- Dynamical Systems
- Inverse Problems

**Dynamical Systems **(Christine Nowak, Christian Pötzsche)

Dynamical systems are mathematical models of real-world processes, which change over time. From simple movements of a pendulum, like known from high school, the examples range from chemical reactions, biological interactions or sociological interplay, to complex climate models and thus in virtually all areas of our lives, and indeed on any scale, from micro- to the macrocosm, as well as from the simplest linear models to now much-discussed non-linear, chaotic and random systems.

The equations used to describe dynamical systems (differential and difference equations) are typically so complicated that they cannot be solved exactly. This is particularly true for problems coming from applications and which are subject to influences that can not be overlooked in the smallest details. For a treatment of these equations it is therefore essential to employ thorough so-called geometrical-qualitative methods to obtain information on the solution behavior without knowing the exact solutions. For example this is done by trying to identify a geometrical fine structure for the space of all possible states of a dynamical system. This, in turn, yields detailed information on the temporal evolution of the system, i.e. its dependence on initial conditions and external parameters.

The focus of our research are theory and applications of models that are subject to time-varying external influences - one speaks of nonautonomous systems. Their analysis requires the interaction of a variety of mathematical tools ranging from the classical theory of dynamical systems to nonlinear analysis and numerics. Moreover, we are interested in questions on nonuniqueness of solutions to differential equations.

**Inverse Problems **(Barbara Kaltenbacher, Elena Resmerita)

Inverse problems deal with the "art" of deducing quantities that are not accessible to direct measurements, from indirect observations. Therefore they have many applications ranging from medical imaging via material characterization to parameter identification in systems biology.

Many complex processes from natural sciences, medicine and technology are described by ordinary or partial differential equations. These models usually contain unknown parameters such as space dependent coefficient functions, source terms, initial- and boundary data whose determination from measured data leads to large scale inverse problems.

The particular challenges in the numerical solution of inverse problems lies in their inherent instability in the sense that small perturbations in the measured data can lead to large deviations in the solution. Therefore regularization methods have to be developed in order to approch the solution to an inverse problem along a stable path, taking into account the mentioned instability as well as the fact that the data are usually contaminated with noise. Another crucial question in inverse problems is identifiability, i.e., whether the given data really determines the searched for quantities in a unique manner.

Our research in Inverse Problems focuses on the development and implementation of regularization methods, as well as the analysis of their convergence. This also involves mathematical modelling and numerical simulation of forward problems, as well as optimization in the context of partial differential equations. Examples of application fields we are currently working in are medical imaging, characterization of smart materials, including hysteresis modelling (piezoelectricity, electromagnetics), nonlinear acoustics in medical applications of high intensity ultrasound, and parameter identification in systems biology.

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