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

通过基于软件的 Frechet 微分进行偏微分方程数值模拟的自动侵入算法

• 批准号: 0830655
• 负责人: Robert Kirby
• 金额: $ 27万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-10-01 至 2011-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0830655&HistoricalAwards=false
• 关键词: Automated; Intrusive; Algorithms; Numerical; Simulation

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.

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是为了自动化繁琐且容易出错的数值计算而发明的。具有讽刺意味的是，计算机编程本身就是一项单调乏味、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，计算机是否能改善这些程序的开发和执行是值得研究的。这个项目的重点是开源软件Sundance，它可以自动从高级数学抽象转换到高性能的并行偏微分方程（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的开发时间从几个月或几年缩短到几天甚至几个小时。不太明显，但同样重要的是，将软件牢固地建立在数学抽象上的好处是，内部性能改进可以自动化，而且侵入式算法——需要对方程集进行转换以产生非标准运算符的算法——可以更容易地实现，因为这种转换可以自动进行。这是通过将编程任务表述为数学问题，然后将其解决方案自动化来实现的。其核心是一个定理，将Frechet微分建立为高级符号编程和高性能数值计算之间的“桥梁”。之前的工作已经为此奠定了基础；这将这些结果扩展到PDE模拟和非PDE范式（如密度泛函理论（DFT））的其他方面，并研究耦合多物理场问题的侵入性前置条件。自动高性能和预先调节、灵敏度分析和pde约束优化的有效侵入算法的组合使得像Sundance这样的高级通用工具实际上优于手工编码的专用模拟器成为可能。具有非线性耦合系统的有限元离散化的先进优化算法的现成可用性，所有这些都具有有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。