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

基本信息

批准号：
1911019
负责人：
Andreas Kloeckner
金额：
$ 31.96万
依托单位：
University of Illinois at Urbana-Champaign
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 2024-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1911019&HistoricalAwards=false
关键词：
SHF Small Collaborative Research Transform

项目摘要

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.

基于偏微分方程 (PDE) 的计算机模拟在科学和工程中无处不在，为天气预报、汽车和飞机设计过程以及海啸预测等用例提供支持。它们基于数学导数，因此仅考虑物理原理和相互作用的“局部”描述。局部模型无法捕捉某些自然过程（如反常扩散），因此计算科学家越来越多地考虑“非局部”模型。其中包括涉及更一般相互作用（例如分数导数）的积分方程公式和模型。虽然非局部方法的有效数值方法是已知的并且是正在进行的积极研究的主题，但软件支持远不如局部运营商成熟。对这两种方法耦合的支持虽然非常重要，但基本上不存在。在该项目中，研究人员将数值方法和软件工具方面的专业知识结合起来，供本地和非本地运营商扩展 Firedrake 项目 (https://www.firedrakeproject.org)（一个高级偏微分方程工具集），以服务于这一重要任务。需要。这种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高效实现。这些工具将为科学家和工程师提供一种支持技术，使其能够可靠、高效地解决比当前可用的更广泛的模型。该项目下开发的所有软件都将在开源许可证下免费分发，获得的知识将通过会议演示、出版物和教学进行传播。这项工作将利用并建立在研究人员在每个抽象层（运算符、算法、循环嵌套等）开发一套表示形式和工具的工作基础上，以将这些抽象向下转换为机器代码。研究人员将规划并扩展有限元算法的范围，以包括新的非局部算法；提供一个统一的框架来推理这些算法、设计语言和编译器基础，从而允许矩阵汇编和算子应用任务的完整规范；并在工具包中部署自动化非本地操作员，该工具包已包含经典有限元方法，并且能够针对特定架构进行定位。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查进行评估，被认为值得支持标准。