Overcoming the Bottlenecks in Polynomial Chaos: Algorithms and Applications to Systems Biology and Fluid Mechanics

PI建议开发新的求解随机偏微分方程（SPDEs）的有效方法。特别是，PI将解决多项式混沌（PC）方法在物理和生物系统的计算机模拟中建模不确定性的两个突出问题。第一个问题是如何有效地处理多个随机维度，第二个问题是如何准确地建模白噪声。在相对相关长度较小或独立随机参数较多的应用中，会出现这样的问题。这两种方法是互补的，因为相关长度很小的问题可以用白噪声过程有效地建模。新的思想是使用方差分析分解和引入适当的加权维纳混沌空间和随机卷积积。方差分析提供了一种利用系统有效维度的分层功能分解。这种依维分解方法可以有效地打破某些近似问题中有效维数远低于标称维数的维数诅咒。在初步工作中，PI已经证明了新方法在有效逼近500维以上问题方面的有效性。拟议的工作将具有重大和广泛的影响，因为它将为许多物理和生物系统的不确定性量化，数据同化和敏感性分析奠定严格的基础。例如，在计算流体动力学中，它将建立一个鲁棒和高效的框架，赋予模拟一个复合误差条，该误差条超越数值精度，包括操作条件，物理参数和域的不确定性。提议的工作是变革性的，因为它将使随机模拟成为标准而不是例外。它还将从根本上影响新实验的设计方式和可以解决的问题类型，而模拟和实验之间的互动将变得更有意义和更动态。PI计划将这些新想法纳入布朗大学的工程和应用数学课程。受资助的研究生和本科生将参与这项研究，并将与包括一些国际访问者在内的所有高级人员进行互动。PI将与本科生密切合作，这些本科生通过布朗大学两个非常有效的组织参与拓展活动，这些组织的目标是科学和工程领域的女性以及中学生。他还计划通过与教师一起开发基于计算机的交互式数学学习策略，为市中心的高中开展外展活动。与MET学校合作的初步结果非常令人鼓舞，PI计划将这项活动扩展到全国。

The PI proposes to develop new effectrive methods for solving stochastic partial differential equations (SPDEs). In particular, the PI will address two outstanding issues in polynomial chaos (PC) methods for modeling uncertainty in computer simulations of physical and biological systems. The first one is related to treating effectively many stochastic dimensions while the second one is related to modeling accurately white noise. Such problems arise in applications with small relative correlation length or large number of independent random parameters. The two approaches are complementary to each other as problems with very small correlation length can be effectively modeled by white noise processes.The new ideas are the use of ANOVA decomposition and the introduction of proper weighted Wiener chaosspaces and stochastic convolution products. ANOVA provides a hierarchical functional decomposition that exploits the effective dimensionality of the system. This type of dimension-wise decomposition can effectively break the curse of dimensionality in certain approximation problems in which the effective dimensionality is much lower than the nominal dimensionality. In preliminary work, the PI has demonstrated the effectiveness of the new approach in approximating efficiently problems with more than 500 dimensions.The proposed work will have significant and broad impact as it will set rigorous foundations in uncertainty quantification, data assimilation and sensitivity analysis for many physical and biological systems. For example, in computational fluid dynamics, it will establish a robust and efficient framework to endow simulations with a composite error bar that goes beyond numerical accuracy and includes uncertainties in operating conditions, the physical parameters, and the domain.The proposed work is transformative as it will make stochastic simulations the standard rather than the exception. It will also affect fundamentally the way new experiments are designed and the type of questions that can be addressed, while the interaction between simulation and experiment will become more meaningful and more dynamic.The PI plans to incorporate these new ideas in engineering and applied mathematics courses at Brown. Sponsored graduate and undergraduate students will be involved in this research and will interact withall senior personnel that includes several international visitors. The PI will work closely with undergraduate students who are involved with outreach activities through two very effective organizations at Brown that target women in science and engineering and also middle school students. He also plans outreach activities for inner-city high schools by developing along with the teachers computer-based interactive math learning strategies. Preliminary results working with the MET school have been very encouraging, and the PI plans to expand this activity nationwide.