Fast solver of optimal control problems constrained by Ohta-Kawasaki equations
- PDF / 4,336,803 Bytes
- 23 Pages / 439.642 x 666.49 pts Page_size
- 8 Downloads / 212 Views
Fast solver of optimal control problems constrained by Ohta-Kawasaki equations Rui-Xia Li1 · Guo-Feng Zhang1,2 · Zhao-Zheng Liang1 Received: 21 May 2019 / Accepted: 23 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is concerned with fast solver of distributed optimal control problems constrained by a nonlocal Cahn-Hilliard equation. By eliminating the control variable, a linear system on four-by-four block matrix form is obtained after discretization. Deforming the corresponding coefficient matrix into a form with special structure, an efficient preconditioner that can be utilized in an inner-outer way is designed, which leads to a fast Krylov subspace solver, that is robust with respect to mesh sizes, model parameters, and regularization parameters. Moreover, we prove that the eigenvalues of the corresponding preconditioned system are all real. Numerical experiments are presented to illustrate the robustness of the proposed solution methods. Keywords Cahn-Hilliard equation · Optimal control · Preconditioning · Iterative solution method · Spectral analysis
1 Introduction The Cahn-Hilliard (CH) equation is the major mathematical tool used in the phasefield model, which is based on minimization of the so-called free energy functional E as follows: E(C) = f (C)d,
Guo-Feng Zhang
gf [email protected] Rui-Xia Li [email protected] 1
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China
2
Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu Province, People’s Republic of China
Numerical Algorithms
for some spatial domain ∈ Rd , d = 1, 2, 3. Depending on the particular physical phenomena, the free energy density function f may have different forms. In the study of interface dynamics, phase-field models have drawn a considerable amount of attention in recent years. In 1958, the equations that describe phase transition were proposed by Cahn and Hilliard [10], that is, the so-called CH equations. Such models have been widely applied to many applications, for example, fluid dynamics, image processing, reaction pathways controlling the structural evolution of complex material mixtures and many more [21]. Research on CH equations has been extensively developed recently. Theoretical research on global existence and the uniqueness of the solutions of the CH equations has been shown [31]. Computational research on the numerical methods using the finite element approximation [5] and the finitedifference method [20] was investigated. For the literature on numerical issues, we also refer to [8, 9], to mention just a few. However, in many applications, it is practical and meaningful to influence the phase transition in such a way that a scheduled goal is achieved, that is the optimal control problem of the CH equation. The optimal control of two-phase systems has been studied in various papers. For the optimal control problem constrained by the viscous CH equations, Zhao et al. proved the existe
Data Loading...
