Numerical Solution of Constrained Optimization Problems Governed by Partial Differential Equations with Uncertain Parameters

本课题将为不确定参数偏微分方程优化问题的求解提供新的数学算法和理论分析。这些问题出现在许多科学和工程决策应用中，其中必须在可以观察到影响决策结果的不确定输入实现之前做出决策。使用所谓的风险度量，可以将不确定性纳入优化公式，这些度量通常涉及利息数量的期望值及其与期望值的偏差度量。原则上，这些公式允许人们计算出平衡预期结果最大化和不确定性风险最小化的决策。然而，这些问题的数值解提出了许多理论和算法上的挑战。例如，数值解需要对随机输入进行某种采样，这使得这些PDE约束优化问题的求解成本非常高。为了解决上述几个挑战，本研究将对一类半线性椭圆型PDE约束优化问题的适定性和最优性条件进行理论分析，并将推导稀疏网格和准蒙特卡罗离散化应用于PDE约束优化的离散化误差界。此外，它将开发和分析自适应方法，以减少所需的样本总数，或纳入降阶模型。本研究是随机规划、确定性偏微分方程约束优化和随机输入偏微分方程求解的交叉领域，将对这些领域做出算法和理论贡献。将理论和数值方法应用于实例问题将为其他研究人员和决策者提供一个模型，并将为不确定性下重要的决策类别带来更有效的算法。结果将通过出版算法和结果来传播。此外，项目成果将用于定期开设的优化理论与应用课程，以及针对数工科学生的不确定条件下PDE约束优化专题课程。

This project will provide new mathematical algorithms and theoretical analyses for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain parameters. These problems arise in many science and engineering decision making applications, where a decision has to be made before the realization of uncertain inputs can be observed that will impact the outcome of the decision. The uncertainty can be incorporated into the optimization formulation using so-called risk measures, which typically involve the expected value of the quantity of interest and a measure of its deviation from the expected value. In principle, these formulations allow one to compute decisions that balance maximization of their expected outcome and minimization of the risk due to uncertainty. However, the numerical solution of these problems presents many theoretical and algorithmic challenges. For example, the numerical solution requires some sort of sampling of the random inputs, which can make these PDE constrained optimization problems extremely expensive to solve. To address several of the above mentioned challenges, this research will provide theoretical analyses of the well-posedness and of optimality conditions for a class of semilinear elliptic PDE constrained optimization problems, and it will derive discretization error bounds for sparse grid and quasi Monte Carlo discretizations applied to PDE constrained optimization. Furthermore, it will develop and analyze adaptive methods which reduce the total number of samples needed, or incorporate reduced order models. This research is at the interface between stochastic programming, deterministic PDE constrained optimization, and solution of PDEs with random inputs, and it will make algorithmic and theoretical contributions to these areas. The application of theories and numerical methods to example problems will serve as a model for other researchers and decision makers, and will lead to more efficient algorithms for important classes of decision making under uncertainty. Results will be disseminated through publication of algorithms and results. In addition, the results of the project will be used in regularly offered courses on the theory and applications of optimization as well as in special courses on PDE constrained optimization under uncertainty aiming at students in both mathematics and engineering.