PDE-constrained parameter optimizations appear in a diverse array of scientific and engineering problems. Especially in applications involving many parameters, solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves. To address this challenge, this work uses the reduced-basis method (RBM) in conjunction with a trust region optimization framework.

While many different model order reduction techniques exist, a key feature of RBM is the existence of rigorous a posteriori bounds on the error in the reduced model. In this approach, reduced-basis models are used in place of expensive full PDE solves in determining steps taken by the optimization algorithm. The a posteriori error bounds are used to (i) decide whether to accept and reject optimization iterates, (ii) limit the number of full solves needed in the optimization by intelligently determining when to update the reduced model, and (iii) guarantee convergence of the algorithm to a stationary point of the true (unreduced) model.

This talk will begin with a brief overview of RBM before delving into the details of the framework, highlighting how RBM is leveraged to increase rigor in the often finicky art of optimization. The approach is demonstrated on a small (6 parameter) optimization inspired by inference problems, and results are compared to those obtained using a canonical quasi-Newton method.