Reduced-Order and Low-Rank Methods for Parameter-Dependent Partial Differential Equations

这项研究的目标是开发新的计算机算法，用于模拟工程模型和物理现象，目的是预测系统在物理世界中的行为。使用这些技术的例子包括对地下水中污染物流动的建模，评估飞机机翼等结构在面临高温或压力等压力时的稳定性，以及模拟磁场对半导体性能的影响或核聚变中的反应。这样的算法有助于工程师和科学家就新技术的建设和使用做出正确的决定，但只有在证明有效(即使用适量的计算机时间和内存)和准确的情况下，它们才在实际中有用。这项工作的目的是促进这类高效算法的发展，并展示它们在流体流动和物理结构稳定性等瞬变现象模型中的应用。本项目的技术目标是通过构造低阶结构的近似解来研究参数依赖偏微分方程组的求解算法。当系统的微分算子中的基本项依赖于一组未知或随机参数时，就会出现这种类型的参数化问题。例如，受温度影响的扩散模型或速度场中的未知渗透率。在这种情况下，寻求的解也取决于参数，但它们通常可以在低维空间中很好地逼近，即所有参数值的解可以表示为小的有限函数集的线性组合。这种减少的表示提供了显著降低计算解决方案所需成本的前景。该项目将探索这种方法在两类问题上的效用：(I)动力系统，对时间的依赖需要新的方法来开发长期准确的降阶模型，以及(Ii)本征值问题，重点是动力系统的稳定性分析技术。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。

The goal of this study is to develop new computer algorithms to be used to simulate engineering models and physical phenomena, with the aim of predicting behavior of systems in the physical world. Examples of the use of these techniques include modeling of the flow of pollutants in groundwater, assessing the stability of structures such as airplane wings in the face of stresses such as high temperatures or pressures, and simulating the effects of magnetic fields on performance of semiconductors or reactions in nuclear fusion. Such algorithms help engineers and scientists to make good decisions about construction and use of new technology, but they are only practically useful if demonstrated to be efficient (that is, use modest amounts of computer time and memory) and accurate. The aim of this work is to advance the development of efficient algorithms of this type and to demonstrate their utility for models of transient phenomena such as fluid flows and stability of physical structures.The technical goals of the project are to study solution algorithms for parameter-dependent partial differential equations by constructing approximate solutions of low-rank structure. Parameterized problems of this type arise when underlying terms figuring in the differential operators of the system depend on a set of unknown or random parameters. Examples include unknown permeabilities in models of diffusion or velocity fields that are affected by temperatures. In this scenario, the solutions sought also depend on parameters, but they can often be approximated well in a space of low dimension, i.e., solutions for all parameter values can be represented as a linear combination of a small finite set of functions. Such a reduced representation offers the prospect of significant reduction of costs required to compute solutions. The project will explore the utility this approach for two types of problems: (i) dynamical systems, for which the dependence on time requires new methods to develop reduced-order models that are accurate over long time periods, and (ii) eigenvalue problems, with emphasis on techniques for stability analysis of dynamical systems.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.