A Function-Space Framework for BDDC Preconditioning in Control- and State-Constrained Sparse Optimal Control
Abstract
We develop a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for the interface systems arising from active-set semi-smooth Newton linearizations of elliptic optimal control problems with box-constrained controls, L1-sparsity, and Moreau--Yosida regularized state constraints. The domain decomposition and BDDC construction are formulated directly at the infinite-dimensional level. Eliminating the interior variables yields a Schur complement for the state and adjoint traces assembled from local subdomain operators. We prove a global--local equivalence result, establish well-posedness of the local and partially assembled problems under verifiable conditions on the primal constraints. The BDDC operator admits a two-level additive Schwarz representation with independent local solves and a finite-dimensional coarse solve. We implement the preconditioner using conforming finite elements and test it for distributed and Neumann boundary control. For a fixed ratio of subdomain diameter H to mesh size h, the GMRES iteration counts are nearly independent of mesh refinement and grow only moderately as this ratio increases. Enriching the coarse space substantially improves convergence and reduces sensitivity to the control regularization and sparsity weights. For the state-constrained problem, scaling the Moreau--Yosida penalty parameter proportionally to h2 yields nearly constant GMRES iteration counts as the mesh is refined.
Related
- Optimal Control of State Constrained Systems via Measure Relaxations and Polynomial Optimization
- Robust preconditioners for finite element approximations of control-state constrained optimal control problems governed by elliptic equations
- A Preconditioned Semismooth Newton Method for Box-Constrained Elliptic Optimal Control Problems with Random Coefficients
- Iterative Method for Stochastic Optimal Control Problem Constrained by Parabolic Partial Differential Equation
- Trajectory Parameterization and Evolutionary Search for Constrained Optimal Control
BibTeX
@misc{antil2026functionspace,
title = {{A Function-Space Framework for BDDC Preconditioning in Control- and State-Constrained Sparse Optimal Control}},
author = {Harbir Antil and Keegan L. A. Kirk and Umarkhon Rakhimov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09541},
eprint = {2609.09541},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09541},
}
RIS
TY - UNPB
TI - A Function-Space Framework for BDDC Preconditioning in Control- and State-Constrained Sparse Optimal Control
AU - Harbir Antil
AU - Keegan L. A. Kirk
AU - Umarkhon Rakhimov
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09541
UR - https://arxiv.org/abs/2609.09541
AB - We develop a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for the interface systems arising from active-set semi-smooth Newton linearizations of elliptic optimal control problems with box-constrained controls, L^1-sparsity, and Moreau--Yosida regularized state constraints. The domain decomposition and BDDC construction are formulated directly at the infinite-dimensional level. Eliminating the interior variables yields a Schur complement for the state and adjoint traces assembled from local subdomain operators. We prove a global--local equivalence result, establish well-posedness of the local and partially assembled problems under verifiable conditions on the primal constraints. The BDDC operator admits a two-level additive Schwarz representation with independent local solves and a finite-dimensional coarse solve. We implement the preconditioner using conforming finite elements and test it for distributed and Neumann boundary control. For a fixed ratio of subdomain diameter H to mesh size h, the GMRES iteration counts are nearly independent of mesh refinement and grow only moderately as this ratio increases. Enriching the coarse space substantially improves convergence and reduces sensitivity to the control regularization and sparsity weights. For the state-constrained problem, scaling the Moreau--Yosida penalty parameter proportionally to h^2 yields nearly constant GMRES iteration counts as the mesh is refined.
ER -