Computer Science Technical Reports
CS at VT

On the discrete adjoints of adaptive time stepping algorithms

Alexe, Mihai and Sandu, Adrian (2008) On the discrete adjoints of adaptive time stepping algorithms. Technical Report TR-08-08, Computer Science, Virginia Tech.

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We investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation (AD). By differentiating the time step controller and the error estimator of the original algorithm, reverse mode AD generates spurious adjoint derivatives of the time steps. The resulting discrete adjoint models become inconsistent with the adjoint ODE, and yield incorrect derivatives. To regain consistency, one has to cancel out the contributions of the non-physical derivatives in the discrete adjoint model. We demonstrate that the discrete adjoint models of one-step, explicit adaptive algorithms, such as the Runge--Kutta schemes, can be made consistent with their continuous analogs using simple code modifications. Furthermore, we extend the analysis to cover second order adjoint models derived through an extra forward-mode differentiation of the discrete adjoint code. Two numerical examples support the mathematical derivations.

Item Type:Departmental Technical Report
Keywords:adaptive adjoints, automatic differentiation, Runge - Kutta methods, sensitivity analysis
Subjects:Computer Science > Mathematical Software
ID Code:1028
Deposited By:Alexe, Mihai
Deposited On:28 April 2008