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On Properties of Adjoint Systems for Evolutionary PDEs

Brian K. Tran, Ben S. Southworth, Melvin Leok

TL;DR

The paper develops an infinite-dimensional Hamiltonian framework for adjoint systems of evolution equations and uses it to analyze discretize-then-optimize versus optimize-then-discretize questions. It shows that adjoint systems possess a Hamiltonian structure that persists across fully continuous, semi-discrete, and fully discrete levels, and introduces adjoint-variational conservation laws that govern sensitivity analysis. For Galerkin semi-discretizations and one-step time integrators, commuting discretization and adjoint operations are established, with the cotangent lift playing a central role in time integration. The results yield naturality diagrams linking levels via duality pairings, establish uniqueness of dual discretizations under conservation laws, and reveal practical implications for structure-preserving discretization and preconditioning. These advances provide a geometric basis for designing DtO and OtD schemes that preserve essential adjoint structure and enable accurate gradient computations for optimization and control of PDEs.

Abstract

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.

On Properties of Adjoint Systems for Evolutionary PDEs

TL;DR

The paper develops an infinite-dimensional Hamiltonian framework for adjoint systems of evolution equations and uses it to analyze discretize-then-optimize versus optimize-then-discretize questions. It shows that adjoint systems possess a Hamiltonian structure that persists across fully continuous, semi-discrete, and fully discrete levels, and introduces adjoint-variational conservation laws that govern sensitivity analysis. For Galerkin semi-discretizations and one-step time integrators, commuting discretization and adjoint operations are established, with the cotangent lift playing a central role in time integration. The results yield naturality diagrams linking levels via duality pairings, establish uniqueness of dual discretizations under conservation laws, and reveal practical implications for structure-preserving discretization and preconditioning. These advances provide a geometric basis for designing DtO and OtD schemes that preserve essential adjoint structure and enable accurate gradient computations for optimization and control of PDEs.

Abstract

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.
Paper Structure (13 sections, 10 theorems, 133 equations)

This paper contains 13 sections, 10 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Discretize-then-Optimize versus Optimize-then-Discretize
  3. Main Results
  4. Adjoint Systems for Evolutionary PDEs
  5. Geometry of Finite-Dimensional Adjoint Systems
  6. Semilinear Evolutionary PDEs
  7. Infinite-dimensional Hamiltonian Systems
  8. Adjoint Systems for Evolutionary PDEs
  9. Discretization of Adjoint Systems
  10. Semi-Discretization of Adjoint Systems
  11. Time Integration
  12. Naturality of the Full Discretization
  13. Computational Implications

Key Result

Proposition 2.1

For the Hamiltonian $H(q,p) = \langle p, f(q)\rangle_q$, the corresponding Hamiltonian vector field $X_H$ is equal to the cotangent lift $\widehat{f}$ of $f$.

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Proposition 2.2: Proposition 2.2 of TrLe2024
  • Proposition 2.3: Proposition 2.3 of TrLe2024
  • Example 2.1: Linear Case
  • Proposition 2.4
  • proof
  • Remark 2.1
  • Proposition 3.1
  • proof
  • ...and 17 more