Volume 44, pp. 367-400, 2015.

Efficient preconditioners for PDE-constrained optimization problem with a multilevel sequentially semiseparable matrix structure

Yue Qiu, Martin B. van Gijzen, Jan-Willem van Wingerden, Michel Verhaegen, and Cornelis Vuik

Abstract

PDE-constrained optimization problems yield a linear saddle-point system that has to be solved. We propose a preconditioner that makes use of the global MSSS structure and a preconditioner that exploits the block MSSS structure of the saddle-point system. For the computation of preconditioners based on MSSS matrix computations, model order reduction algorithms are essential to obtain a low computational complexity. We study two different model order reduction approaches, one is the new approximate balanced truncation with low-rank approximated Gramians for SSS matrices and the other is the standard Hankel blocks approximation algorithm. We test our preconditioners on the problems of optimal control of the convection-diffusion equation in 2D and of the Poisson equation in 3D. For 2D problems, numerical experiments illustrate that both preconditioners have linear computational complexity and the global MSSS preconditioner reduces the number of iterations significantly and needs less computation time. Moreover, the approximate balanced truncation algorithm is computationally cheaper than the Hankel blocks approximation algorithm. Besides the mesh size independent convergence, the global MSSS preconditioner also gives the regularization parameter independent convergence, while the block MSSS preconditioner just gives mesh size independent convergence. For 3D problems, both the block MSSS preconditioner and global MSSS preconditioner give virtually mesh size independent convergence. Furthermore, the global MSSS preonconditioner reduces the number of iterations dramatically compared with the block MSSS preconditioner.

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Key words

PDE-constrained optimization, saddle-point problem, preconditioners, multilevel sequentially semiseparable matrix, model order reduction, low-rank approximation

AMS subject classifications

15B99, 65Fxx, 93C20, 65Y20

Links to the cited ETNA articles

[9]	Vol. 38 (2011), pp. 113-145 Younès Chahlaoui: Two efficient SVD/Krylov algorithms for model order reduction of large scale systems

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