Optimisation and calculus of variations

In the field of Inverse Problems we are concerned with several topics: We prove exact reconstruction results (also known as superresolution) as well as reconstruction error and convergence estimates based on source conditions. We are also interested in dynamic inverse problems, in which the state to be reconstructed changes during the measurements so that these are particularly ill-posed. In this context, especially for particle tracking we devise and analyse optimal transport regularization techniques for these problems in cooperation with medical physicists (PET and MRI specialists) in Münster.

We devise mathematical image processing methods (e.g. registration or online motion correction) for microscopy setups.

We analyse spatial patterns observed in different contexts such as microstructure in materials. One example is compliance minimization, which depending on the load case exhibits a plethora of different regimes with very different optimal microstructure patterns. Our tools here are energy scaling laws, Gamma-convergence, as well as analytic continuation and Fourier space methods.

We work in PDE-constrained shape optimization as well as the analysis of shape spaces, which are infinite-dimensional Riemannian manifolds. For instance, we prove existence and regularity of shortest geodesics and devise variational discretization of geodesics and other Riemannian objects, for which we prove convergence via variational methods.

What is ... ?

• Energy scaling law

  A number of difficult, nonconvex minimization problems $\min_x J^\epsilon(x)$ from physics or other contexts depend on a small parameter $\epsilon$. These problems are too complex to be solved exactly, but sometimes they can be understood by proving an energy scaling law, i.e. an estimate $c\epsilon^\alpha\leq\min_x J^\epsilon(x)-\inf_x J^0(x)\leq C\epsilon^\alpha$ for some constants $c,C,\alpha$. The upper bound is proved by construction, i.e. one carefully devises an $x$ whose cost satisfies the upper bound. The lower bound in contrast is ansatz-free, i.e. one shows that no other $x$ (whatever you might come up with) can do better, at least up to a constant factor. This way you proved that your constructed $x$ is near-optimal (optimal up to a constant factor), so one may expect to see similar $x$ in nature. To refine the understanding one needs to go beyond energy scaling laws and really exploit the optimality conditions, e.g. sometimes one can prove asymptotic self-similarity of minimizers.

• Compliance minimisation

  This is a classical shape and topology optimisation problem: One optimises the geometry of a mechanical device (e.g. a bridge or a lever) such that it minimises a weighted sum of volume (or equivalently weight) and compliance under a fixed mechanical load. The compliance is the work performed by the load when deforming the device, and it is a measure of the weakness of the device. Without additional regularisation (e.g. penalisation of the device perimeter), the problem leads to microstructure, which is mostly understood. However, there are many variations that are still poorly understood, such as the case of multiple stochastic load distributions, stochastic material properties, different objective functions (such as maximum stress), etc.

• Inverse problem

  A problem in which one attempts to reconstruct something from indirect measurements. The most prominent examples come from medical imaging, such as computed tomography, where one reconstructs the 3D anatomy of a patient based on a sequence of 2D X-ray images (which is an indirect measurement since one cannot access the 3D anatomy directly). The (linear or nonlinear) process that generates the measurement from the object is called the "forward operator", and the goal is essentially to invert that operator. Inverse problems are ill-posed, which, in mathematical terms, means that the (infinite-dimensional) forward operator has no continuous inverse. As a result, measurement noise is typically overamplified when attempting to invert the forward operator, and one needs to employ so-called regularisation (which must be chosen appropriately with respect to the noise level) in order to approximate the solution to inverse problems.

• Superresolution

  Some infinite-dimensional objects can be characterised by, and therefore reconstructed from, finitely many measurements. This concept became particularly famous in fluorescence microscopy over the past decades, culminating in a Nobel Prize in 2014. In this context, the optics of the microscope have only finite resolution, which mathematically means that, during the recording of an image, the high Fourier frequencies are lost (leaving one with a finite truncated Fourier series of the image). However, if the image consists of Dirac masses (i.e. single fluorescent points corresponding, for example, to individual molecules), the exact locations of these masses can be reconstructed despite the finitely many measurements.

• Shape space

  An infinite-dimensional space of objects (e.g. 2D surfaces embedded in 3D or closed planar curves in 2D, known as "shapes") that is endowed with the structure of an infinite-dimensional Riemannian manifold. Such spaces are used in computer graphics, inverse problems, computer vision, shape optimisation, etc., whenever shapes need to be found or generated. Typically, shape spaces have additional structure; for example, they are invariant under rigid motions of their elements, and some even possess a Lie group structure. Many of the typical shape spaces are still not fully understood (e.g. their completeness properties, the existence and regularity of geodesics, numerical discretisation), and some are linked (via the equations satisfied by geodesics) to PDEs and fluid dynamics.