ITR/AP: Realistic Uncertainty Bounds for Complex Dynamic Models

目前社会面临的问题，如全球变暖，地震防备，核废料运输的安全性，以及汽车发动机的污染物排放，要求集成各种计算机程序，每个程序解决不同学科的数值问题。 对这些复杂模型的首要关注是它们的可靠性：可预测性、真实性和不确定性。 目前工作的重点是模拟现实的不确定性，从多响应，大规模，非线性动态模型的预测，使用一种新的策略来攻击这个problem.This工作重新审视与复杂的物理系统相关联的数学模型的概念，考虑实验和理论是一个不可分割的一部分的模型和治疗模型的不确定性参数作为内部，“状态”变量。 通过这种方式，实验和理论基础的不确定性被“直接”转化为模型预测的不确定性。 建立这种直接关系还可以解决相反的问题：确定哪些具体数据对预测不确定性贡献最大;确定使预测不确定性达到给定水平所需的实验精度;或者评估计划的实验是否能够改善预测的不确定性。这是通过将控制理论中的凸松弛与解决方案映射开发和应用于典型的化石燃料燃烧的化学动力学的数值模拟。 解决方案映射技术使用计算机实验的统计设计，以替代多项式模型取代复杂的ODE模型。 这些更简单但精确的代数模型更适合于数值优化。 凸松弛允许由多项式目标和多项式约束（通常为非凸的）描述的优化问题被凸优化攻击，即具有线性矩阵不等式约束的线性目标。 在鲁棒控制中近二十年的应用表明，这些松弛在各种各样的物理激励问题和应用中非常有用。在这种方法中，代理模型被开发为所有的响应，无论是从训练集还是从预测集。 每个代理模型表示为一个二次形式的内部模型参数，开发了一系列的直接ODE集成进行根据因子设计，涵盖了一个子空间的参数不确定性。 然后通过优化算法探索替代响应模型的二次形式。 在最初的努力中，在天然气燃烧模型中的不确定性的传播的问题是铸造的形式服从的S-程序，凸优化方法广泛应用于控制理论。 更复杂的凸松弛最近已经开发出来，集中在观察，确定一个给定的多项式是否是平方和（因此全局非负），可以转换为凸可行性问题，并在多项式时间（多项式的顺序）验证。 这项工作将探讨这种可能性，探索新的途径，数值经济的评估复杂的动态模型的现实误差范围。

The present problems facing society-such as global warming, earthquake preparedness, safety of transport of nuclear waste, and pollutant emission from automobile engines-call for integration of a variety of computer programs, each solving numerical problems in a different discipline. The overriding concern for such complex models is their reliability: predictability, authenticity, and uncertainty. The focus of the present effort is modeling realistic uncertainty in predictions from multi-response, large-scale, nonlinear dynamic models, using a new strategy to attack this problem.This work re-examines the concept of a mathematical model associated with complex physical systems, considering experiment and theory to be an integral part of the model and treating uncertain parameters of the model as internal, "state" variables. In this way, the uncertainties of the experimental and theoretical foundation are transferred "directly" into uncertainties of model predictions. Establishing this direct relationship allows one also to address the reverse problem: to identify which specific data contribute the most to the prediction uncertainty; to determine the required accuracy of an experiment to bring the prediction uncertainty to a given level; or to assess whether a planned experiment will be able to improve the prediction uncertainty.This is accomplished by merging convex relaxations from control theory with the technique of solution mapping developed and applied to numerical modeling of chemical kinetics typical of fossil-fuel combustion. The solution mapping technique uses statistical design of computer experiments to replace complex ODE models with surrogate polynomial models. These simpler, though accurate, algebraic models are more suited to numerical optimization. The convex relaxations allow for optimization problems described by a polynomial objective and polynomial constraints (generally nonconvex) to be attacked by convex optimization, namely linear objectives with linear-matrix-inequality constraints. Nearly twenty years of use in robust control has shown these relaxations to be remarkably useful in a wide variety of physically motivated problems and applications.In this approach, surrogate models are developed for all responses, both from the training set and from the prediction set. Each surrogate model is expressed as a quadratic form in terms of internal model parameters, developed in a series of direct ODE integrations performed according to a factorial design covering a subspace of parameter uncertainties. The quadratic form of the surrogate response models is then explored by an optimization algorithm. In the initial effort, the problem of propagation of uncertainties in a natural-gas-combustion model is cast in the form amenable for the S-procedure, a method of convex optimization widely used in control theory. Even more sophisticated convex relaxations have recently been developed, centering on the observation that determining if a given polynomial is a sum-of-squares (and hence globally nonnegative), can be cast as a convex feasibility problem, and verified in polynomial-time (in the order of the polynomial). This work will investigate such possibilities for exploring novel avenues for numerically economical assessment of realistic error bounds of complex dynamic models.