Automated Intrusive Algorithms for Numerical Simulation of Partial Differential Equations via Software-Based Frechet Differentiation

摘要：自动侵入算法的数值模拟偏微分方程通过软件为基础的Frechet微分计算机的发明自动化繁琐和容易出错的数值计算;具有讽刺意味的是，编程计算机本身是一个乏味和容易出错的任务。考虑到开发科学模拟程序的重要性和费用，研究计算机是否能改进这些程序的开发和执行是值得的。 该项目的重点是开源软件圣丹斯，它自动从高级数学抽象到高性能，并行偏微分方程（PDE）仿真代码的过渡，使用户从低级编程的负担。 这种方法可以将模拟器的开发时间从几个月或几年减少到几天甚至几小时。不太明显但同样重要的是，将软件牢固地建立在抽象基础上的好处是，内部性能改进可以自动化，以及侵入式算法--需要对方程组进行转换以产生非标准运算符的算法--可以更容易地实现，因为这样的转换可以自动进行。这是通过将编程任务制定为数学问题，自动的这一点的核心是一个定理，建立弗雷歇微分作为一个“桥梁”之间的高级符号编程和高性能numericalcomputing。 以前的工作奠定了基础，这将这些结果的工作扩展到其他方面的PDE模拟和tonon-PDE范例，如密度泛函理论（DFT），并调查侵入预条件耦合multiphysicsproblems。 自动高性能和有效的预处理，灵敏度分析和PDE约束优化的侵入式算法的现成可用性相结合，使高层次的通用工具，如圣丹斯，实际上优于手工编码的专用模拟器。 先进的优化算法与有限元离散的非线性耦合系统的现成可用性，所有与有效的实施和较短的开发周期，将是变革如何计算科学家的工作。

Abstract: Automated intrusive algorithms for numericalsimulation of partial differential equations via software-basedFrechet differentiation Computers were invented to automate tedious and error-prone numericalcomputations; ironically, programming computers is itself a tediousand error-prone task. Given the importance and expense of developingscientific simulation programs, it is worth studying whether computerscan improve the development, as well as the execution, of theseprograms. This project focuses on the open-source software Sundance,which automates transition from high-level mathematical abstractionsto high-performance, parallel partial differential equation (PDE)simulation code, freeing users from the burden of low-levelprogramming. This approach can reduce simulator development time frommonths or years to days or even hours. Less obvious, but equallyimportant, benefits of basing software firmly on mathematicalabstractions are that internal performance improvements can beautomated, and that intrusive algorithms -- algorithms that requiretransformation of the equation set to produce nonstandard operators --can be implemented much more easily because such transformations canbe carried out automatically.This is enabled by formulating programming tasks as mathematicalproblems whose solutions are then automated. Central to this is a theorem establishing Frechet differentiation as a ``bridge'' betweenhigh-level symbolic programming and high-performance numericalcomputing. Previous work has laid the foundations for this; thisextends those results work to other aspects of PDE simulation and tonon-PDE paradigms such as density functional theory (DFT), and investigates intrusive preconditioners for coupled multiphysicsproblems. The combination of automatic high performance andthe ready availability of efficient intrusive algorithms forpreconditioning, sensitivity analysis, and PDE-constrainedoptimization makes it possible for a high-level, general-purpose toolsuch as Sundance to actually outperform hand-coded special-purposesimulators. The ready availability of advanced optimizationalgorithms with finite element discretizations for nonlinear coupledsystems, all with efficient implementation and a short developmentcycle, will be transformative to how computational scientists work.