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

基于偏微分方程（PDEs）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、海啸预测以及其他用例。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。局部模型无法捕捉某些自然过程（如异常扩散），因此计算科学家越来越多地考虑“非局部”模型。这些包括积分方程公式和涉及更一般的相互作用的模型，如分数导数。虽然非局部方法的有效数值方法是已知的，并且正在进行积极的研究，但软件支持远不如局部操作成熟。对耦合这两种方法的支持虽然非常重要，但基本上不存在。在这个项目中，研究人员正在结合数值方法和软件工具方面的专业知识，为本地和非本地运营商扩展Firedrake项目（https://www.firedrakeproject.org），这是一个高级PDE工具集，以满足这一重要需求。这个扩展涵盖了相应软件的所有方面，从用于问题描述的计算机语言到算法和有效实现。这些工具将为科学家和工程师提供一种使能技术，以可靠和有效地解决比目前可用的更广泛的模型。在这个项目下开发的所有软件都将在开源许可下自由发布，获得的知识将通过会议演讲、出版物和教学传播。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、循环巢等）开发一套表示和工具的基础上，以将这些抽象向下转换为机器代码。研究人员将绘制和扩展有限元算法的景观，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，允许矩阵组装和操作符应用任务的完整规范；并在工具包中部署自动化的非本地操作符，该工具包已经包含了经典的有限元方法，并且能够针对特定的体系结构进行定位。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.