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

摘要：基于软件的Frechet微分计算机用于偏微分方程数值模拟的自动侵入算法的发明是为了自动进行乏味和容易出错的数值计算；具有讽刺意味的是，对计算机编程本身就是一项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成本，计算机是否可以改善这些程序的开发和执行，这是值得研究的。这个项目的重点是开源软件Sundance，它自动从高级数学抽象转换为高性能、并行的偏微分方程(PDE)模拟代码，将用户从低级编程的负担中解放出来。这种方法可以将模拟器开发时间从几个月或几年减少到几天甚至几个小时。不那么明显，但同样重要的是，将软件牢牢建立在数学抽象之上的好处是，内部性能改进可以自动进行，而侵入性算法--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作为数学问题来实现的，然后这些问题的解决方案就会自动进行。这一点的核心是一个定理，它将Frechet区分确立为高级符号编程和高性能数值计算之间的“桥梁”。前人的工作已经为此奠定了基础；这将这些结果扩展到偏微分方程组模拟和Tonon-PDE范式的其他方面，如密度泛函理论(DFT)，并研究了耦合多物理问题的介入式预条件。自动的高性能和用于预处理、灵敏度分析和PDE约束的优化的高效侵入式算法的现成可用性相结合，使得像Sundance这样的高级、通用工具实际上可以超越手工编码的专用模拟器。用于非线性耦合系统的先进的有限元离散优化算法的现成现成，所有这些算法都具有高效的实现和较短的开发周期，将改变计算科学家的工作方式。

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.