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

Author

Pearson, John W., Porcelli, Margherita, and Stoll, Martin

Subjects
Mathematics - Optimization and Control, Mathematics - Numerical Analysis, 65F08, 65F10, 65K05, 76D55, 90C20, 93C20
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 $\rm L^1$ 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.

Published
2018

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

Author

Martin Stoll, John W. Pearson, Margherita Porcelli, Pearson J.W., Porcelli M., and Stoll M.

Subjects
Scheme (programming language), Mathematical optimization, Optimization problem, Discretization, 010103 numerical & computational mathematics, 01 natural sciences, interior-point method, preconditioning, 65F08, 65F10, 65K05, 76D55, 90C20, 93C20, box constraint, FOS: Mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Mathematics - Optimization and Control, Mathematics, computer.programming_language, Algebra and Number Theory, Applied Mathematics, sparsity, Perspective (graphical), saddle-point system, State (functional analysis), Numerical Analysis (math.NA), Term (time), 010101 applied mathematics, Constrained optimization problem, Optimization and Control (math.OC), PDE-constrained optimization, computer, Interior point method
Abstract

Partial differential equation (PDE)–constrained optimization problems with control or state constraints are challenging from an analytical and 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 that 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.

Published
2020

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