Matrix-Free Jacobian Chaining

TL;DR

This paper introduces a matrix-free approach to efficiently compute Jacobians in large-scale simulations by reformulating the matrix chain product problem, leveraging algorithmic differentiation and limited memory techniques.

Contribution

It presents a novel matrix-free Jacobian chaining method that reduces memory usage and improves efficiency in large-scale modular simulations.

Findings

Open-source implementation available for reproducibility

Significant memory savings demonstrated in numerical experiments

Applicable to large-scale scientific computing problems

Abstract

The efficient computation of Jacobians represents a fundamental challenge in computational science and engineering. Large-scale modular numerical simulation programs can be regarded as sequences of evaluations of in our case differentiable subprograms with corresponding elemental Jacobians. The latter are typically not available. Tangent and adjoint versions of the individual subprograms are assumed to be given as results of algorithmic differentiation instead. The classical (Jacobian) Matrix Chain Product problem is reformulated in terms of matrix-free Jacobian-matrix (tangents) and matrix-Jacobian products (adjoints), subject to limited memory for storing information required by latter. All numerical results can be reproduced using an open-source reference implementation.