A Solve-Then-Discretize Paradigm for Spectral Methods
谱方法的求解然后离散范式
基本信息
- 批准号:1818757
- 负责人:
- 金额:$ 30万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2018
- 资助国家:美国
- 起止时间:2018-07-01 至 2022-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Spectral methods are one of the big three technologies (along with finite differences and finite element methods) for the numerical solution of partial differential equations (PDEs) and are particularly powerful for fluid flow and airfoil simulations. This research project aims to develop a new infinite-dimensional framework for solving PDEs to derive competitive computational algorithms that preserve the continuum structure of differential operators, promising to overcome many of the hard-and-fast computational barriers with spectral discretizations. We aim to produce a collection of adaptive, robust, and industrial-strength iterative solvers for spectral methods to allow for the accurate resolution of fluid flows. We will also develop tools for computing the pseudospectra and continuous spectra of differential operators, facilitating improved understanding of inelastic scattering. The results will help to demonstrate that spectrally-accurate methods, when done carefully, are flexible, general, and powerful numerical tools in computational mathematics and engineering.The standard paradigm for solving a PDE is to first discretize the equation and then solve the resulting linear system. This approach has a number of drawbacks for spectral methods related to the design of preconditioners, the introduction of non-normality, and the perturbation of spectra. The infinite-dimensional framework under development in this project preserves the continuum structure of PDEs by avoiding the discretization of differential operators, and instead only discretizes smooth functions, such as the solution and the source terms of the PDE. Not working with finite sections of differential operators promises to enable us to develop robust Krylov-based iterative solvers, motivate preconditioners directly from the differential operator, compute the continuous part of the spectrum of operators, and develop a theoretical foundation for the adaptive resolution of solutions and eigenfunctions based on error analysis. We will apply these new tools to the numerical simulation of advection-dominated fluid flow as well as inelastic scattering.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.
谱方法是数值求解偏微分方程组的三大技术之一(与有限差分法和有限元方法并列),特别适用于流体流动和翼型模拟。这一研究项目旨在开发一个新的无限维框架来求解偏微分方程组,以推导出具有竞争力的计算算法,保持微分算子的连续结构,承诺通过谱离散来克服许多硬而快速的计算障碍。我们的目标是为频谱方法生产一系列自适应、健壮和工业强度的迭代求解器,以允许准确地解析流体流动。我们还将开发工具来计算微分算子的伪谱和连续谱,以促进对非弹性散射的更好理解。结果将有助于证明,谱精度方法,如果仔细操作,是计算数学和工程中灵活、通用和强大的数值工具。求解偏微分方程组的标准范例是首先离散方程,然后求解得到的线性系统。这种方法对于谱方法有许多缺点,涉及预条件的设计、非正态分布的引入和谱的扰动。这个项目中开发的无限维框架通过避免对微分算子的离散化来保持偏微分方程组的连续结构,而只离散化光滑函数,例如偏微分方程组的解和源项。不使用有限段的微分算子有望使我们能够开发健壮的基于Krylov的迭代求解器,直接从微分算子激励预条件算子,计算算子谱的连续部分,并为基于误差分析的解和特征函数的自适应求解发展理论基础。我们将把这些新工具应用于以平流为主的流体流动以及非弹性散射的数值模拟。这一奖项反映了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。
项目成果
期刊论文数量(13)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Fast Poisson solvers for spectral methods
- DOI:10.1093/imanum/drz034
- 发表时间:2017-10
- 期刊:
- 影响因子:2.1
- 作者:D. Fortunato;Alex Townsend
- 通讯作者:D. Fortunato;Alex Townsend
Bounding Zolotarev Numbers Using Faber Rational Functions
使用 Faber 有理函数限制 Zolotarev 数
- DOI:
- 发表时间:2022
- 期刊:
- 影响因子:2.7
- 作者:Daniel Rubin, Alex Townsend
- 通讯作者:Daniel Rubin, Alex Townsend
The ultraspherical spectral element method
超球形光谱元法
- DOI:10.1016/j.jcp.2020.110087
- 发表时间:2021
- 期刊:
- 影响因子:4.1
- 作者:Fortunato, Dan;Hale, Nick;Townsend, Alex
- 通讯作者:Townsend, Alex
Rational neural networks
- DOI:
- 发表时间:2020-04
- 期刊:
- 影响因子:0
- 作者:N. Boull'e;Y. Nakatsukasa;Alex Townsend
- 通讯作者:N. Boull'e;Y. Nakatsukasa;Alex Townsend
Computing Spectral Measures of Self-Adjoint Operators
- DOI:10.1137/20m1330944
- 发表时间:2021-09-01
- 期刊:
- 影响因子:10.2
- 作者:Colbrook, Matthew;Horning, Andrew;Townsend, Alex
- 通讯作者:Townsend, Alex
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Alex Townsend其他文献
COMPUTING WITH FUNCTIONS IN THE BALLast
使用 BALLast 中的函数进行计算
- DOI:
- 发表时间:
2020 - 期刊:
- 影响因子:0
- 作者:
Alex Townsend - 通讯作者:
Alex Townsend
A generalization of the randomized singular value decomposition
随机奇异值分解的推广
- DOI:
- 发表时间:
2022 - 期刊:
- 影响因子:0
- 作者:
Nicolas Boulle;Alex Townsend - 通讯作者:
Alex Townsend
Are sketch-and-precondition least squares solvers numerically stable?
草图和前提条件最小二乘求解器数值稳定吗?
- DOI:
- 发表时间:
2023 - 期刊:
- 影响因子:1.5
- 作者:
Maike Meier;Y. Nakatsukasa;Alex Townsend;M. Webb - 通讯作者:
M. Webb
Circulant networks of identical Kuramoto oscillators: Seeking dense networks that do not globally synchronize and sparse ones that do
相同仓本振荡器的循环网络:寻求不全局同步的密集网络和全局同步的稀疏网络
- DOI:
- 发表时间:
2019 - 期刊:
- 影响因子:0
- 作者:
Alex Townsend;M. Stillman;S. Strogatz - 通讯作者:
S. Strogatz
Learning Elliptic Partial Differential Equations with Randomized Linear Algebra, Foundations of Computational Mathematics
用随机线性代数学习椭圆偏微分方程,计算数学基础
- DOI:
- 发表时间:
2022 - 期刊:
- 影响因子:0
- 作者:
Nicolas Boulle;Alex Townsend - 通讯作者:
Alex Townsend
Alex Townsend的其他文献
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{{ truncateString('Alex Townsend', 18)}}的其他基金
CAREER: Computing with Rational Functions
职业:使用有理函数进行计算
- 批准号:
2045646 - 财政年份:2021
- 资助金额:
$ 30万 - 项目类别:
Continuing Grant
Collaborative Research: Optimal-Complexity Spectral Methods for Complex Fluids
合作研究:复杂流体的最优复杂谱方法
- 批准号:
1952757 - 财政年份:2020
- 资助金额:
$ 30万 - 项目类别:
Standard Grant
Advancements in the Ultraspherical Spectral Method
超球面光谱方法的进展
- 批准号:
1645445 - 财政年份:2016
- 资助金额:
$ 30万 - 项目类别:
Standard Grant
Advancements in the Ultraspherical Spectral Method
超球面光谱方法的进展
- 批准号:
1522577 - 财政年份:2015
- 资助金额:
$ 30万 - 项目类别:
Standard Grant