulator sensitivity to perturbations in the physical observations. Asymptotic properties of the selected optimal simulator are discussed. It is shown that the proposed procedure includes a conventional calibration method as a special case. The finite sample performance of the proposed procedure is demonstrated through numerical examples. In the application of cell biology, an optimal simulator is selected, which can shed light on the T cell recognition mechanism in the human immune system. Supplementary materials for this article are available online.

The numerical accuracy of finite element analysis (FEA) depends on the number of finite elements used in the discretization of the space, which can be varied using the mesh size. The larger the number of elements, the more accurate the results are. However, the computational cost increases with the number of elements. In current practice, the experimenter chooses a mesh size that is expected to produce a reasonably accurate result, and for which the computer simulation can be completed in a reasonable amount of time. Improvements to this approach have been proposed using multifidelity modeling by choosing two or three mesh sizes. However, mesh size is a continuous parameter, and therefore, multifidelity simulations can be performed easily by choosing a different value for the mesh size for each of the simulations. In this article, we develop a method to optimally find the mesh sizes for each simulation and satisfy the same time constraints as a single or a double mesh size experiment. A range of different mesh sizes used in the proposed method allows one to fit multifidelity models more reliably and predict the outcome when meshes approach infinitesimally small, which is impossible to achieve in actual simulations. We illustrate our approach using an analytical function and a cantilever beam finite element analysis experiment.

Thompson sampling is a popular algorithm for tackling multi-armed bandit problems, and has been applied in a wide range of applications, from website design to portfolio optimization. In such applications, however, the number of choices (or arms) N can be large, and the data needed to make adaptive decisions require expensive experimentation. One is then faced with the constraint of experimenting on only a small subset of K≪N arms within each time period, which poses a problem for traditional Thompson sampling. We propose a new Thompson Sampling under Experimental Constraints (TSEC) method, which addresses this so-called “arm budget constraint”. TSEC makes use of a Bayesian interaction model with effect hierarchy priors, to model correlations between rewards on different arms. This fitted model is then integrated within Thompson sampling, to jointly identify a good subset of arms for experimentation and to allocate resources over these arms. We demonstrate the effectiveness of TSEC in two applications with arm budget constraints. The first is a simulated website optimization study, where TSEC shows considerable improvements over industry benchmarks. The second is a portfolio optimization application on industry-based exchange-traded funds, where TSEC provides more consistent and greater wealth accumulation over standard investment strategies.

Kriging (or Gaussian process regression) is a popular machine learning method for its flexibility and closed-form prediction expressions. However, one of the key challenges in applying kriging to engineering systems is that the available measurement data is scarce due to the measurement limitations and high sensing costs. On the other hand, physical knowledge of the engineering system is often available and represented in the form of partial differential equations (PDEs). We present in this work a PDE Informed Kriging model (PIK), which introduces PDE information via a set of PDE points and conducts posterior prediction similar to the standard kriging method. The proposed PIK model can incorporate physical knowledge from both linear and nonlinear PDEs. To further improve learning performance, we propose an Active PIK framework (APIK) that designs PDE points to leverage the PDE information based on the PIK model and measurement data. The selected PDE points not only explore the whole input space but also exploit the locations where the PDE information is critical in reducing predictive uncertainty. Finally, an expectation-maximization algorithm is developed for parameter estimation. We demonstrate the effectiveness of APIK in two synthetic examples, a shock wave case study, and a laser heating case study.

In this paper, we investigate Gaussian process regression with input location error, where the inputs are corrupted by noise. Here, we consider the best linear unbiased predictor for two cases, according to whether there is noise at the target untried location or not. We show that the mean squared prediction error does not converge to zero in either case. We investigate the use of stochastic Kriging in the prediction of Gaussian processes with input location error, and show that stochastic Kriging is a good approximation when the sample size is large. Several numeric examples are given to illustrate the results, and a case study on the assembly of composite parts is presented. Technical proofs are provided in the Appendix.

With the rapid development of metro systems, it has become increasingly important to study phenomena such as passenger flow distribution and passenger boarding behavior. It is difficult for existing methods to accurately describe actual situations and to extend to the whole metro system due to the limitations from parameter uncertainties in their mathematical models. In this article, we propose a passenger-to-train assignment model to evaluate the probabilities of individual passengers boarding each feasible train for both no-transfer and one-transfer situations. This model can be used to understand passenger flows and crowdedness. The input parameters of the model include the probabilities that the passengers take each train and the probability distribution of egress time, which is the time to walk to the tap-out fare gate after alighting from the train. We present the likelihood method to estimate these parameters based on data from the automatic fare collection and automatic vehicle location systems. This method can construct several nonparametric density estimates without assuming the parametric form of the distribution of egress time. The EM algorithm is used to compute the maximum likelihood estimates. Simulation results indicate that the proposed estimates perform well. By applying our method to real data in Beijing metro system, we can identify different passenger flow patterns between peak and off-peak hours.

