AF: Small: Metanumerical Computing for Emerging Architectures: Automated Embedded Algorithms for Partial Differential Equations on Multicore Platforms

计算机的发明是为了自动化繁琐和容易出错的任务，特别是在科学模拟的数值计算方面。考虑到开发这些模拟的重要性和困难性，研究计算机是否也可以帮助它们的生产是有意义的。 本项目的重点是开源项目圣丹斯，它提供了高级语法来描述有限元方法（FEM）的偏微分方程的数值解，使用户从底层编程的负担。这种方法可以将开发时间从几个月缩短到几天甚至几小时，使科学家和工程师能够更快地获得正确的答案。圣丹斯是一个有用的软件，从坚实的理论基础出发，其中有限元算子的结构是从基于软件的变分形式的Frechet微分推导和分析的。通过这个特定领域的嵌入式语言，圣丹斯用户（领域科学家和工程师）可以表达和模拟非线性过程，比线性过程更容易;雅可比矩阵是内部自动计算的。 这个强大的系统自动化不仅标准的模拟，而且灵敏度计算，eigenvalueproblems，和偏微分方程约束优化，所有通过微分的语言，提供了一个简单的网关，有效的“嵌入式”优化和控制算法。 此外，在这个项目中，圣丹斯将被扩展到运行在新兴的架构，如多核CPU和GPU机器。 基于伯恩斯坦多项式的新的低复杂度有限元算法将被开发并包含在圣丹斯中，以进一步利用这种新的硬件。 因此，该项目代表了具体的变革性贡献，不仅自动化数学软件，而且在核心数值算法，扩大了领域专家的圈子，获得了领先的仿真技术。

Computers were invented to automate tedious and error-pronetasks, especially in the context of numerical calculations forscientific simulation. Given the importance and difficulty of developing these simulations, it makes sense to study whether computers can assist in their production as well. This project focuses on the open-source project Sundance, which provideshigh-level syntax to describe finite element methods (FEM) for thenumerical solution of partial differential equations, freeing users from the burdens of low-level programming. This approach can reduce development time from months to days or even hours, allowing scientists and engineers to get correct answers more quickly. Sundance is useful software starting from a solid theoretical basis,where the structure of finite element operators are derived andanalyzed from software-based Frechet differentiation of variationalforms.Via this domain-specific embedded language, Sundance users (domainscientistis and engineers) can express and simulate nonlinear processes with little more difficulty than linear ones; Jacobians are calculated internally and automatically. This robust system automates not only standard simulations, but also sensitivity calculations, eigenvalueproblems, and PDE-constrained optimization, all through the language ofdifferentiation, providing a simple gateway to efficient ``embedded''algorithms for optimization and control. Furthermore, in this project, Sundance will be extended to run on emerging architectures such as multicore CPU and GPU machines. New low-complexity finite element algorithms based on Bernstein polynomials will be developed and included within Sundance to further exploit this new hardware. Thus, this project represents concrete transformational contributions not only automatingmathematical software but in core numerical algorithms, broadening thecircle of domain specialists with access to leading-edge simulationtechnology.