Inexact Newton with Iterated Sensitivities for optimization of elliptic PDE

Master Thesis Presentation

Justin Pearse-Danker

Tuesday, March 24, 2020, 11:00

Room 01-012, Georges-Köhler-Allee 102, Freiburg 79110, Germany

The presentation will be held online.



Abstract of the Talk:

Solving optimal control problems subject to partial differential equations (PDEs) is a
challenging but important task, for instance, optimal heating or cooling.
The problem arising from the discretization can easily have 10^6 decision variables and
therefore it is important to provide methods that can solve such problems efficiently.
Thus, the interaction between the optimization method and the numerical simulation
is crucial for such problems.
The Inexact Newton with Iterated Sensitivities (INIS) method solves a particular
class of nonlinear programming (NLP) problems, which arise from optimal control
formulations where a subset of the variables are implicitly defined by nonlinear equality
constraints. The system of nonlinear equality constraints is called the forward problem.
In contrast to other inexact Newton-type optimization methods, the INIS method
preserves the local convergence properties and the asymptotic contraction rate of
the inexact Newton-type method with the same Jacobian approximation applied to
the forward problem. The INIS method is especially suited for problems where the
number of states is significantly larger than the number of controls. This is the case
for PDE constrained optimal control problems with boundary controls, which we
regard in this thesis.
Moreover, we consider a forward problem that arises from the discretization of a PDE
defined on a 2-dimensional domain. An efficient method for solving linear systems
that result from the discretization of PDEs is the multi-grid (MG) method.
This thesis shows that the MG method can be combined with the INIS method
resulting in the INIS-MG algorithm. The algorithm is applied to a test problem
and compared to ipopt, a software package for large-scale nonlinear optimization.
Furthermore, the theoretical properties of the INIS method are verified for the PDE
constrained case.

The numerical experiments show that even a MATLAB implementation of the INIS-
MG algorithm can outperform state of the art NLP solvers like ipopt.