Automated derivation of the adjoint of high-level transient finite element programs (1204.5577v2)

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.

• The paper presents a framework that automates the derivation of adjoint models from finite element programs using symbolic representations of the forward problem.
• It leverages FEniCS and libadjoint to simplify differentiation by abstracting complex tasks like checkpointing and parallel communication.
• The study demonstrates efficient, accurate sensitivity analysis and optimization in real-world applications while outlining potential improvements for future work.

Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs

The computational science community has placed significant emphasis on the development of adjoint models due to their utility in numerous scientific and engineering applications, such as parameter estimation, sensitivity analysis, and optimization. The paper "Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs" by Farrell et al. presents a novel framework for the automated derivation of adjoint and tangent linear models, particularly focusing on their application within finite element systems.

The authors introduce a computational approach that hinges on the symbolic representation of the forward problem. By leveraging the capabilities of a high-level finite element environment, FEniCS, and integrating it with the libadjoint library, the framework facilitates the derivation of adjoint models with minimal manual intervention. This integration, encapsulated within the software package dolfin-adjoint, permits the automatic generation of adjoint and tangent linear models from forward models coded in FEniCS, requiring only trivial code annotations.

The essence of the proposed method lies in its abstraction of mathematical problems into sequences of equation solves. This abstraction effectively disambiguates the mathematical formulation of the problem from its implementation, thereby aligning closely with the principles of finite element modeling. The libadjoint library plays a critical role by enabling symbolic manipulation of the annotated systems to determine the structure of the adjoint equations. This integration eschews the complexities involved in algorithmic differentiation approaches that deal with low-level code components, such as memory management and parallel communications, simplifying the user's task in developing adjoint models.

Key advantages of the framework include its efficiency in handling complex and nonlinear time-dependent models, facilitated by optimal checkpointing schemes that manage memory requirements without detracting from computational throughput. Moreover, the framework seamlessly extends to parallel computing environments, circumventing the challenges typically associated with differentiating parallel communications.

The framework also extends its utility to matrix-free solvers, often prevalent in large-scale scientific computations such as mantle convection models. By abstracting the problem at a high level, the tool can handle matrix-free implementations without additional complications.

The paper presents a series of numerical examples across diverse applications, demonstrating the framework's wide applicability and verifying the accuracy of the generated adjoint solutions through meticulous convergence tests. In these examples, the efficiency of adjoint computations closely approaches theoretical expectations underlining the robustness of the dolfin-adjoint system.

However, the approach presents limitations, particularly the restriction to exclusive use of the Python interface to DOLFIN and the inability to differentiate functions affiliated directly with manipulated mesh parameters. These constraints do present areas for exploration and potential enhancement in future work.

The implications of this research are profound, as it provides a systematic, automated pathway to efficient adjoint computation in finite element modeling. This development holds promise for enhancing computational capabilities in scientific disciplines that rely heavily on optimization and sensitivity studies. Future work could aim to extend the capabilities of dolfin-adjoint to broader contexts and explore more generalized adjoint strategies across other computational frameworks within scientific computing.