TY - JOUR
T1 - Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem
AU - Li , Mengyu
AU - Liu , Tiegang
AU - Cao , Kui
AU - Feng , Chengliang
AU - Zhang , Bin
AU - Yuan , Weixiong
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 817
EP - 844
PY - 2025
DA - 2025/09
SN - 18
DO - http://doi.org/10.4208/nmtma.OA-2024-0135
UR - https://global-sci.org/intro/article_detail/nmtma/24329.html
KW - Singularly perturbed equation-constrained optimization problem, exponential-type finite difference scheme, Il’in-Allen-Southwell scheme, entirely exponential-type scheme.
AB - <p style="text-align: justify;">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.</p>