We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the adjoint system associated with an evolutionary partial differential equation has an infinite-dimensional Hamiltonian structure, which is useful for connecting the fully continuous, semi-discrete, and fully discrete levels. We subsequently address the question of discretize-then-optimize versus optimize-then-discrete for both semi-discretization and time integration, by characterizing the commutativity of discretize-then-optimize methods versus optimize-then-discretize methods uniquely in terms of an adjoint-variational quadratic conservation law. For Galerkin semi-discretizations and one-step time integration methods in particular, we explicitly construct these commuting methods by using structure-preserving discretization techniques.

翻译：本文研究了演化偏微分方程伴随系统在全连续、半离散及全离散层面上的几何结构，以及这些层面之间的关联。我们证明了演化偏微分方程的伴随系统具有无穷维哈密顿结构，该结构有助于连接全连续、半离散与全离散层面。随后，我们通过以伴随变分二次守恒律唯一刻画"先离散后优化"方法与"先优化后离散"方法的交换性，探讨了半离散化与时间积分中这两类方法的对比问题。特别针对伽辽金半离散化与单步时间积分方法，我们利用结构保持离散化技术显式构造了这些可交换方法。