Approximation of weak adjoints by reverse automatic differentiation of BDF methods

TL;DR

This paper explores the relationship between discrete adjoints of BDF methods and the solutions of adjoint differential equations, introducing a weak adjoint framework and proving convergence properties.

Contribution

It develops a functional-analytic framework for weak adjoints, linking discrete adjoints of BDF methods with continuous adjoint solutions, and proves their convergence.

Findings

Finite element approximation of weak adjoints converges asymptotically.

Discrete adjoints converge to classical adjoints on the inner time interval.

Numerical results demonstrate the framework's effectiveness for adaptive BDF schemes.

Abstract

With this contribution, we shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions. We devise a finite element Petrov-Galerkin interpretation of the BDF method together with its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the finite element approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its asymptotic convergence in the space of normalized functions of bounded variation. We also obtain asymptotic convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over the existing theory on global error estimation techniques from finite element methods to BDF methods.