Computing Derivatives for PETSc Adjoint Solvers using Algorithmic Differentiation

TL;DR

This paper explores using algorithmic differentiation with ADOL-C to efficiently compute Jacobians for nonlinear PDEs in PETSc, offering a problem-independent alternative to hand-coding or finite differences.

Contribution

It demonstrates the application of ADOL-C for Jacobian computation in PDE solvers, comparing strategies like compressed and matrix-free approaches against traditional methods.

Findings

ADOL-C provides accurate Jacobians comparable to analytic derivations.

Matrix-free and compressed strategies improve computational efficiency.

Numerical experiments validate the effectiveness of AD-based Jacobian computation.

Abstract

Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as finite differences. For complex applications, hand-coding the Jacobian can be time-consuming and error-prone, yet computationally efficient. Whilst finite difference approximations are straight-forward to implement, they have high arithmetic complexity and low accuracy. Alternatively, one may compute Jacobians using algorithmic differentiation (AD), yielding the same derivatives as an analytic derivation, with the added benefit that the implementation is problem independent. In this work, the operator overloading AD tool ADOL-C is applied to generate Jacobians for time-dependent, nonlinear PDEs and their adjoints. Various strategies are considered, including compressed and matrix-free approaches. In numerical experiments with a 2D diffusion-reaction model, the performance of these strategies has been studied and compared to the hand-derived version.