KMS Of Academy of mathematics and systems sciences, CAS
| Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems | |
| Wu, Shu-Lin1; Zhou, Tao2,3 | |
| 2020-11-13 | |
| Source Publication | ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
 |
| ISSN | 1292-8119 |
| Volume | 26Pages:26 |
| Abstract | Solving parabolic PDE-constrained optimization problems requires to take into account the discrete time points all-at-once, which means that the computation procedure is often time-consuming. It is thus desirable to design robust and analyzable parallel-in-time (PinT) algorithms to handle this kind of coupled PDE systems with opposite evolution directions. To this end, for two representative model problems which are, respectively, the time-periodic PDEs and the initial-value PDEs, we propose a diagonalization-based approach that can reduce dramatically the computational time. The main idea lies in carefully handling the associated time discretization matrices that are denoted by B-per and B-ini for the two problems. For the first problem, we diagonalize B-per directly and this results in a direct PinT algorithm (i.e., non-iterative). For the second problem, the main idea is to design a suitable approximation B-per B per of B-ini, which naturally results in a preconditioner of the discrete KKT system. This preconditioner can be used in a PinT pattern, and for both the Backward-Euler method and the trapezoidal rule the clustering of the eigenvalues and singular values of the preconditioned matrix is justified. Compared to existing preconditioners that are designed by approximating the Schur complement of the discrete KKT system, we show that the new preconditioner leads to much faster convergence for certain Krylov subspace solvers, e.g., the GMRES and BiCGStab methods. Numerical results are presented to illustrate the advantages of the proposed PinT algorithm. |
| Keyword | Parabolic PDE-constrained optimization PinT algorithm diagonalization technique preconditioner GMRES BiCGStab |
| DOI | 10.1051/cocv/2020012 |
| Indexed By | SCI |
| Language | 英语 |
| Funding Project | NSF of China[11822111] ; NSF of China[11688101] ; NSF of China[11731006] ; NSF of China[11771313] ; NSF of Sichuan Province[2018JY0469] ; Science Challenge Project[TZ2018001] ; Science Challenge Project[TZ2016002] ; National Key Basic Research Program[2018YFB0704304] ; Youth Innovation Promotion Association (CAS) |
| WOS Research Area | Automation & Control Systems ; Mathematics |
| WOS Subject | Automation & Control Systems ; Mathematics, Applied |
| WOS ID | WOS:000594831500004 |
| Publisher | EDP SCIENCES S A |
| Citation statistics | |
| Document Type | 期刊论文 |
| Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/57784 |
| Collection | 中国科学院数学与系统科学研究院 |
| Corresponding Author | Zhou, Tao |
| Affiliation | 1.Northeast Normal Univ, Sch Math & Stat, Changchun 130024, Peoples R China 2.Chinese Acad Sci, NCMIS, Inst Computat Math & Sci Engn Comp, Acad Math & Syst Sci, Beijing 100190, Peoples R China 3.Chinese Acad Sci, LSEC, Inst Computat Math & Sci Engn Comp, Acad Math & Syst Sci, Beijing 100190, Peoples R China |
| Recommended Citation GB/T 7714 | Wu, Shu-Lin,Zhou, Tao. Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems[J]. ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS,2020,26:26. |
| APA | Wu, Shu-Lin,&Zhou, Tao.(2020).Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems.ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS,26,26. |
| MLA | Wu, Shu-Lin,et al."Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems".ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 26(2020):26. |
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