Multigrid Preconditioning of Linear Systems for Interior Point Methods Applied to a Class of Box-constrained Optimal Control Problems
- Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States). Dept. of Mathematics and Statistics
- Argonne National Lab. (ANL), Argonne, IL (United States). Mathematics and Computer Science Division
We construct and analyze multigrid preconditioners for discretizations of operators of the form $${\mathcal D}_{\lambda}+{\mathcal K}^*{\mathcal K}$$, where $$D_{\lambda}$$ is the multiplication with a relatively smooth function $$\lambda>0$$ and $${\mathcal K}$$ is a compact linear operator. These systems arise when applying interior point methods to the minimization problem $$\min_{u} \frac{1}{2}(|\!|{\mathcal K} u-f|\!|^2 +\beta|\!|u|\!|^2)$$ with box-constraints $$\underline{u}\leqslant u\leqslant\overline{u}$$ on the controls. The presented preconditioning technique is closely related to the one developed by Draganescu and Dupont [Math. Comp., 77 (2008), pp. 2001–2038] for the associated unconstrained problem and is intended for large-scale problems. As in that work, the quality of the resulting preconditioners is shown to increase as $$h\downarrow 0$$, but it decreases as the smoothness of $$\lambda$$ declines. We test this algorithm on a Tikhonov-regularized backward parabolic equation with box-constraints on the control and on a standard elliptic-constrained optimization problem. In both cases it is shown that the number of linear iterations per optimization step, as well as the total number of finest-scale matrix-vector multiplications, is decreasing with increasing resolution, thus showing the method to be potentially very efficient for truly large-scale problems.
- Research Organization:
- Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
- Grant/Contract Number:
- SC0005455; DMS-1016177; DMS-0821311; CCF-0728878
- OSTI ID:
- 1493159
- Journal Information:
- SIAM Journal on Numerical Analysis, Vol. 50, Issue 1; ISSN 0036-1429
- Publisher:
- Society for Industrial and Applied MathematicsCopyright Statement
- Country of Publication:
- United States
- Language:
- English
Web of Science
Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization
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journal | May 2017 |
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