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

Texas Tech University, Lubbock TX

Investigators

Abstract

Abstract: Automated intrusive algorithms for numerical simulation of partial differential equations via software-based Frechet differentiation Computers were invented to automate tedious and error-prone numerical computations; ironically, programming computers is itself a tedious and error-prone task. Given the importance and expense of developing scientific simulation programs, it is worth studying whether computers can improve the development, as well as the execution, of these programs. This project focuses on the open-source software Sundance, which automates transition from high-level mathematical abstractions to high-performance, parallel partial differential equation (PDE) simulation code, freeing users from the burden of low-level programming. This approach can reduce simulator development time from months or years to days or even hours. Less obvious, but equally important, benefits of basing software firmly on mathematical abstractions are that internal performance improvements can be automated, and that intrusive algorithms -- algorithms that require transformation of the equation set to produce nonstandard operators -- can be implemented much more easily because such transformations can be carried out automatically. This is enabled by formulating programming tasks as mathematical problems whose solutions are then automated. Central to this is a theorem establishing Frechet differentiation as a ``bridge'' between high-level symbolic programming and high-performance numerical computing. Previous work has laid the foundations for this; this extends those results work to other aspects of PDE simulation and to non-PDE paradigms such as density functional theory (DFT), and investigates intrusive preconditioners for coupled multiphysics problems. The combination of automatic high performance and the ready availability of efficient intrusive algorithms for preconditioning, sensitivity analysis, and PDE-constrained optimization makes it possible for a high-level, general-purpose tool such as Sundance to actually outperform hand-coded special-purpose simulators. The ready availability of advanced optimization algorithms with finite element discretizations for nonlinear coupled systems, all with efficient implementation and a short development cycle, will be transformative to how computational scientists work.