Reduced-order finite element approximation based on POD for the parabolic optimal control problem

• Published in: Numerical algorithms Vol. 95; no. 3; pp. 1189 - 1211
• Main Authors: Song, Junpeng, Rui, Hongxing
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2024; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computer Science; Finite element method; Inequality; Mathematical analysis; Mathematical models; Methods; Numeric Computing; Numerical Analysis; Optimal control; Original Paper; Proper Orthogonal Decomposition; Theory of Computation; Reduced-order finite element method; Proper orthogonal decomposition; Computational costs; Error estimates; Accurate and efficient; Parabolic optimal control
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-023-01605-x

In this paper, we construct a reduced-order finite element (ROFE) method holding seldom unknowns for the parabolic optimal control problem. We apply the proper orthogonal decomposition (POD) technique to develop two unsteady systems about state and co-state approximations, which efficiently reduces the number of unknowns and computational costs. Optimal a priori error estimates for the state, co-state and control approximations are derived. Finally, numerical examples are presented to verify that the ROFE method is accurate and efficient for solving the parabolic optimal control problem.