Title: Multilevel Preconditioning Algorithms for 
       Large-scale Nonconvex PDE-Constrained Optimization

Optimal control, optimal design 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 in 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 symmetric indefinite linear systems quickly
and reliably, which requires specialized iterative linear solvers. Novel
strategies and algorithms will be presented in this talk to solve convex and
nonconvex optimization problems from three-dimensional PDE-constrained
optimization with more than 30 million state variables and control variables
with both lower and upper bound on the desktop. Results for biomedical
hyperthermia cancer simulations as well as other PDE examples will be
presented.