Nonlinear preconditioning techniques for full-space Lagrange-Newton solution of PDE-constrained optimization problems

Haijian Yang, Feng Nan Hwang, Xiao Chuan Cai

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)A2756-A2778
JournalSIAM Journal on Scientific Computing
Volume38
Issue number5
DOIs
StatePublished - 2016

Keywords

  • Flow control
  • Inexact Newton method
  • Nonlinear elimination preconditioner
  • PDE-constrained optimizations
  • Sequential quadratic programming

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