Automated derivation of the adjoint of high-level transient finite element programs

TL;DR

This paper introduces a high-level, automated method for deriving adjoint and tangent linear models of finite element programs, significantly improving efficiency and ease of use over traditional techniques.

Contribution

The paper presents a novel high-level symbolic approach for automatic adjoint derivation in finite element models, implemented in the dolfin-adjoint software package based on FEniCS.

Findings

Automated adjoint derivation is more efficient than standard methods.

The approach works with complex, nonlinear, time-dependent models.

It automatically manages checkpointing and parallel execution.

Abstract

In this paper we demonstrate a new technique for deriving discrete adjoint and tangent linear models of finite element models. The technique is significantly more efficient and automatic than standard algorithmic differentiation techniques. The approach relies on a high-level symbolic representation of the forward problem. In contrast to developing a model directly in Fortran or C++, high-level systems allow the developer to express the variational problems to be solved in near-mathematical notation. As such, these systems have a key advantage: since the mathematical structure of the problem is preserved, they are more amenable to automated analysis and manipulation. The framework introduced here is implemented in a freely available software package named dolfin-adjoint, based on the FEniCS Project. Our approach to automated adjoint derivation relies on run-time annotation of the temporal structure of the model, and employs the FEniCS finite element form compiler to automatically generate the low-level code for the derived models. The approach requires only trivial changes to a large class of forward models, including complicated time-dependent nonlinear models. The adjoint model automatically employs optimal checkpointing schemes to mitigate storage requirements for nonlinear models, without any user management or intervention. Furthermore, both the tangent linear and adjoint models naturally work in parallel, without any need to differentiate through calls to MPI or to parse OpenMP directives. The generality, applicability and efficiency of the approach are demonstrated with examples from a wide range of scientific applications.