Scalable Computational Methods for Large-Scale Stochastic Optimization under High-Dimensional Uncertainty

在计算科学和工程中，大规模模拟不仅是为了深入了解一个系统，也是为了作为决策的基础。当决策变量代表工程或自然系统的设计或控制时，系统由偏微分方程（PDEs）控制，由于缺乏知识或内在可变性，输入不确定，确定最优设计或控制的任务导致pde约束的随机优化问题。这样的问题在科学和工程的各个领域比比皆是。例子包括对地下流动、等离子体聚变反应堆以及化学和材料过程的最佳控制；航空航天、汽车和民用基础设施系统的优化结构设计；以及生物医学、电子和纳米结构器件的形状、布局或拓扑优化。解决这类问题有几个关键的挑战，包括由不确定性和/或优化变量空间引起的高维，以及需要解决具有大量不确定参数样本的大规模偏微分方程。本项目将开发、分析和实现可扩展的计算方法，使高维不确定性下的大规模pde约束随机优化问题的求解变得易于处理。这些方法将应用于具有社会影响的地下水流问题；软件将以开放源代码的形式开发和广泛传播。研究生将参与并接受跨学科培训。该项目利用了随机优化问题的内在结构——特别是随机参数到目标映射的内在低维性、平滑性和几何性。具体而言，研究内容包括：(1)分析该映射的Hessian的秩或谱衰减，以证明若干经典随机pde约束优化问题的内在低维性。(2)将基于局部二次逼近的随机优化推广到基于Hessian近似作为平移不变算子、高阶Taylor近似和混合模型的多点Taylor近似的随机优化。(3)应用于具有随机渗透率场的地下多孔介质中大规模且具有挑战性的最优流动控制问题。在这个项目中开发的方法将适用于一类广泛的pde约束随机优化问题。为了使更广泛的社区能够使用这些方法，并允许随机优化专家对新算法进行原型设计并快速运行实验，将实现并发布一个Python库SOUPy （Python中高维不确定性下的随机优化）。用户将能够在SOUPy中快速原型化新的PDE模型和目标函数，以及快速实现新的算法，进行数值实验，并解决新领域中的挑战性问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Large-scale simulation in computational science and engineering is often carried out not only to obtain insight about a system, but also as a basis for decision-making. When the decision variables represent the design or control of an engineered or natural system, and the system is governed by partial differential equations (PDEs) with uncertain input due to lack of knowledge or intrinsic variability, the task of determining the optimal design or control leads to a PDE-constrained stochastic optimization problem. Such problems abound across all areas of science and engineering. Examples include optimal control of subsurface flows, plasma fusion reactors, and chemical and materials processes; optimal structural design of aerospace, automotive, and civil infrastructure systems; and shape, layout, or topology optimization of biomedical, electronic, and nano-structured devices. There are several critical challenges in solving such problems including high dimensionality stemming from uncertainty and/or optimization variable spaces, and the need to solve large-scale PDEs with numerous samples of the uncertain parameters. This project will develop, analyze, and implement scalable computational methods to make tractable the solution of large-scale PDE-constrained stochastic optimization problems under high-dimensional uncertainty. These methods will be applied to subsurface flow problems with societal impact; software will be developed and disseminated widely in open source form. Graduate students will be involved and will receive interdisciplinary training. This project exploits the intrinsic structure of the stochastic optimization problems--in particular the intrinsic low dimensionality, smoothness, and geometry of the random parameter-to-objective map. Specifically, the components of the research include: (1) Analysis of the rank or spectrum decay of the Hessian of this map to prove intrinsic low-dimensionality for several classical stochastic PDE-constrained optimization problems. (2) Extension of local quadratic approximation-based stochastic optimization to that based on approximation of the Hessian as a translation invariant operator, higher order Taylor approximation, and multi-point Taylor approximation with mixture models. (3) Application to a specific large-scale and challenging problem of optimal flow control in a subsurface porous medium with a random permeability field. The methods developed in this project will apply to a wide class of PDE-constrained stochastic optimization problems. To make the methods accessible to broader communities and allow stochastic optimization specialists to prototype new algorithms and quickly run experiments, a Python library, SOUPy (Stochastic Optimization under high-dimensional Uncertainty in Python), will be implemented and released. Users will be able to rapidly prototype new PDE models and objective functions, as well as quickly implement new algorithms, conduct numerical experiments, and solve challenging problems in new domains in SOUPy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.