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

In this paper we demonstrate the capability of automatically deriving the discrete adjoint and tangent linear models from a forward model written in the high-level FEniCS finite element computing environment. 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. Our approach to automated adjoint derivation relies on run-time annotation of the temporal structure of the model, and employs the same 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 and applicability of the approach is demonstrated with examples from a wide range of scientific applications.