A splitting algorithm for constrained optimization problems with parabolic equations

TL;DR

This paper introduces a parallel splitting algorithm for constrained optimal control problems governed by parabolic equations, combining finite element discretization and Jacobian decomposition to enhance computational efficiency and ensure convergence.

Contribution

The paper presents a novel parallel splitting method with Jacobian decomposition for parabolic control problems, improving efficiency and providing convergence analysis.

Findings

The proposed method significantly reduces computation time.

Numerical simulations confirm the effectiveness and accuracy of the algorithm.

The approach achieves global convergence with discretization and iteration error estimates.

Abstract

In this paper, an efficient parallel splitting method is proposed for the optimal control problem with parabolic equation constraints. The linear finite element is used to approximate the state variable and the control variable in spatial direction. And the Crank-Nicolson scheme is applied to discretize the constraint equation in temporal direction. For consistency, the trapezoidal rule and midpoint rule are used to approximate the integrals with respect to the state variable and the control variable of the objective function in temporal direction, respectively. Based on the separable structure of the resulting coupled discretized optimization system, a full Jacobian decomposition method with correction is adopted to solve the decoupled subsystems in parallel, which improves the computational efficiency significantly. Moreover, the global convergence estimate is established using the discretization error by the finite element and the iteration error by the full Jacobian decomposition method with correction. Finally, numerical simulations are carried out to verify the efficiency of the proposed method.