**Series in Computational Science
edited by Illia Horenka, Rolf
Krause, Olaf Schenk
Volume 3**

Johannes Huber

**Interior-Point
Methods for**

**PDE-Constrained
Optimization**

First Edition 2013. 150 pages. EUR 64,00.

ISBN
978-3-86628-463-0

In applied sciences
PDEs model an extensive variety of phenomena. Typically the final goal of
simulations is a system which is optimal in a certain sense. In these
optimization problems PDEs appear as equality constraints. PDE-constrained optimization
problems are large-scale and often nonconvex. Their
numerical solution leads to large ill-conditioned linear systems. In many
practical problems inequality constraints implement technical limitations or
prior knowledge.

In this thesis
interior-point (IP) methods are considered to solve nonconvex
large-scale PDE-constrained optimization problems with inequality constraints.
To cope with enormous fill-in of direct linear solvers, inexact search
directions are allowed in an inexact IP method. SMART tests cope with the lack
of inertia information to control Hessian modification and also specify
termination tests for the iterative linear solver.

The original inexact
IP method needs to solve two sparse large-scale linear systems in each
optimization step. This is improved to only a single linear system solution in
most optimization steps. Within this improved framework, two iterative linear
solvers are evaluated: A general purpose algebraic multilevel preconditioned
SQMR method is applied to PDE-constrained optimization problems for optimal
server room cooling in three space dimensions and to compute the ambient
temperature for optimal cooling. The results show robustness and efficiency of
the inexact IP method when compared with the exact IP method. These advantages
are even more evident for a reduced-space preconditioned (RSP) GMRES solver
which takes advantage of the linear system's structure. This RSP-IIP method is
studied on the basis of control problems originating from superconductivity and
from two-dimensional and three-dimensional parameter estimation problems in
groundwater modeling. The numerical results exhibit
the improved efficiency especially for multiple PDE constraints.

An inverse medium
problem for the Helmholtz equation with pointwise box
constraints is solved by IP methods. The ill-posedness
of the problem is explored numerically and different regularization strategies
are compared. The impact of box constraints and the importance of Hessian
modification is demonstrated. A real world seismic
imaging problem is solved successfully by the RSP-IIP method.

About the Author:

**Johannes Huber
**studied Physics at the University for Applied Scienes in Isny and Mathematics
at the University of Karlsruhe. During this time he took part in the Fulbright
program and studied abroad at the NCAT&T University in Greensboro, USA. In
2008 he received the faculty price for mathematics at the University of
Karlsruhe. In his doctoral studies at the University of Basel he worked as an
assistant in the field of numerical PDE simulation and numerical optimization.
In his multifacetted career he worked as software developer in the industry
for nine years and took a research internship at the IBM T.J. Watson research center in Yorktown Heights, USA.

**Keywords: **large-scale optimization,
PDE-constrained optimization, optimal control, inverse problems, interior-point
methods, nonconvex programming, line search, inexact
Newton method, inexact linear system solvers, Krylov
subspace methods.

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