WIAS Research Highlights

Modern image acquisition technologies allow clinicians to record detailed information not only on patient anatomy, but also related to biophysical processes. We focus on three selected applications of mathematical methods in this context.

Inverse problems are ubiquitous in all areas of science where measurements and data play a role. We focus on mathematical foundations for applications as magnetic resonance imaging (MRI) and quantitative MRI (qMRI).

On a model that describes the deformation of rocks in the lithosphere as the flow of a fluid with viscous, elastic, and plastic properties is studied and a generalized solution concept is proposed.

Ultrashort optical pulses are of great theoretical and practical interest. Practical applications of these pulses clearly benefit from the reduction in pulse duration, which enhances data transmission capacity and enable the observation of extremely fast processes.

We identify simple mathematical models that explain how active particles or agents can interact with each other to form large-scale structures and patterns (like models that are appropriate on the time and space scales of insects such as ants).

A journey from a novel abstract thermodynamic description, to nonlinear coupled systems that capture the various phase transitions in hydrogels, to agent models for cell motion in these materials.

Research results are highlighted that have been achieved with the goal of formulating of a general mathematical structure that supports the mathematical modeling and analysis of processes with bulk-interface coupling in a variational framework

We deal with phase transitions arising from random graphs occurring in diverse contexts. The models we consider reflect many real-world properties, such as clustering and being scale free, and often exhibit important features associated with the networks.

On a novel regularization method for the problem of calibrating local stochastic volatility models. Numerical results suggest that the approach is efficient for the calibration of local stochastic volatility models and can outperform widely used local kernel methods.

Electron shuttling devices are new functional elements in modular concepts for spin-qubit based quantum computers that have promising prospects for scalability due to direct compatibility with industrial fabrication techniques. Numerical device simulation is crucial for understanding the limiting factors an

Convolutional neural networks are an efficient tool to solve pPDEs and are amenable to a thorough mathematical analysis. Small approximation errors can be achieved with network sizes growing only logarithmically with the inverse of the required error bound.

Thermodynamic structures give rise to effective methods to analyze cross-diffusion, including change to entropy variables and symmetrization.