Inverse problems involve the “art” of deducing sizes of which the direct measurements are not accessible from indirect observations. This means they have numerous areas of application, ranging from medical imaging and material characterisation to parameter identification in system biology.
Many complex processes in sciences, medicine and technology are described by common or partial differential equations. These models often contain unknown parameters such as spatially varying coefficients, source terms, initial or boundary conditions, whose determination from measured data after discretisation results in high-dimensional inverse problems.
The particular challenge with inverse problems is associated with their inherent instability, in the sense that minor disruptions in the data can lead to major deviations in the reconstruction. For this reason, regularisation exercises must be developed and applied in order to approach the solution of an inverse problem along a stable path, taking into consideration the mentioned instability and the fact that the data entered is usually subject to measurement errors. A further crucial issue is the identifiability each time, that is, whether the sought quantities are uniquely determined by the data provided at all.
Our research in the field of inverse problems focuses on the development, analysis and implementation of regularisation exercises. This includes mathematical modelling and numerical simulation of the respective direct problems along with optimisation in the context of partial differential equations.
Examples of areas of application in which we are currently working include medical imaging, characterising intelligent materials, including modelling hysteresis (piezoelectricity, electromagnetics), non-linear acoustics in medical applications of high-intensity ultrasound and parameter identification in system biology.