NOTES AND CORRESPONDENCE Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part V: Coarse-Grain Adjoint and Problems in Gradient Check

When on/off switches are triggered at discrete time levels by a threshold condition in a traditionally discretized model, the model solution is not continuously dependent on the initial state and this causes problems in tangent linearization and adjoint computations. It is shown in this paper that the problems can be avoided by introducing coarse-grain tangent linearization and adjoint without modifying the traditional discretization, although the coarsegrain gradient check can be performed only for finite perturbations.