CAREER: High Performance Computational Method for Stochastic Design Problems

近年来，不确定性分析和随机建模的重要性受到越来越多的关注，特别是在计算数学学会，人们对不确定性的量化和随机计算进行了大量的研究，并开发了新的数值方法来有效地处理具有不确定性的大型复杂系统。虽然这些方法已经被证明在预测复杂随机系统的行为方面是非常有效的，但设计业并没有充分利用这些发展，许多学科的设计过程几乎完全是确定性的。这一建议的研究目标是为具有不确定输入的多学科设计问题发展新的数学和数值方法，重点是新方法的效率和精度，以便它们适用于大规模、现实的工程应用。这一目标将通过三大努力来实现：(1)利用严格的数学理论形成一个统一的框架，以便进行系统分析和误差估计；(2)利用最先进的随机算法，为基于不确定性的两大类设计：稳健设计和可靠性设计构建一套高性能设计算法；(3)将新方法扩展到大规模复杂系统，并开发快速并行求解。它在现代战略规划和风险管理中具有特别重要的意义，因为决策是在一个不断变化的环境中做出的，有许多未知因素。疫情控制、极端条件下的飞机优化、自然灾害后的最优应对等例子比比皆是。这类问题本质上是在一个复杂的、多学科的环境中进行设计和优化的，在系统和参数中，大量的不确定性以高度非线性的方式相互作用。虽然基于模拟的设计工具继续以快速的速度发展，但很少有人注意将最先进的数学技术纳入随机分析和不确定性量化。传统的方法，例如通过使用安全系数来粗略地适应不确定性的方法，正变得越来越过时，往往导致过于保守的决定。这个项目的价值在于它的多学科影响，它对基于不确定性的设计问题的根本贡献，更重要的是，它的高性能随机设计算法为决策者提供了更好和更敏锐的分析。该项目将把分散在各种工程应用中的基于不确定性的设计技术与尖端的随机计算框架结合起来，是计算数学工具和实际需求的真正协同，可以扩展到一大类设计和优化问题。

The importance of uncertainty analysis and stochastic modelinghas received increasing amount of attention in recent years,especially from the Computational Mathematics society.Extensive research efforts have been devoted to uncertaintyquantification and stochastic computations, and novel numerical methods have been developed to efficiently deal with large and complex systems with uncertainty. Although these methods have been demonstrated to be highly effective in predicting the behavior of complex stochastic systems, the design industry has not taken full advantage of these developments, and the design procedures in many disciplines remain almost exclusively deterministic. The research objective of this proposal is to develop new mathematical and numerical methods for multidisciplinary design problems with uncertain inputs, with an emphasis on the efficiency and accuracyof the new methods so that they are applicable to large-scale, realistic engineering applications. This objective will be attained through three major efforts: (1) employing rigorous mathematical theory to form a unified framework that allows one to conduct systematical analysis and error estimates; (2) employing the state-of-the-art stochastic algorithms to construct a set of high performance design algorithms for two major classes of uncertainty-based design: robust design and reliability design;and (3) extending the new methods to large-scale complex systemsand developing fast and parallel solvers.Quantifying uncertainty is of paramount importance in almost all aspects of science and engineering. It is of particular significance in modern-day strategic planning and risk management where decisions are made in a constantly changing landscape with many unknown factors. Examples such as epidemic control, aircraft optimization under extreme conditions, optimal response following natural disasters, etc, are abundant. Such problems are essentially design and optimization in a complex and multidisciplinary environment, with substantial uncertainty interacting in a highly nonlinear fashion in the systems and parameters. While simulation based design tools continue to be advanced at rapid rates, little attention has been paid to incorporation of state-of-the-art mathematical techniques in stochastic analysis and uncertainty quantification. The traditional approaches, e.g., those by using safety factors to accommodate uncertainty in a gross manner, are becoming increasingly obsolete and often result in overly conservative decisions. This project is valuable for its multi-disciplinary influence, its fundamental contribution to uncertainty-based design problems, and more importantly, its high performance stochastic design algorithms the provide better and sharper analysis for decision makers. The project will unite a collection of uncertainty-based design techniques scattered over various engineering applications with the cutting-edge stochastic computation framework.It is a true synergy of Computational Mathematics tools and practicaldemands, and can be extended to a large class of design and optimizationproblems.