A multi-level ADMM algorithm for elliptic PDE-constrained optimization problems
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A multi-level ADMM algorithm for elliptic PDE-constrained optimization problems Xiaotong Chen1 · Xiaoliang Song1
· Zixuan Chen2 · Bo Yu1
Received: 9 May 2020 / Revised: 19 October 2020 / Accepted: 5 November 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, elliptic PDE-constrained optimization problems with box constraints on the control are considered. To numerically solve the problems, we apply the ‘optimize-discretizeoptimize’ strategy. Specifically, the alternating direction method of multipliers (ADMM) algorithm is applied in function space first, and then, the standard piecewise linear finiteelement approach is employed to discretize the subproblems in each iteration. Finally, some efficient numerical methods are applied to solve the discretized subproblems based on their structures. Motivated by the idea of the multi-level strategy, instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, the subproblems in each iteration are solved by appropriate inexact methods. Based on the strategies above, an efficient convergent multi-level ADMM (mADMM) algorithm is proposed. We present the convergence analysis and the iteration complexity results o(1/k) for the mADMM algorithm. Some numerical experiments are done and the numerical results show the high efficiency of the mADMM algorithm. Keywords PDE-constrained optimization · ADMM · Multi-level · Convergence analysis Mathematics Subject Classification 49J20 · 49N05 · 49M25 · 68W01
Communicated by Jinyun Yuan.
B
Xiaoliang Song [email protected] Xiaotong Chen [email protected] Zixuan Chen [email protected] Bo Yu [email protected]
1
School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China
2
College of Sciences, Northeastern University, Shenyang, Liaoning 110819, China 0123456789().: V,-vol
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X. Chen et al.
1 Introduction In this paper, we consider the elliptic PDE-constrained optimization problem with box constraints on the control: min
(y,u)∈Y ×U
s.t.
J (y, u) =
1 α y − yd 2L 2 (Ω) + u2L 2 (Ω) 2 2
L y = u + yr y=0
(P)
in Ω, on ∂Ω,
u ∈ Uad = {v(x)|a ≤ v(x) ≤ b, a.e on Ω} ⊆ U , where Y := H01 (Ω), U := L 2 (Ω), Ω ⊆ Rn (n = 2, 3) is a convex, open, and bounded domain with C 1,1 -or polygonal boundary; the desired state yd ∈ H 1 (Ω) and the source term yr ∈ H 1 (Ω) are given; parameters α > 0, −∞ < a < b < +∞; L is the uniformly elliptic differential operator given by: L y := −
n
(ai j yxi )x j + c0 y,
i, j=1
where ai j , c0 ∈ L ∞ (Ω), c0 ≥ 0, ai j = a ji and there is a constant θ > 0, such that: n
ai j (x)ξi ξ j ≥ θ ξ 2 , a.a. x ∈ Ω, ∀ξ ∈ Rn .
i, j=1
As is known to all, there are two possible approaches to tackle optimization problems with PDE constraints numerically. One is First discretize, then optimize, another is First optimize, then discretize Hinze et al. (2009). In the first approach, the discretization is applied
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