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

克服多项式混沌的瓶颈：系统生物学和流体力学的算法和应用

• 批准号: 0915077
• 负责人: George Karniadakis
• 金额: $ 20万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0915077&HistoricalAwards=false
• 关键词: Overcoming; Bottlenecks; Polynomial; Chaos; Algorithms

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.

PI提议开发新的效应方法来求解随机部分微分方程（SPDE）。特别是，PI将解决多项式混乱（PC）方法中的两个杰出问题，用于建模物理和生物系统计算机模拟中的不确定性。第一个与有效处理许多随机维度有关，而第二个随机维度与准确的白噪声有关。这些问题在相对长度较小或大量独立随机参数的应用中出现。两种方法相互互补，因为可以通过白噪声过程有效地建模具有很小相关长度的问题。新想法是使用方差分析的分解以及适当加权的维纳尔·韦纳（Wiener Chaosspaces）和随机卷积产品。 ANOVA提供了分层功能分解，可利用系统的有效维度。在某些近似问题中，这种类型的尺寸分解可以有效地打破维数的诅咒，在某些近似问题中，有效维度远低于标称维度。在初步工作中，PI证明了新方法在近似500多个维度的有效问题方面的有效性。拟议的工作将对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设定严格的基础，将产生重大和广泛的影响。例如，在计算流体动力学中，它将建立一个强大而有效的框架，以赋予模拟范围超出数值准确性的复合误差栏，并在操作条件，物理参数和域上包括不确定性。提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。它还将从根本上影响新实验的设计方式以及可以解决的问题的类型，而模拟与实验之间的相互作用将变得更有意义，更具动态性。PI计划将这些新想法纳入Brown的工程和应用数学课程。赞助的毕业生和本科生将参与这项研究，并将与包括几个国际游客在内的所有高级人员进行互动。 PI将与本科生紧密合作，他们通过布朗的两个非常有效的组织参与外展活动，该组织针对科学和工程学的女性以及中学生。他还通过与教师基于计算机的互动数学学习策略一起开发城市中学的外展活动。与大都会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项活动。