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Adaptive High Order Low-Rank Tensor Methods for High-Dimensional Partial Differential Equations with Application to Kinetic Simulations

高维偏微分方程的自适应高阶低阶张量方法及其在动力学模拟中的应用

基本信息

批准号：
2111383
负责人：
Wei Guo
金额：
$ 23.71万
依托单位：
Texas Tech University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111383&HistoricalAwards=false
关键词：
Adaptive Order Low Rank Tensor

项目摘要

Kinetic models find a wide range of applications in science and engineering. One celebrated example is the Vlasov-Maxwell system which plays a fundamental role in plasma physics. Among many existing challenges for deterministic kinetic simulations, the curse of dimensionality and the associated huge computational cost have been a key obstacle for realistic high-dimensional simulations. The project focuses on the development of a novel computational framework for kinetic simulations by integrating the emerging low-rank tensor decompositions with advanced high order discretization schemes. The framework can lead to a class of accurate and efficient algorithms that are resistant to the curse of dimensionality for solving challenging high-dimensional problems. This project will support one graduate student during each of the three years of the project. The main objective of the project is to develop a novel computational framework for simulating high-dimensional kinetic models with high order accuracy, numerical stability, and manageable cost. The algorithms incorporate the emerging tensor approach and advanced high order discretization schemes, aiming to accurately and efficiently extract the low-rank structure of the solution data with significantly reduced complexity and potentially break the curse of dimensionality. Another project focus is to develop a novel low-rank tensor approach that can mitigate possible severe rank increase for transport-dominated simulations, which is known as a major limitation for the existing low-rank tensor methods. The key component in the algorithm design is to consider the underlying flow map system. Several computational and theoretical aspects of the proposed algorithms will be investigated, including structure preservation, complexity, stability and accuracy analysis, application to the rotating Vlasov model, and beyond. Through this research, both numerical and analytical techniques for solving challenging high-dimensional problems will be advanced.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.

动力学模型在科学和工程中有着广泛的应用。一个著名的例子是弗拉索夫-麦克斯韦系统，它在等离子体物理学中起着重要的作用。在确定性动力学仿真的诸多挑战中，维数的诅咒和随之而来的巨大的计算成本一直是实现真实高维仿真的主要障碍。该项目的重点是通过将新兴的低秩张量分解与先进的高阶离散化方案相结合，开发一种新的动力学模拟计算框架。该框架可以导致一类精确和高效的算法，这些算法可以抵抗维度的诅咒，用于解决具有挑战性的高维问题。该项目将在项目的三年中每年资助一名研究生。该项目的主要目标是开发一种新的计算框架，用于模拟高维动力学模型，具有高阶精度，数值稳定性和可管理的成本。该算法结合了新兴的张量方法和先进的高阶离散化方法，旨在准确、高效地提取解数据的低秩结构，显著降低了复杂度，并有可能打破维数诅咒。另一个项目重点是开发一种新的低秩张量方法，可以减轻运输主导模拟可能出现的严重秩增加，这被称为现有低秩张量方法的主要限制。算法设计的关键部分是考虑底层流程图系统。提出的算法的几个计算和理论方面将进行研究，包括结构保存，复杂性，稳定性和精度分析，在旋转Vlasov模型中的应用，等等。通过这项研究，解决高维问题的数值和分析技术将得到发展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。