Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem

We found that no convergence to the correct solution can happen when a popular method is applied to discretize the derivative appearing in the objective function for optimization problems with singularly perturbed ODE constraints. The non-convergence mentioned above can occur even if the error bound of the numerical solution of the state equation has nothing to do with the small parameter. We disclose that the underlying reason for non-convergence to the correct solution is an inaccurate derivative calculation in the objective function for a model problem, which is solvable mathematically. To ensure correct convergence regardless of the small parameter, we propose an entirely exponential-type scheme for solving the optimization problem, in which an exponential-type scheme is used for the derivative in the objective function, together with an exponential-type finite difference scheme for the state equation. Both theoretical analysis and numerical experiments can verify the correct convergence of E2S in solving the singularly perturbed equation-constrained optimization problem.