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Interior‐point methods and preconditioning for PDE‐constrained optimization problems involving sparsity terms
Author(s) -
Pearson John W.,
Porcelli Margherita,
Stoll Martin
Publication year - 2020
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2276
Subject(s) - discretization , interior point method , mathematics , mathematical optimization , partial differential equation , optimization problem , scheme (mathematics) , point (geometry) , function (biology) , mathematical analysis , geometry , evolutionary biology , biology
Summary 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 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 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.