Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 38, Issue 5, Pages A2756-A2778Publisher
SIAM PUBLICATIONS
DOI: 10.1137/15M104075X
Keywords
PDE-constrained optimizations; flow control; nonlinear elimination preconditioner; inexact Newton method; sequential quadratic programming
Categories
Funding
- National Natural Science Foundation of China [11571100, 91530103, 91330111]
- Ministry of Science and Technology, Taiwan [MOST-103-2115-M-008-007]
- National Science Foundation [CCF-1216314]
- Direct For Computer & Info Scie & Enginr
- Division of Computing and Communication Foundations [1216314] Funding Source: National Science Foundation
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The full-space Lagrange-Newton algorithm is one of the numerical algorithms for solving problems arising from optimization problems constrained by nonlinear partial differential equations. Newton-type methods enjoy fast convergence when the nonlinearity in the system is well-balanced; however, for some problems, such as the control of incompressible flows, even linear convergence is difficult to achieve and a long stagnation period often appears in the iteration history. In this work, we introduce a nonlinearly preconditioned inexact Newton algorithm for the boundary control of incompressible flows. The system has nine field variables, and each field variable plays a different role in the nonlinearity of the system. The nonlinear preconditioner approximately removes some of the field variables, and as a result, the nonlinearity is balanced and inexact Newton converges much faster when compared to the unpreconditioned inexact Newton method or its two-grid version. Some numerical results are presented to demonstrate the robustness and efficiency of the algorithm.
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