A1 Journal article (refereed)
A nonsmooth primal-dual method with interwoven PDE constraint solver (2024)

Jensen, B., & Valkonen, T. (2024). A nonsmooth primal-dual method with interwoven PDE constraint solver. Computational Optimization and Applications, Early online. https://doi.org/10.1007/s10589-024-00587-3

JYU authors or editors

Publication details

All authors or editorsJensen, Bjørn; Valkonen, Tuomo

Journal or seriesComputational Optimization and Applications



Publication year2024

Publication date08/06/2024

VolumeEarly online


Publication countryUnited States

Publication languageEnglish


Publication open accessOpenly available

Publication channel open accessPartially open access channel

Publication is parallel published (JYX)https://jyx.jyu.fi/handle/123456789/95907

Publication is parallel publishedhttps://arxiv.org/abs/2211.04807


We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss–Seidel, conjugate gradients), always taking only one step of the linear system solver for each step of the optimization method. The control parameter is updated on each iteration as determined by the optimization method. We prove linear convergence under a second-order growth condition, and numerically demonstrate the performance on a variety of PDEs related to inverse problems involving boundary measurements.

Keywordspartial differential equationsmathematical optimisationcontrol theorynumerical analysisnumerical methods

Free keywordsprimal-dual; nonsmooth; PDE-constrained; splitting; Jacobi; Gauss–Seidel

Contributing organizations

Ministry reportingYes

Reporting Year2024

Preliminary JUFO rating1

Last updated on 2024-03-07 at 00:46