A Scalable Interior-Point Gauss-Newton Method for PDE-Constrained Optimization with Bound Constraints

Abstract

We present a scalable approach to solve a class of elliptic partial differential equation (PDE)-constrained optimization problems with bound constraints. This approach utilizes a robust full-space interior-point (IP)-Gauss-Newton optimization method. To cope with the poorly-conditioned IP-Gauss-Newton saddle-point linear systems that need to be solved, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE-constrained optimization problems. A block Gauss-Seidel preconditioner is proposed for the GMRES-based solution of the IP-Gauss-Newton linear systems. It is shown, for a large-class of PDE- and bound-constrained optimization problems, that the spectrum of the block Gauss-Seidel preconditioned IP-Gauss-Newton matrix is asymptotically independent of discretization and is not impacted by the ill-conditioning that notoriously plagues interior-point methods. We propose a regularization and log-barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)-based solution of the related IP-Gauss-Newton-Schur complement linear systems. The scalability of the approach is demonstrated on an example problem with bound and nonlinear elliptic PDE constraints. The numerical solution of the optimization problem is shown to require a discretization independent number of IP-Gauss-Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill-conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of preconditioner and linear system matrix applies, achieved with algebraic multigrid based solvers, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE- and bound-constrained problems.