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：我们研究了与演化型偏微分方程相关联的伴随系统在完全连续、半离散和完全离散三个层次上的几何结构，以及这些层次之间的相互关系。我们证明，与演化型偏微分方程相关联的伴随系统具有无穷维哈密顿结构，这一性质有助于连接完全连续、半离散和完全离散这三个层次。随后，我们通过唯一地表征离散后优化方法与优化后离散方法在伴随-变分二次守恒律方面的交换性，探讨了半离散化和时间积分中的"离散后优化"与"优化后离散"问题。特别地，对于Galerkin半离散化和单步时间积分方法，我们利用保结构离散技术显式构建了这些可交换方法。