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

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，支持天气预报、汽车和飞机设计过程以及海啸预测等用例。它们基于数学导数，因此只考虑物理原理和相互作用的“局部”描述。局部模型无法捕捉到某些自然过程(如异常扩散)，因此计算科学家们越来越多地考虑“非局部”模型。其中包括涉及更一般相互作用的积分方程式和模型，如分数导数。虽然非本地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，但软件支持远远不像本地运营商那样成熟。对这两种方法的耦合支持虽然非常重要，但基本上是不存在的。在这个项目中，研究人员正在结合本地和非本地运营商在数值方法和软件工具方面的专业知识来扩展Firedrake项目(https://www.firedrakeproject.org)，)，以满足这一重要需求。这一扩展涵盖了相应软件的所有方面，从用于描述问题的计算机语言到算法和有效实施。这些工具将为科学家和工程师提供一种使其能够可靠和高效地处理比目前可用的更广泛的模型的技术。在该项目下开发的所有软件将在开放源码许可下免费分发，所获得的知识将通过会议报告、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(操作符、算法、循环嵌套等)开发一套表示法的工作基础上。以及将这些抽象向下转换为机器代码的工具。调查人员将规划和扩展有限元算法的版图，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的推理提供统一的框架，允许完整规范矩阵汇编和运算符应用任务；并在工具包中部署自动化的非局部运算符，该工具包已经包括经典的有限元方法，并能够针对特定架构。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。

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.