Unified Embedded Parallel Finite Element Computations via Software-Based Fréchet Differentiation

Abstract

Computational analysis of systems governed by partial differential equations (PDEs) requires not only the calculation of a solution but the extraction of additional information such as the sensitivity of that solution with respect to input parameters or the inversion of the system in an optimization or design loop. Moving beyond the automation of discretization of PDEs by finite element methods, we present a mathematical framework that unifies the discretization of PDEs with these additional analysis requirements. In particular, Fréchet differentiation on a class of functionals together with a high-performance finite element framework has led to a code, called Sundance, that provides high-level programming abstractions for the automatic, efficient evaluation of finite variational forms together with the derived operators required by engineering analysis.