A Model Reduction Approach to Stochastic PDEs: Forward Uncertainty Propagation and Stochastic Homogenization

该项目旨在开发一种新的理论、算法和计算工具，以实现具有高维随机输入的偏微分方程组(PDE)的线性可扩展表征和解。在现实情况下，系统的动力学本质上是可变的，或者对潜在的物理规律的理解是不完整的，需要表示和量化这种不确定性对感兴趣的量的影响。当不确定的独立来源的数量很大时，就会出现根本的困难。在这种情况下，具有不确定输入的偏微分方程解的标准方法会遇到计算复杂性的指数增长，即所谓的维灾。所提出的工作通过一种新的基于多变量解的低阶分离表示的随机模型降阶方法来解决这一问题。通过数值均匀化的降阶表征和不确定性的正向传播都将被考虑。虽然这种方法适用于各种基于随机偏微分方程的模型，但本项目主要集中在具有高维随机速度场和扩散场的非线性平流-反应-扩散方程的求解。具有高维不确定性的多尺度、多物理系统的预测是当前科学和工程领域关注的主要问题之一。例如，在燃烧、储能系统、聚变能源等领域很常见。这一奖项的新的数值技术，基于多变量函数的非线性、多线性和稀疏逼近的最新思想，将显著促进此类问题的计算机模拟的最新技术。为了进一步扩大这一项目的影响，投资促进局将开设研究生水平的服务课程，向从事复杂系统研究的研究人员介绍新的不确定性量化方法。

This project seeks to develop a new theory, algorithms, and computational tools to enable a linearly scalable characterization and solution of partial differential equations (PDEs) with high-dimensional random inputs. In realistic situations where the dynamics of a system is intrinsically variable or understanding of the underlying physical laws is incomplete, there is a need to represent and quantify the impact of such uncertainties on quantities of interest. A fundamental difficulty arises when the number of independent sources of uncertainty is large. In these situations, standard approaches to the solution of PDEs with uncertain inputs encounter an exponential growth of computational complexity, i.e., the so-called curse-of-dimensionality. The proposed effort tackles this issue through a new approach to stochastic model reduction based on low-rank separated representation of multi-variate solutions. Both reduced-order characterization, via numerical homogenization, and forward propagation of uncertainties will be considered. While this approach is applicable to a wide variety of stochastic PDE-based models, this project is primarily focused on the solution of nonlinear, advection-reaction-diffusion equations with high-dimensional stochastic velocity and diffusion fields. One of the major problems of interest in science and engineering is the prediction of multi-scale and multi-physics systems with high-dimensional uncertainties. Examples are common in combustion, energy storage systems, fusion energy, among others. The new numerical techniques of this award, based upon recent ideas from nonlinear, multi-linear, and sparse approximation of functions with many variables, will significantly advance the state-of-the-art in computer simulation of such problems. To further broaden the impact of this project, the PIs will develop graduate level service courses to introduce the new uncertainty quantification methods to researchers working on complex systems.