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

参数相关偏微分方程的降阶和低秩方法

基本信息

批准号：
1819115
负责人：
Howard Elman
金额：
$ 20万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-01-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819115&HistoricalAwards=false
关键词：
Reduced Order Low Rank Methods

项目摘要

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.

本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，目的是预测物理世界中系统的行为。使用这些技术的例子包括模拟地下水中污染物的流动，评估飞机机翼等结构在高温或高压等应力下的稳定性，以及模拟磁场对半导体性能或核聚变反应的影响。这些算法可以帮助工程师和科学家在新技术的构建和使用方面做出正确的决策，但它们只有在被证明是有效的（即使用适量的计算机时间和内存）和准确的情况下才有实际用处。这项工作的目的是推进这种类型的高效算法的发展，并证明其效用的瞬态现象，如流体流动和物理结构的稳定性的模型。该项目的技术目标是通过构造低秩结构的近似解来研究参数相关偏微分方程的求解算法。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。例子包括受温度影响的扩散或速度场模型中的未知渗透率。在这种情况下，寻求的解决方案也取决于参数，但它们通常可以在低维空间中很好地近似，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。这种简化的表示提供了显著降低计算解所需成本的前景。该项目将探讨这种方法对两类问题的效用：（i）动态系统，对于动态系统，对时间的依赖性需要新的方法来开发在长时间段内精确的降阶模型，以及（ii）本征值问题，该奖项反映了美国国家科学基金会的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。