Estimating Expected Information Gains for Experimental Designs With Application to the Random Fatigue-Limit Model

Expected gain in Shannon information is commonly suggested as a Bayesian design evaluation criterion. Because estimating expected information gains is computationally expensive, examples in which they have been successfully used in identifying Bayes optimal designs are both few and typically quite simplistic. This article discusses in general some properties of estimators of expected information gains based on Markov chain Monte Carlo (MCMC) and Laplacian approximations. We then investigate some issues that arise when applying these methods to the problem of experimental design in the (technically nontrivial) random fatigue-limit model of Pascual and Meeker. An example comparing follow-up designs for a laminate panel study is provided.     

Simulation-Based Optimal Design Using a Response Variance Criterion

Classical optimal design criteria suffer from two major flaws when applied to nonlinear problems. First, they are based on linearizing the model around a point estimate of the unknown parameter and therefore depend on the uncertain value of that parameter. Second, classical design methods are unavailable in ill-posed estimation situations, where previous data lack the information needed to properly construct the design criteria. Bayesian optimal design can, in principle, solve these problems. However, Bayesian design methods are not widely applied, mainly due to the fact that standard implementations for efficient and robust routine use are not available. In this article, we point out a concrete recipe for implementing Bayesian optimal design, based on the concept of simulation-based design introduced by Muller, Sanso, and De Iorio (2004 Muller, P., Sanso, B. and De Iorio, M. 2004. Optimal Bayesian Design by Inhomogeneous Markov Chain Simulation. Journal of the American Statistical Association, 99(467): 788–798. [Taylor & Francis Online] , [Google Scholar]). We develop further a predictive variance criterion and introduce an importance weighting mechanism for efficient computation of the variances. The simulation-based approach allows one to start the model-based optimization of experiments at an early stage of the parameter estimation process, in situations where the classical design criteria are not available. We demonstrate that the approach can significantly reduce the number of experiments needed to obtain a desired level of accuracy in the parameter estimates. A computer code package that implements the approach in a simple case is provided as supplemental material (available online).     

A generalized structured doubling algorithm for the numerical solution of linear quadratic optimal control problems

We propose a generalization of the structured doubling algorithm to compute invariant subspaces of structured matrix pencils that arise in the context of solving linear quadratic optimal control problems. The new algorithm is designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because the stable and unstable invariant subspaces are not well separated (because of eigenvalues near or on the imaginary axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence of the method and compare the new method with the classical structured doubling algorithm as well as some structured QR methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Playing with models and optimization to overcome the tragedy of the commons in groundwater

Groundwater is the natural resource most extracted in the world. It supplies 50% of the total potable water requirements, 40% of the industry take, and 20% of agriculture groundwater is a strategic resource for every country. That common‐pool resources are highly susceptible to lead to a tragedy of the commons is a well‐known fact. We claim that a combination of groundwater modeling, optimization, and a game theoretical analysis may in fact avoid the tragedy. A groundwater model in MODFLOW from Zamora aquifer in Mexico was used as input of a basic but instructive, optimization problem: extract the greatest possible volume of water, but at the same time minimizing the drawdown and drawdown velocity. The solutions of the optimization problem were used to construct the payoffs of a hypothetical game among the aquifer users, the resource's administrator, and a resource protector entity. We show that the success of the optimal management program depends heavily on the information that the users have about the resource. Therefore, better decision‐making processes are a consequence of sustainability literacy. Particularly, water literacy could lead to the usage of water considering it as a part of an ecosystem and not only as a natural resource. Additionally, a new non‐classical equation for underground flow was derived, that may be specially important to understand and predict the groundwater flow in highly heterogeneous conditions as in karstic aquifers or fractured media. © 2013 Wiley Periodicals, Inc. Complexity 19: 9–21, 2013     

Optimal Market Making in the Foreign Exchange Market

Abstract

This paper is concerned with optimal market making in the foreign exchange market. The market maker's holdings in the different currencies are modelled as stochastic processes that are influenced by both the stochastic exchange rates and the stochastic customer buy and sell orders. The market maker can control their own bid and ask price quotes and, additionally, can buy and sell at other market participants' quotes. The resulting stochastic control problem consists of a controlled diffusion problem for the optimal quotes and a singular control problem for optimal trades at other market participants' quotes. A Markov chain approximation is used to derive optimal strategies.     

Spatiotemporal dynamics and feedforward control of an infinite-dimensional,time-variant,thermal-coating process

In this contribution, a model for the spatiotemporal dynamics of a thermal-coating process is derived from first principles (the conservation of energy). The model is a dynamical system consisting of a system of hyperbolic, partial differential equations (PDEs), describing the evolution of the temperature distribution of the substrate. By studying the C ₀- semigroup generated by the system operator we find that in discrete time the infinite-dimensional plant may conveniently be represented by finite-dimensional operators defined on appropriately chosen Euclidean spaces. This representation provides the basis for numerical efficient solution of several optimal feedforward control problems associated with set point changes and launching of the process. Numerical and experimental studies highlight the value of this approach.     

Mathematical analysis of an influenza epidemic model,formulation of different controlling strategies using optimal control and estimation of basic reproduction number

In this article, a deterministic model is formulated to perform a thorough investigation of the transmission dynamics of influenza. In particular, our model takes into account the effects of medication as well as hospitalization. An in-depth stability analysis of the model is performed, and it is subsequently shown that the model is locally, as well as globally asymptotically stable, when R₀ > 1. It is also shown that there exists a unique endemic equilibrium whenever R₀ > 1. After estimating the effective contact rate, we estimate the basic reproduction number, using both an ordinary least squares and generalized least squares methodology. We also estimated confidence intervals for the effective contact rate using parametric bootstrapping. Furthermore, we perform uncertainty and sensitivity analysis to recognize the impact of crucial model parameters on R₀. In addition, using ideas from the optimal control theory, optimal medication and hospitalization strategies are proposed to eliminate the disease.     

制造商检验的最优质量水平及影响因素研究

为提高我国食品质量安全水平,对小型、分散的单个供应商和一个制造商构成的食品供应链中,制造商对原材料和最终食品进行质量检测的情况,建立了以供应商和制造商各自的质量水平为决策变量的优化模型,分析了最优质量水平的影响因素和影响效应.并依据结论提出相应建议,如提高消费者对优劣食品的识别能力,发挥制造商在供应链中的监督检验作用等.