Approximation of weak adjoints by reverse automatic differentiation of BDF methods
This work provides a theoretical foundation for applying finite element error estimation techniques to BDF methods, benefiting numerical analysts and practitioners in scientific computing.
The authors establish a theoretical connection between discrete adjoints of BDF methods and weak adjoint solutions via a variational framework, proving asymptotic convergence of discrete adjoints to classical adjoints. Numerical results for non-adaptive and adaptive BDF schemes are provided.
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.