Approximation of weak adjoints by reverse automatic differentiation of BDF methods

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.

Discrete Adjoint Implicit Peer Methods in Optimal Control

“…Here, the discrete adjoint schemes of linear multistep methods are in general not consistent or show a significant decrease of the approximation order, see Sandu [13] and Albi et al [1]. Backward differentiation formula (BDF) and Peer methods [2,15] which are particularly suitable for large-scale, nonlinear and stiff systems of ODEs keep their high order in the interior of the time domain, but the adjoint initialization steps are usually inconsistent approximations [3,17] and the numerical approximation of missing starting values has to be done with care. These inherent difficulties have limited the application of multistep methods for optimal control problems in a first-discretizethen-optimize solution strategy.…”

“…Aiming at high-order convergence it will be shown in Section 4 that for an s-step method order O(h s ) can be shown for the y-variable if the solution for the p-variable has O(h s−1 ) convergence at least. Accordingly, the construction of 3-stage methods is based on a thorough discussion of methods with the global order pair (3,2) for the solution y and the adjoint p. This is also motivated by the fact that the order pair (3,3) can not be satisfied in our present setting. Since this discussion shows a certain preference for nodes with flip symmetry this question is pursued in Section 6 in detail by combining the forward and adjoint order conditions.…”

Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems

“…We are interested in the numerical solution of the following ODE-constrained nonlinear optimal control problem: minimize C y(T ) (4) subject to y (t) = f y(t), u(t) , u(t) ∈ U ad , t ∈ (0, T ],…”

“…There are one-step as well as multistep time integrators in common use to solve ODE constrained optimal control problems. Symplectic Runge-Kutta methods [5,20,25] and backward differentiation formulas [1,4] are prominent classes, but also partitioned and implicit-explicit Runge-Kutta methods [14,22] and explicit stabilized Runge-Kutta-Chebyshev methods [2] have been proposed. However, fully implicit one-step methods often request the solution of large systems of coupled stages and might suffer from serious order reduction due to their lower stage order.…”