Features of dolfin-adjoint


dolfin-adjoint works for both steady and time-dependent models, and for both linear and nonlinear models. The user interface is exactly the same in both cases. For an example of adjoining a nonlinear time-dependent model, see the tutorial.

Ease of use

dolfin-adjoint has been carefully designed to try to make its use as easy as possible. In many cases the only change to the forward model is to add

from dolfin_adjoint import *

at the top of the model. For example, deriving the adjoint of the tutorial example requires adding precisely three lines to the forward model. dolfin-adjoint also makes it extremely easy to verify the correctness of the adjoint model. It offers a powerful syntax for expressing general functionals.


Efficiency of the resulting model is absolutely crucial for real applications. The efficiency of an adjoint model is measured as (time for forward and adjoint run)/(time for forward run). Naumann (2011) states that a typical value for this ratio when using algorithmic differentiation tools is in the range 3–30. By contrast, dolfin-adjoint is extremely efficient; consider the following examples from the papers:

PDE Theoretical optimum Achieved efficiency
Cahn-Hilliard 1.2 1.22
Stokes 2.0 1.86
Viscoelasticity 2.0 2.029
Gross-Pitaevskii 1.5 1.54
Gray-Scott 2.0 2.04
Navier-Stokes 1.33 1.41
Mathematical programming with equilibrium constraints 1.125 1.126
Shallow water 1.125 1.125
Wetting and drying 1.5 1.55


Parallelism is ubiquitous in modern computational science. However, applying algorithmic differentiation to parallel codes is still a major research challenge. Algorithmic differentiation tools must be specially modified to understand MPI and OpenMP directives, and translate them into their parallel equivalents. By contrast, because of the high-level abstraction taken in libadjoint, the problem of parallelism simply disappears. In fact, there is no code whatsoever in either dolfin-adjoint or pyadjoint to handle parallelism; by deriving the adjoint at the right level of abstraction, the problem no longer exists. If the forward model runs in parallel, the adjoint model also runs in parallel, with no modification.

For more details, see the manual section on parallelism and the dolfin-adjoint paper.