SHF: Small: Collaborative Research: Transform-to-perform: languages, algorithms, and solvers for nonlocal operators

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.