A nonsmooth primal-dual method with interwoven PDE constraint solver

TL;DR

This paper presents a novel primal-dual optimization method for nonsmooth PDE-constrained problems that interweaves a simple linear solver, avoiding full PDE solves each iteration, and demonstrates linear convergence and practical efficiency.

Contribution

It introduces a first-order primal-dual method that interleaves PDE linear system solving, reducing computational cost while maintaining convergence guarantees.

Findings

Achieves linear convergence under second-order growth conditions.

Demonstrates efficiency on inverse PDE problems with boundary measurements.

Avoids solving PDEs fully at each iteration, saving computational resources.

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.