SHF: Small: Collaborative Research: Transform-to-Perform: Languages, Algorithms, and Solvers for Nonlocal Operators

SHF：小型：协作研究：从转换到执行：非本地算子的语言、算法和求解器

• 批准号: 1909176
• 负责人: Robert Kirby
• 金额: $ 17.9万
• 依托单位: Baylor University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-10-01 至 2023-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1909176&HistoricalAwards=false
• 关键词: SHF; Small; Collaborative; Research; Transform

Computer simulations based on partial differential equations (PDEs) are ubiquitous in science and engineering, underpinning weather forecasts, car and airplane design processes, and tsunami predictions, among other use cases. They are based on mathematical derivatives and thus only consider "local" descriptions of physical principles and interactions. Local models fail to capture certain natural processes (like anomalous diffusion), so computational scientists are increasingly considering "nonlocal" models. These include integral equation formulations and models involving more general interactions such as fractional derivatives. While effective numerical methods for nonlocal methods are known and the subject of ongoing active research, software support is far less mature than for local operators. Support for coupling these two approaches, while very important, is basically nonexistent. In this project, the researchers are combining expertise in numerical methods and software tools for both local and nonlocal operators to extend the Firedrake project (https://www.firedrakeproject.org), a high-level PDE tool set, to serve this important need. This extension spans all aspects of the corresponding software, ranging from the computer language used for problem description to algorithms and efficient implementation. These tools will provide an enabling technology for scientists and engineers to reliably and efficiently address a much broader range of models than currently available. All software being developed under this project will be freely distributed under open-source licenses, and knowledge gained will be disseminated through conference presentations, publications, and teaching. This work will leverage and build upon the researchers' work on developing a suite of representations at each layer of abstraction (operators, algorithms, loop nests, etc.) and tools to transform these abstractions downward towards machine code. The investigators will map out and extend the landscape of finite element algorithms to include new nonlocal algorithm; provide a unifying framework for reasoning about these algorithms, design language and compiler foundations that allow the complete specification of matrix assembly and operator application tasks; and deploy automated non-local operators in a toolkit that already includes classical finite element methods and is capable of architecture-specific targeting.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.

基于部分微分方程（PDE）的计算机仿真在科学和工程，底层天气预报，汽车和飞机设计过程以及海啸预测等用例中无处不在。它们基于数学导数，因此仅考虑对物理原理和相互作用的“局部”描述。本地模型无法捕获某些自然过程（例如异常扩散），因此计算科学家越来越考虑“非本地”模型。这些包括积分方程式公式和涉及更通用相互作用（例如分数衍生物）的模型。尽管已知有效的非局部方法的数值方法，并且是正在进行的主动研究的主题，但软件支持的成熟程度远不如本地运营商。支持这两种方法虽然非常重要，但基本上是不存在的。在该项目中，研究人员正在将数值方法和软件工具方面的专业知识结合起来，以供本地和非本地操作员扩展Firedrake项目（https://www.firedrakeproject.org），这是一种高级PDE工具集，以服务于这一重要需要。该扩展涵盖了相应软件的各个方面，从用于问题描述的计算机语言到算法和有效的实现。这些工具将为科学家和工程师提供一项促成技术，以可靠，高效地解决比目前可用的更广泛的模型。该项目下开发的所有软件将在开源许可下自由分发，并且获得的知识将通过会议演示，出版物和教学传播。这项工作将利用研究人员的工作，以在每一层抽象（运营商，算法，环巢等）和工具上开发一套表示形式，并将这些抽象向下转换为机器代码。研究人员将绘制并扩展有限元算法的景观，以包括新的非局部算法；为这些算法，设计语言和编译器基础提供一个统一的框架，以完整规范矩阵汇编和操作员应用程序任务；并将自动化的非本地运营商部署在已经包括经典有限元方法的工具包中，并且能够具有特定于建筑的针对性。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子的评估和更广泛的影响评论来评估的。标准。

项目成果

Integral equation methods for the Morse-Ingard equations

Morse-Ingard 方程的积分方程方法

• DOI: 10.1016/j.jcp.2023.112416
• 发表时间: 2023
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: Wei, Xiaoyu;Klöckner, Andreas;Kirby, Robert C.
• 通讯作者: Kirby, Robert C.

Finite Elements for Helmholtz Equations with a Nonlocal Boundary Condition

具有非局部边界条件的亥姆霍兹方程的有限元

• DOI: 10.1137/20m1368100
• 发表时间: 2021
• 期刊: SIAM Journal on Scientific Computing
• 影响因子: 3.1
• 作者: Kirby, Robert C.;Klöckner, Andreas;Sepanski, Ben
• 通讯作者: Sepanski, Ben