  PDE-constrained optimization problems find many applications in medical image
analysis, for example, neuroimaging, cardiovascular imaging, and oncological
imaging. We review related literature and give examples on the formulation,
discretization, and numerical solution of PDE-constrained optimization problems
for medical imaging. We discuss three examples. The first one is image
registration. The second one is data assimilation for brain tumor patients, and
the third one data assimilation in cardiovascular imaging. The image
registration problem is a classical task in medical image analysis and seeks to
find pointwise correspondences between two or more images. The data
assimilation problems use a PDE-constrained formulation to link a biophysical
model to patient-specific data obtained from medical images. The associated
optimality systems turn out to be sets of nonlinear, multicomponent PDEs that
are challenging to solve in an efficient way.
  The ultimate goal of our work is the design of inversion methods that
integrate complementary data, and rigorously follow mathematical and physical
principles, in an attempt to support clinical decision making. This requires
reliable, high-fidelity algorithms with a short time-to-solution. This task is
complicated by model and data uncertainties, and by the fact that
PDE-constrained optimization problems are ill-posed in nature, and in general
yield high-dimensional, severely ill-conditioned systems after discretization.
These features make regularization, effective preconditioners, and iterative
solvers that, in many cases, have to be implemented on distributed-memory
architectures to be practical, a prerequisite. We showcase state-of-the-art
techniques in scientific computing to tackle these challenges.
