A nonsmooth primal-dual method with interwoven PDE constraint solver

• Published in: Computational optimization and applications Vol. 89; no. 1; pp. 115 - 149
• Main Authors: Jensen, Bjørn, Valkonen, Tuomo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2024; Springer Nature B.V
• Subjects: Algorithms; Convex and Discrete Geometry; Hilbert space; Inverse problems; Linear systems; Management Science; Mathematics; Mathematics and Statistics; Methods; Operations Research; Operations Research/Decision Theory; Optimization; Partial differential equations; Solvers; Statistics; nonsmooth; Splitting; PDE-constrained; Jacobi; Primal-dual; Gauss–Seidel
• ISSN: 0926-6003; 1573-2894; 1573-2894
• DOI: 10.1007/s10589-024-00587-3

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.