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

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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.