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Block triangular preconditioning for elliptic boundary optimal control with mixed boundary conditions

Chaojie Wang

TL;DR

This work tackles efficiently solving the saddle-point linear systems from the elliptic boundary optimal control with mixed boundary conditions. It proposes a permuted block triangular preconditioner using a Schur-complement approximation hatS = K and analyzes the spectrum of the preconditioned operator to show eigenvalue clustering and a fixed multiplicity of 1. Numerical experiments on the unit square with varying beta and boundary sets demonstrate mesh-size robustness and reduced iterations for GMRES with the proposed preconditioner compared to diagonal MINRES preconditioners. The results indicate the method is scalable for large-scale PDE-constrained optimization with mixed Neumann/Dirichlet boundaries.

Abstract

In this paper, preconditioning the saddle point problem arising from the elliptic boundary optimal control problem with mixed boundary conditions is considered. A block triangular reconditioning method is proposed based on permutations of the saddle point problem and approximations of the corresponding Schur complement. The spectral properties of the preconditioned matrix is analyzed. Numerical experiments are conducted to demonstrate the effectiveness of the proposed preconditioning method.

Block triangular preconditioning for elliptic boundary optimal control with mixed boundary conditions

TL;DR

This work tackles efficiently solving the saddle-point linear systems from the elliptic boundary optimal control with mixed boundary conditions. It proposes a permuted block triangular preconditioner using a Schur-complement approximation hatS = K and analyzes the spectrum of the preconditioned operator to show eigenvalue clustering and a fixed multiplicity of 1. Numerical experiments on the unit square with varying beta and boundary sets demonstrate mesh-size robustness and reduced iterations for GMRES with the proposed preconditioner compared to diagonal MINRES preconditioners. The results indicate the method is scalable for large-scale PDE-constrained optimization with mixed Neumann/Dirichlet boundaries.

Abstract

In this paper, preconditioning the saddle point problem arising from the elliptic boundary optimal control problem with mixed boundary conditions is considered. A block triangular reconditioning method is proposed based on permutations of the saddle point problem and approximations of the corresponding Schur complement. The spectral properties of the preconditioned matrix is analyzed. Numerical experiments are conducted to demonstrate the effectiveness of the proposed preconditioning method.
Paper Structure (11 sections, 1 theorem, 51 equations, 2 figures, 6 tables)

This paper contains 11 sections, 1 theorem, 51 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Discretization
  4. Previous work
  5. Preconditioning
  6. Block triangular preconditioner
  7. Analysis of the preconditioner
  8. Numerical results
  9. Test problem 1
  10. Test problem 2
  11. Conclusion

Key Result

Proposition 3.1

If the block triangular matrix $\mathcal{P}_{T}$ in (eq15) is taken as a preconditioner for the matrix $\mathcal{A}$ in (eq12), then 1 is an eigenvalue of the preconditioned matrix $\mathcal{P}^{-1}_{T}\mathcal{A}$ with $2n$ multiplicity and the other $m_{B}$ eigenvalues are bounded by $[1+\frac{1}{

Figures (2)

  • Figure : Fig. 1. Computed state (a) and control (b) for Test problem 1 with $\beta=10^{-6}$ and $\Gamma=\Gamma_{3}$.
  • Figure : Fig. 2. Computed state (a) and control (b) for Test problem 2 with $\beta=10^{-6}$ and $\Gamma=\Gamma_{3}$.

Theorems & Definitions (1)

  • Proposition 3.1