We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on Griewank and Walther's state transformations, synthesizes the calculation of the Hessian as a formula. These dual points of view, graphical and algebraic, lead to a new framework for Hessian computation. This is illustrated by developing edge_pushing, a new truly reverse Hessian computation algorithm that fully exploits the Hessian's symmetry. Computational experiments compare the performance of edge_pushing on sixteen functions from the CUTE collection against two algorithms available as drivers of the software ADOL-C, and the results are very promising.

中文翻译：

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一种新的 Hessians 计算框架

我们使用图模型和代数模型通过自动微分研究 Hessian 矩阵的计算。图模型揭示了计算 Hessian 所涉及的固有对称性。代数模型基于 Griewank 和 Walther 的状态变换，将 Hessian 的计算综合为一个公式。这些图形和代数的双重观点导致了 Hessian 计算的新框架。这可以通过开发 edge_pushing 来说明，这是一种新的真正反向 Hessian 计算算法，它充分利用了 Hessian 的对称性。计算实验将 CUTE 集合中的 16 个函数的 edge_pushing 性能与可用作软件 ADOL-C 驱动程序的两种算法进行了比较，结果非常有希望。