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Volume 18, Issue 3
Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem

Mengyu Li, Tiegang Liu, Kui Cao, Chengliang Feng, Bin Zhang & Weixiong Yuan

DOI: 10.4208/nmtma.OA-2024-0135

Numer. Math. Theor. Meth. Appl., 18 (2025), pp. 817-844.

Published online: 2025-09

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  • Abstract

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.

  • Keywords

Singularly perturbed equation-constrained optimization problem, exponential-type finite difference scheme, Il’in-Allen-Southwell scheme, entirely exponential-type scheme.

  • AMS Subject Headings

34E15, 49M41, 65L09

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-18-817, author = {Li , MengyuLiu , TiegangCao , KuiFeng , ChengliangZhang , Bin and Yuan , Weixiong}, title = {Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2025}, volume = {18}, number = {3}, pages = {817--844}, abstract = {

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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0135}, url = {http://global-sci.org/intro/article_detail/nmtma/24329.html} }
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 -

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.

Li , MengyuLiu , TiegangCao , KuiFeng , ChengliangZhang , Bin and Yuan , Weixiong. (2025). Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem. Numerical Mathematics: Theory, Methods and Applications. 18 (3). 817-844. doi:10.4208/nmtma.OA-2024-0135
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