CAREER: Randomized Multiscale Methods for Heterogeneous Nonlinear Partial Differential Equations

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。具有多尺度显著特征的异质系统普遍存在于科学和工程领域。一个旨在捕捉所有尺度上的相关现象的直接数值模拟通常需要大量的计算时间。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，因此考虑了各种尺度。例如，在对由复合材料制成的风力涡轮机进行建模时，可以模拟部分风力涡轮机叶片在运行过程中的变形。然后由这些局部解建立了整个风力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小于给定容差的多尺度方法尤其令人感兴趣。本项目的目标是设计和分析这种用于数值求解非线性偏微分方程组的多尺度方法，用于模拟(现实中)风力发电机组的变形。预计新方法将对建立数字孪生兄弟至关重要，即可以实时使用的物理对象的数学模型，例如，评估系统的结构健康。将该成果应用于风力涡轮机的数字双胞胎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，通过(I)为高中生设计和指导课程，帮助他们通过创造性的和基于项目的学习技术，帮助他们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)为来自未被充分代表的群体的本科生建立一个指导计划。为了开发所需的多尺度方法，在该项目中，局部ANSATZ函数将被构造为(准)最佳地逼近偏微分方程(PDE)的局部解的非线性集合。为了接近后者，将开发模型降阶方法的随机化版本。虽然确定性模型降阶算法可证明地构造了最优空间来逼近依赖于参数(这里是任意Dirichlet边界数据)的偏微分方程解的集合，但对于高维参数集，它们遭受了维度诅咒。将这些方法随机化有望打破维度诅咒，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：开发和分析(I)椭圆型和(Ii)抛物型非线性偏微分方程组的随机多尺度方法，其中局部ANSATZ函数可以在时间上并行构造，以及(Iii)应用于风力涡轮机变形的模拟。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.