Optimal and Robust Operations of Complex Processes withNon-Gaussian Distributed Uncertain Variables under ChanceConstraints - Extension to Model Predictive Control of Parabolic Partial Differential Equation Systems

许多工程过程都是用偏微分方程（PDEs）来建模的，模型参数和运行条件都存在不确定性。不确定条件下约束偏微分方程的优化问题由于其数学和数值上的复杂性一直是一个挑战。对空间分布输出变量施加机会约束的随机PDE系统的优化问题目前还没有得到很好的研究。本项目的目标是开发具有高斯和非高斯分布不确定性的PDE系统的机会约束优化的有效方法。第一个具有挑战性的任务是通过输入不确定性适当地描述空间分布的输出变量。与机会约束一致，将构造正交多项式的适当集，将约束的输出变量展开为无限和。将研究降维策略，截断随机PDE模型的无限维数。第二个目标是发展有效计算PDE系统机会约束的数学和数值方法。多维稀疏网格技术将用于随机偏微分方程的确定性变换以及有效地计算机会约束及其梯度的高维积分。一般来说，空间分布的机会约束的评估带来了巨大的困难。因此，我们将设计一个易于处理的、单调的解析近似的机会约束，保证先验的可行性。为了减轻计算负担，将开发和实现适合并行计算的策略。整体的理论和数值发展将通过对内源性和外源性不确定性热力学过程优化的案例研究来展示。

Many engineering processes are modeled by partial differential equations (PDEs) with uncertainties both in model parameters and in operating conditions. Optimization of constrained PDEs under uncertainties remains a challenge due to its mathematical and numerical complexity.Optimization of stochastic PDE systems where chance constraints are to be imposed on spatially distributed output variables has not been properly investigated until now. The objective of this proposed project is to develop efficient approaches to chance constrained optimization of PDE systems with Gaussian and non-Gaussian distributed uncertainties. The firstchallenging task is to appropriately describe spatially distributed output variables through the input uncertainties. Consistent with chance constraints, proper sets of orthogonal polynomials will be constructed for the expansion of the constrained output variables into infinite sums.Dimension reduction strategies will be investigated to truncate the infinite dimensionality of the stochastic PDE model. The second target is to develop mathematical and numerical methods for efficient computation of chance constraints of the PDE systems. Multidimensional sparsegrid techniques will be developed for deterministic transformation of stochastic PDEs as well as for efficiently computing high-dimensional integrals of chance constraints and their gradients.In general, the evaluation of spatially distributed chance constraints poses enormous difficulties. Hence, we will design a tractable and monotonic analytic approximation for chance constraints which guarantees a priori feasibility. In order ameliorate computational burdens suitable strategiesfor parallel computing will be developed and implemented. The overall theoretical and numerical development will be demonstrated through case studies on optimization of thermodynamic processes with endogenous and exogenous uncertainties.