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Innovative Numerical Methods for High-Dimensional Applications

高维应用的创新数值方法

基本信息

批准号：
2012286
负责人：
Jianfeng Lu
金额：
$ 29.31万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012286&HistoricalAwards=false
关键词：
Innovative Numerical Methods Dimensional Applications

项目摘要

This project aims to develop efficient numerical algorithms for a class of important systems in science and engineering that are modeled with high-dimensional partial differential equations (PDE) such as the many-body Schrödinger equation. Examples of such systems include many-body quantum mechanics, dynamics of chemical systems, learning and control of complex systems, and spectral methods for high-dimensional data. The numerical solution of high-dimensional PDE has been one of the greatest challenges in computational science and remains a formidable task even with today's computational power and algorithmic advances. Efficient numerical simulations present opportunities for major breakthroughs in scientific understanding. This project will employ modern techniques to develop novel efficient numerical algorithms for such important systems. Graduate students will be trained through involvement in the research.The research project combines mathematical analysis and algorithmic design to make progress in numerical methods for high-dimensional PDE. The research will draw from and further develop ideas and tools from recent advances in computational physics, quantum chemistry, and machine learning. In particular, the project will use modern techniques for large-scale optimization and nonlinear parameterization of high-dimensional functions. Specifically, the PI will (1) develop novel highly efficient coordinate algorithms for large-scale eigenvalue problems, and (2) develop and analyze efficient methods based on neural-network parameterization of the solutions for high-dimensional PDE.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.

该项目旨在为一类科学和工程中的重要系统开发高效的数值算法，这些系统是用高维偏微分方程组(PDE)建模的，例如多体薛定谔方程。这类系统的例子包括多体量子力学、化学系统的动力学、复杂系统的学习和控制，以及用于高维数据的光谱方法。高维偏微分方程组的数值解一直是计算科学中最大的挑战之一，即使在今天的计算能力和算法进步的情况下，仍然是一项艰巨的任务。高效的数值模拟为科学理解的重大突破提供了机会。这个项目将利用现代技术为这些重要的系统开发新的高效的数值算法。通过参与研究培养研究生。研究项目结合数学分析和算法设计，在高维偏微分方程组的数值方法方面取得进展。这项研究将借鉴并进一步开发计算物理、量子化学和机器学习方面的最新进展的思想和工具。特别是，该项目将使用现代技术进行大规模优化和高维函数的非线性参数化。具体地说，PI将(1)开发用于大规模特征值问题的新型高效坐标算法，以及(2)开发和分析基于高维PDE解的神经网络参数化的高效方法。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。