# Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems

@article{Pearson2012RegularizationRobustPF, title={Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems}, author={John W. Pearson and M. Stoll and Andrew J. Wathen}, journal={SIAM J. Matrix Anal. Appl.}, year={2012}, volume={33}, pages={1126-1152} }

In this article, we motivate, derive, and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems which arise in PDE-constrained optimization problems. We consider the distributed control problem involving the heat equation and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the effectiveness of our preconditioners in each case is an effective approximation of the Schur complement of the…

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## References

SHOWING 1-10 OF 62 REFERENCES

### One-shot solution of a time-dependent time-periodic PDE-constrained optimization problem

- Computer Science, Mathematics
- 2014

The efficient solution of a PDE-constrained optimization problem subject to the time-periodic heat equation is described and a space-time formulation for which a monolithic solver is proposed, which allows the use of a FFT based preconditioner.

### Robust Iterative Solution of a Class of Time‐Dependent Optimal Control Problems

- Computer Science, Mathematics
- 2012

This paper develops a strategy to approximate the (1, 1)‐block and Schur complement of the saddle point system that results from solving this problem, and therefore derive a block diagonal preconditioner to be used within the MINRES algorithm.

### Optimal Solvers for PDE-Constrained Optimization

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2010

Two optimal preconditioners are introduced for large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type, and the theoretical proof indicates that these approaches may have much broader applicability for other PDEs.

### A new approximation of the Schur complement in preconditioners for PDE‐constrained optimization

- Computer ScienceNumer. Linear Algebra Appl.
- 2012

This short manuscript presents a new Schur complement approximation for PDE‐constrained optimization, an important class of these problems with constraints, and designs such preconditioners for which this optimality property holds independently of both the mesh size and the Tikhonov regularization parameter β that is used.

### FAST ITERATIVE SOLVERS FOR CONVECTION-DIFFUSION CONTROL PROBLEMS ∗

- Computer Science, Mathematics
- 2013

Effective solvers for the optimal control of stabilized convectiondiffusion control problems are described and numerical results show that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the step-size h and the regularization parameter β for a range of problems.

### A preconditioning technique for a class of PDE-constrained optimization problems

- Mathematics, Computer ScienceAdv. Comput. Math.
- 2011

We investigate the use of a preconditioning technique for solving linear systems of saddle point type arising from the application of an inexact Gauss–Newton scheme to PDE-constrained optimization…

### All-at-once preconditioning in PDE-constrained optimization

- Computer ScienceKybernetika
- 2010

This paper introduces the basic concepts of PDE-constrained optimization and shows how the all-at-once approach will lead to linear systems in saddle point form and discusses methods and preconditioners.

### All-at-Once Solution if Time-Dependent PDE-Constrained Optimisation Problems

- Computer Science
- 2010

This paper looks at one-shot approaches for the optimal control of time-dependent PDEs and focuses on the fast solution of these problems, using Krylov subspace solvers together with an efficient preconditioner for minimal storage requirements.

### Block‐triangular preconditioners for PDE‐constrained optimization

- Computer ScienceNumer. Linear Algebra Appl.
- 2010

An eigenvalue analysis for the block‐triangular preconditioners that gives convergence bounds in the non‐standard inner product and illustrates their competitiveness on a number of computed examples.

### A Preconditioner for Linear Systems Arising From Interior Point Optimization Methods

- Computer ScienceSIAM J. Sci. Comput.
- 2007

A preconditioning technique applied to the problem of solving linear systems arising from primal-dual interior point algorithms in linear and quadratic programming has the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix.