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.A. MANG, A. GHOLAMI, C. DAVATZIKOS, AND G. BIROS (hard constraints 1 ) to incorporate additional prior knowledge on the expected deformation map. One motivation for adding such constraints is that often, especially in longitudinal studies of the same patient, there is a real deformation by the realization of an actual physical phenomenon. A simple constraint, which nonetheless poses significant numerical challenges, is the incompressibility of tissue [128,129,131]. Examples for more complex biophysical constraints are brain tumor growth models [73,99,174,[210][211][212] or cardiac motion models [188].We will discuss all of these constraints either in the context of diffeomorphic image registration or in the more generic context of data assimilation in medical imaging. This brings us to the second problem we are addressing in this article: Data assimilation in brain tumor imaging [70,100,114,135,138] (see §3). The PDE constraint is, in its simplest form, a nonlinear parabolic differential equation. The inversion variables are, e.g., the initial condition, the growth rate of the tumor, or a diffusion coefficient that controls the net migration of cancerous cells within brain parenchyma. The regularization model is in our case an L 2 -penalty. The third problem is cardiac motion...