Oberseminar Wissenschaftliches Rechnen (Dr. John Pearson)

Abstract:

Optimization problems subject to PDE constraints form a mathematical tool that can be applied
to a wide range of scientific processes, including fluid flow control, medical imaging, option pricing,
biological and chemical processes, and electromagnetic inverse problems, to name a few. These
problems involve minimizing some function arising from a particular physical objective, while at
the same time obeying a system of PDEs which describe the process. It is necessary to obtain ac-
curate solutions to such problems within a reasonable CPU time, in particular for time-dependent
problems, for which the “all-at-once” solution can lead to extremely large linear systems.

In this talk we will consider iterative methods, in particular Krylov subspace methods, to solve
such systems, accelerated by fast and robust preconditioning strategies. In particular, we will
survey several new developments, including block preconditioners for fluid flow control problems,
a circulant preconditioning framework for solving certain optimization problems constrained by
fractional differential equations, and multiple saddle-point preconditioners for block tridiagonal
linear systems. We will illustrate the benefit of using these new approaches through a range of
numerical experiments.

This talk is based on work with Santolo Leveque (Scuola Normale Superiore, Pisa),
Spyros Pougkakiotis (University of Dundee), Jacek Gondzio (University of Edinburgh), and
Andreas Potschka (TU Clausthal).

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