Optimal design, optimal control, and parameter estimation of systems governed by partial differential equations (PDEs) give rise to a class of problems known as PDE-constrained optimization. The size and complexity of the discretized PDE often pose significant challenges for contemporary optimization methods.

Recent advances in algorithms, software and high-performance computing systems have resulted on PDE models that can often scale to millions of variables and there is increasing interest in solving nonlinear optimization problems governed by such large-scale simulations.

Many interior-point optimization algorithms for constrained and unconstrained optimization need to solve a sequence of closely related linear systems. The effectiveness and scalability of large-scale PDE-constrained optimizations ultimately depends on solving these linear systems quickly and reliably, which requires specialized iterative linear solvers. Novel strategies and algorithms will be presented to solve nonconvex optimization problem from three-dimensional PDE-constrained optimization with more than 50 million state variables and control variables with both lower and upper bound. Results for biomedical hyperthermia cancer simulations as well as inverse wave propagation will be presented.