ITR/AP: Realistic Uncertainty Bounds for Complex Dynamic Models

当前社会面临的问题，如全球变暖、地震准备、核废料运输的安全以及汽车发动机的污染物排放，都需要各种计算机程序的整合，每个程序在不同的学科中解决数值问题。对于这样复杂的模型，最重要的问题是它们的可靠性：可预测性、真实性和不确定性。本文的重点是对多响应、大尺度、非线性动态模型预测中的现实不确定性进行建模，并采用一种新的策略来解决这一问题。这项工作重新审视了与复杂物理系统相关的数学模型的概念，将实验和理论视为模型的组成部分，并将模型的不确定参数视为内部“状态”变量。这样，实验和理论基础的不确定性“直接”转化为模型预测的不确定性。建立这种直接关系也使人们能够解决相反的问题：确定哪些特定数据对预测的不确定性贡献最大；确定实验所需的精度，使预测的不确定性达到给定的水平；或者评估一个计划好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映射技术相结合来实现的，该技术已发展并应用于化石燃料燃烧化学动力学的数值模拟。解映射技术利用计算机实验的统计设计，用替代多项式模型代替复杂的ODE模型。这些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（通常非凸）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使用表明，这些松弛在各种各样的物理动机问题和应用中非常有用。在这种方法中，为来自训练集和预测集的所有响应开发代理模型。每个代理模型都以内部模型参数的二次形式表示，根据覆盖参数不确定性子空间的析因设计，在一系列直接ODE积分中开发。然后通过优化算法探索代理响应模型的二次型。在最初的努力中，天然气燃烧模型中不确定性的传播问题以s-过程的形式进行，s-过程是一种广泛应用于控制理论的凸优化方法。甚至更复杂的凸松弛最近也被开发出来了，其核心是观察到，确定一个给定的多项式是否是平方和（因此全局非负），可以作为一个凸可行性问题，并在多项式时间内（按照多项式的顺序）进行验证。这项工作将研究这种可能性，为探索复杂动态模型的实际误差边界的数值经济评估的新途径。

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.