Partial differential equations (PDEs) are widely used for description of physical and engineering phenomena. Some key parameters involved in PDEs, which represents certain physical properties with important scientific interpretations, are difficult or even impossible to be measured directly. Estimation of these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations of numerical solution of PDE through algorithms such as finite element method, which can be time-consuming especially for nonlinear PDEs. In this paper, we propose a novel method for estimating unknown parameters in PDEs, called PDE-Informed Gaussian Process Inference (PIGPI). Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transfers the nonlinear PDE into an equivalent PDE system linear in all derivatives that our PIGPI can handle. PIGPI can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. The method completely bypasses the numerical solver for PDE, thus achieving drastic savings in computation time, especially for nonlinear PDEs. Moreover, the PIGPI method can give the uncertainty quantification for both the unknown parameters and the PDE solution. The proposed method is demonstrated by several application examples from different areas.

Figure 2. Sneezing in a subway car. The sneezing occurs in the middle of the car, towards the right. The particles that are emitted by the sneezing passenger are colored according to the logarithm of the diameter, with red representing the largest particles and blue representing the smallest particles. Note the quick dispersion of particles due to the air conditioning ventilation; some particles enter the left portion of the subway car. The health implications are obvious. Figure courtesy Rainald Löhner and Harbir Antil.

Free Access Upper Percentiles of the F Distribution C. F. Jeff Wu, C. F. Jeff Wu Georgia Institute of Technology, Atlanta, USASearch for more papers by this authorMichael Hamada, Michael Hamada Los Alamos National Laboratory, New Mexico, USASearch for more papers by this author Book Author(s):C. F. Jeff Wu, C. F. Jeff Wu Georgia Institute of Technology, Atlanta, USASearch for more papers by this authorMichael Hamada, Michael Hamada Los Alamos National Laboratory, New Mexico, USASearch for more papers by this author First published: 09 January 2021 https://doi.org/10.1002/9781119470007.app4Book Series:Wiley Series in Probability and Statistics AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onFacebookTwitterLinked InRedditWechat Experiments: Planning, Analysis, and Optimization, Third Edition RelatedInformation

This chapter considers experiments with more than one treatment factor or with blocking. These include paired comparison designs, randomized block designs, two-way layouts with fixed and random effects, multi-way layouts, Latin square designs, Graeco-Latin square designs, balanced incomplete block designs, and split-plot designs. The chapter introduces analysis techniques like data transformation and analysis of covariance. The design employed in the sewage experiment is called a paired comparison design because a pair of treatments are compared in each of the eight samples. An experiment is considered to compare four methods for predicting the shear strength for steel plate girders. This experiment used a randomized block design, which has one experimental factor with k treatments and b blocks of size k. The chapter considers factors whose levels are random; the levels used in the experiment represent a random sample from a large or infinite population of levels.

Motivated by an analysis of single molecular experiments in the study of T-cell signaling, a new model called varying coefficient frailty model with local linear estimation is proposed. Frailty models have been extensively studied, but extensions to nonconstant coefficients are limited to spline-based methods that tend to produce estimation bias near the boundary. To address this problem, we introduce a local polynomial kernel smoothing technique with a modified expectation-maximization algorithm to estimate the unknown parameters. Theoretical properties of the estimators, including their unbiased property near the boundary, are derived along with discussions on the asymptotic bias-variance trade-off. The finite sample performance is examined by simulation studies, and comparisons with existing spline-based approaches are conducted to show the potential advantages of the proposed approach. The proposed method is implemented for the analysis of T-cell signaling. The fitted varying coefficient model provides a rigorous quantification of an early and rapid impact on T-cell signaling from the accumulation of bond lifetime, which can shed new light on the fundamental understanding of how T cells initiate immune responses.

This chapter considers the simplest class of experiments, that with a single factor. It presents modeling and analysis techniques like one-way fixed effects models and random effects models, analysis of variance (ANOVA), multiple comparisons, and residual analysis. The chapter derives expected mean squares in ANOVA tables for sample size determination. It demonstrates, by using a pulp experiment, that the corrected total sum of squares can be split up into two components: treatment sum of squares and residual sum of squares. The corresponding mean squares are obtained by dividing these sum of squares by their respective degrees of freedom. The chapter also provides a discussion on the assessment of model assumption.

This chapter presents practical uses of optimal design and considers optimal design for a specified run size known as exact optimal design. It also considers design criteria motivated by frequentist and Bayesian approaches. A series of examples shows how useful optimal design can be in practice to handle nonstandard design problems. The chapter then considers a design in N points, i.e., a finite set of points, which is referred to as an exact design. Orthogonal arrays with main effect models are D-optimal as shown by A. Dey and R. Mukerjee. In a similar manner, it can be shown that full-factorial designs with main effects and all interactions up to the highest order interaction (i.e., k-order interaction for a k-factor full factorial design) are D-optimal. The chapter includes point and coordinate exchange algorithms and Bayesian versions of them.

Global optimization of expensive functions has important applications in physical and computer experiments. It is a challenging problem to develop efficient optimization scheme, because each function evaluation can be costly and the derivative information of the function is often not available. We propose a novel global optimization framework using adaptive radial basis functions (RBF) based surrogate model via uncertainty quantification. The framework consists of two iteration steps. It first employs an RBF-based Bayesian surrogate model to approximate the true function, where the parameters of the RBFs can be adaptively estimated and updated each time a new point is explored. Then it utilizes a model-guided selection criterion to identify a new point from a candidate set for function evaluation. The selection criterion adopted here is a sample version of the expected improvement criterion. We conduct simulation studies with standard test functions, which show that the proposed method has some advantages, especially when the true function has many local optima. In addition, we also propose modified approaches to improve the search performance for identifying optimal points.