A nonsmooth primal-dual method with interwoven PDE constraint solver
Bjørn Jensen, Tuomo Valkonen
TL;DR
This work develops a first-order primal-dual method for nonsmooth PDE-constrained optimization by interweaving a PDE solver with the optimization loop, so that only a single linear-system step is performed per iteration rather than solving the PDE exactly. The method extends primal-dual proximal splitting to PDE-constrained problems via PDE-splitting operators and a parallel adaptive PDE solver, enabling efficient handling of nonsmooth regularizers like total variation. Global convergence is established under a set of structural and growth conditions, with accelerated $O(1/N)$ rates and linear convergence when strong convexity holds; local convergence results are also discussed for better initialization. Numerical experiments on inverse problems with boundary measurements show that Jacobi or Gauss-Seidel PDE splits outperform full PDE solves per iteration, with Gauss-Seidel often recommended for stability and efficiency, thereby offering a practical, scalable approach for nonsmooth PDE-constrained optimization.
Abstract
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.