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An inexact semismooth Newton-Krylov method for semilinear elliptic optimal control problem

Shiqi Chen, Xuesong Chen

TL;DR

This paper tackles semilinear elliptic PDE-constrained optimal control with box constraints by formulating the discretized first-order optimality system and solving it via an inexact semismooth Newton method that leverages GMRES for Newton steps. The authors introduce a slant-differentiability framework and a merit-based, nonmonotone line search to guarantee global convergence, while proving local superlinear convergence as the Newton residual tolerance $\eta_k$ tends to zero. Numerical experiments on two examples validate the approach, demonstrating high accuracy, reduced memory usage through matrix-free Krylov solves, and robust performance under varying discretizations and initial guesses. The proposed ISSNG-L method offers a scalable and efficient tool for PDE-constrained optimization with nonsmooth, box-bounded controls in large-scale settings.

Abstract

An inexact semismooth Newton method has been proposed for solving semi-linear elliptic optimal control problems in this paper. This method incorporates the generalized minimal residual (GMRES) method, a type of Krylov subspace method, to solve the Newton equations and utilizes nonmonotonic line search to adjust the iteration step size. The original problem is reformulated into a nonlinear equation through variational inequality principles and discretized using a second-order finite difference scheme. By leveraging slanting differentiability, the algorithm constructs semismooth Newton directions and employs GMRES method to inexactly solve the Newton equations, significantly reducing computational overhead. A dynamic nonmonotonic line search strategy is introduced to adjust stepsizes adaptively, ensuring global convergence while overcoming local stagnation. Theoretical analysis demonstrates that the algorithm achieves superlinear convergence near optimal solutions when the residual control parameter $η_k$ approaches to 0. Numerical experiments validate the method's accuracy and efficiency in solving semilinear elliptic optimal control problems, corroborating theoretical insights.

An inexact semismooth Newton-Krylov method for semilinear elliptic optimal control problem

TL;DR

This paper tackles semilinear elliptic PDE-constrained optimal control with box constraints by formulating the discretized first-order optimality system and solving it via an inexact semismooth Newton method that leverages GMRES for Newton steps. The authors introduce a slant-differentiability framework and a merit-based, nonmonotone line search to guarantee global convergence, while proving local superlinear convergence as the Newton residual tolerance tends to zero. Numerical experiments on two examples validate the approach, demonstrating high accuracy, reduced memory usage through matrix-free Krylov solves, and robust performance under varying discretizations and initial guesses. The proposed ISSNG-L method offers a scalable and efficient tool for PDE-constrained optimization with nonsmooth, box-bounded controls in large-scale settings.

Abstract

An inexact semismooth Newton method has been proposed for solving semi-linear elliptic optimal control problems in this paper. This method incorporates the generalized minimal residual (GMRES) method, a type of Krylov subspace method, to solve the Newton equations and utilizes nonmonotonic line search to adjust the iteration step size. The original problem is reformulated into a nonlinear equation through variational inequality principles and discretized using a second-order finite difference scheme. By leveraging slanting differentiability, the algorithm constructs semismooth Newton directions and employs GMRES method to inexactly solve the Newton equations, significantly reducing computational overhead. A dynamic nonmonotonic line search strategy is introduced to adjust stepsizes adaptively, ensuring global convergence while overcoming local stagnation. Theoretical analysis demonstrates that the algorithm achieves superlinear convergence near optimal solutions when the residual control parameter approaches to 0. Numerical experiments validate the method's accuracy and efficiency in solving semilinear elliptic optimal control problems, corroborating theoretical insights.

Paper Structure

This paper contains 8 sections, 12 theorems, 66 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. First-order optimality system and its numerical discretization
  3. Inexact semismooth Newton-Krylov method with nonmonotonic line search
  4. Convergence analysis
  5. Numerical experiments
  6. Example 1
  7. Example 2
  8. Conclusion

Key Result

Lemma 2.1

Using the variational inequality in (equation 2.2), we get an equivalent way to describe the optimal control:

Figures (3)

  • Figure 1: Plots of ISSNG-L results
  • Figure 2: The variations of $\Vert F(z_k)\Vert$ under the ISSNG-L with $c_1=0.5$ and GCSSN with different values of $\alpha$
  • Figure 3: The variations of $\Vert F(z_k)\Vert$ under the ISSNG-L and ISSNG

Theorems & Definitions (14)

  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 4 more