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Inexact Trust-Region Methods for PDE-Constrained Optimization

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Frontiers in PDE-Constrained Optimization

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 163))

Abstract

Numerical solution of optimization problems with partial differential equation (PDE) constraints typically requires inexact objective function and constraint evaluations, derivative approximations, and the use of iterative linear system solvers. Over the last 30 years, trust-region methods have been extended to rigorously, robustly, and efficiently handle various sources of inexactness in the optimization process. In this chapter, we review some of the recent advances, discuss their key algorithmic contributions, and present numerical examples that demonstrate how inexact computations can be exploited to enable the solution of large-scale PDE-constrained optimization problems.

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Notes

  1. 1.

    Clearly, with exact linear system solves we have \(W_k^* W_k g_k = W_k W_k g_k = W_k g_k\), however, in the presence of inexactness the distinction is important.

  2. 2.

    For all details, see the proofs in [20, p. 1536–1538] and [21, p. 295–298].

  3. 3.

    Only the scope of the index k extends from Algorithm 4 to Algorithm 3 – indices i and j are independent, i.e., their scope is local to each algorithm.

  4. 4.

    Under the assumptions of this chapter, augmented systems can always be related to strictly convex quadratic problems of the form \(\min \frac {1}{2} \left \langle s^1,s^1\right \rangle _{X} - \left \langle b^1, s^1\right \rangle _{X} \mbox{ subject to } c_x(x_k)s^1 = b_2\), where (b1b2)T is the right-hand side vector of the augmented system and s1 is the first block of the left-hand side vector.

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Acknowledgements

This work was supported by DARPA EQUiPS grant SNL 014150709 and the DOE NNSA ASC ATDM program.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

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  1. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, 87185-9999, USA

    Drew P. Kouri & Denis Ridzal

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  1. Drew P. Kouri
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Correspondence to Denis Ridzal .

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Editors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Harbir Antil

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Drew P. Kouri

  3. Corporate Strategic Research, ExxonMobil Research and Engineering Company, Annandale, NJ, USA

    Martin-D. Lacasse

  4. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Denis Ridzal

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Kouri, D.P., Ridzal, D. (2018). Inexact Trust-Region Methods for PDE-Constrained Optimization. In: Antil, H., Kouri, D.P., Lacasse, MD., Ridzal, D. (eds) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8636-1_3

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