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Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms

John W. Pearson (j.pearson***at***ed.ac.uk)
Margherita Porcelli (margherita.porcelli***at***unibo.it)
Martin Stoll (martin.stoll***at***mathematik.tu-chemnitz.de)

Abstract: PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting L1 term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method we introduce fast and efficient preconditioners which enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.

Keywords: PDE-constrained optimization, Interior Point methods, Saddle-point systems, Preconditioning, Sparsity, Box constraints

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Category 3: Applications -- Science and Engineering (Control Applications )

Citation: Numerical Linear Algebra with Applications, DOI:10.1002/nla.2276

Download: [PDF]

Entry Submitted: 06/15/2018
Entry Accepted: 06/15/2018
Entry Last Modified: 02/10/2020

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