CAREER: Randomized Multiscale Methods for Heterogeneous Nonlinear Partial Differential Equations

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。在科学和工程中，具有多尺度显著特征的异质系统是普遍存在的。一个直接的数值模拟，旨在捕捉相关的现象，在所有尺度上需要一个经常令人望而却步的大量的计算时间。为了模拟这样的系统，多尺度方法在近似过程中包括数值解的局部行为，从而考虑到各种尺度。例如，在对由复合材料制成的风力涡轮机进行建模时，可以模拟风力涡轮机叶片部分在操作期间的变形。然后从这些局部解建立整个风力涡轮机变形的多尺度近似。能够保证多尺度近似和全局解之间的误差低于给定公差的多尺度方法特别令人感兴趣。该项目的目标是设计和分析用于模拟（现实）风力涡轮机变形的非线性偏微分方程数值解的多尺度方法。预计新方法将在构建数字孪生模型方面发挥关键作用，数字孪生模型是指可以在真实的时间内使用的物理对象的数学模型，用于评估系统的结构健康状况等。在风力涡轮机的数字孪生中应用结果将支持为社会生产可再生能源。该项目包括一个紧密结合的教育计划，以提高在STEM中代表性不足的群体的学生的参与和保留率，方法是（i）为高中生设计和领导课程，帮助他们通过创造性和基于项目的学习技术发现他们正在学习的数学概念如何具有重要的应用;以及（ii）为来自代表性不足群体的本科数学学生建立一个指导计划。为了开发所需的多尺度方法，在本项目中，将构造局部渐近函数以（准）最佳地逼近偏微分方程（PDE）的局部解的非线性集合。为了近似后者，将开发随机版本的模型降阶方法。虽然确定性模型降阶算法构造可证明的最优空间来近似依赖于参数（这里是任意Dirichlet边界数据）的PDE的一组解，但它们遭受高维参数集的维数灾难。随机化这些方法有望打破维数灾难，并允许以适用于非线性系统的新方法分析误差。该项目的三个研究目标是：发展和分析（i）椭圆和（ii）抛物型非线性偏微分方程的随机多尺度方法，其中局部反函数可以在时间上并行构造，以及（iii）该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持和更广泛的影响审查标准。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Heterogeneous systems with salient features at multiple scales are ubiquitous in science and engineering. A direct numerical simulation that aims at capturing relevant phenomena at all scales requires an often prohibitively large amount of computation time. To simulate such systems, multiscale methods include the local behavior of a numerical solution in the approximation process, thus taking into account the various scales. For example, in modeling a wind turbine made from composites, deformations during operation can be simulated for portions of the wind turbine blade. The multiscale approximation for the deformation of the whole wind turbine is then built from these local solutions. Multiscale methods that can guarantee that the error between the multiscale approximation and the global solution is below a given tolerance are of particular interest. The goal of this project is to design and analyze such multiscale methods for the numerical solution of nonlinear partial differential equations that are used in simulating deformations in (realistic) wind turbines. It is anticipated that the new methods will be crucial in building digital twins, that is, mathematical models of physical objects that can be employed in real time to assess, for example, the structural health of a system. Application of the results in digital twins for wind turbines will support the generation of renewable energy for society. The project includes a closely integrated educational plan to increase participation and retention of students from groups underrepresented in STEM by (i) designing and leading courses for high school students, helping them discover via creative and project-based learning techniques how the concepts of mathematics they are learning have important applications; and (ii) establishing a mentoring program for undergraduate mathematics students from underrepresented groups.To develop the desired multiscale methods, in this project, the local ansatz functions will be constructed to (quasi-)optimally approximate the nonlinear set of local solutions of the partial differential equation (PDE). To approximate the latter, randomized versions of model order reduction methods will be developed. While deterministic model reduction algorithms construct provably the optimal space to approximate a set of solutions of a PDE dependent on a parameter (here arbitrary Dirichlet boundary data), they suffer from the curse of dimensionality for high-dimensional parameter sets. Randomizing these methods is expected to break the curse of dimensionality and allow analysis of the error in novel ways suitable for nonlinear systems. The three research objectives of the project are: development and analysis of randomized multiscale methods for (i) elliptic and (ii) parabolic nonlinear PDEs, where the local ansatz functions can be constructed parallel in time, and (iii) application to the simulation of the deformation of wind turbines.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.