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

PI提出发展求解随机偏微分方程的新的有效方法。特别是，PI将解决两个突出的问题，在多项式混沌（PC）的方法，在计算机模拟物理和生物系统的不确定性建模。第一个是有关有效地处理许多随机维度，而第二个是有关准确建模的白色噪声。这样的问题出现在小的相对相关长度或大量的独立随机参数的应用。这两种方法是互补的，因为非常小的相关长度的问题可以有效地建模为白色噪声过程。新的想法是使用方差分解和适当的加权Wiener混沌空间和随机卷积产品的引入。方差分析提供了一个分层的功能分解，利用系统的有效维度。这种类型的维数分解可以有效地打破某些近似问题中的维数灾难，其中有效维数远低于标称维数。在初步工作中，PI已经证明了新方法在有效逼近500多个维度的问题方面的有效性。拟议的工作将产生重大而广泛的影响，因为它将为许多物理和生物系统的不确定性量化，数据同化和灵敏度分析奠定严格的基础。例如，在计算流体力学中，它将建立一个强大而有效的框架，为模拟提供一个复合误差条，该误差条超越了数值精度，包括操作条件、物理参数和域的不确定性。拟议的工作具有变革性，因为它将使随机模拟成为标准而不是例外。它也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。赞助的研究生和本科生将参与这项研究，并将与所有高级人员，包括几个国际游客互动。PI将通过布朗大学两个非常有效的组织与参与外联活动的本科生密切合作，这些组织针对科学和工程领域的妇女以及中学生。他还计划通过与教师一起开发沿着基于计算机的交互式数学学习策略，为市中心的高中开展推广活动。与MET学校合作的初步结果非常令人鼓舞，PI计划在全国范围内扩大这一活动。

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.