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The following eclectic list shows the variety and diversity of the research projects at the Institutes of Mathematics. Information to further projects is available on request at the individual offices. 

Mathematics of Nonlinear Acoustics: Analysis, Numerics, and Optimization (FWF project P24970)
DescriptionResearch on nonlinear acoustics has recently been driven by the increasing number of
industrial and medical applications of high intensity ultrasound ranging from ultrasound
cleaning or welding via sonochemistry to lithotripsy and thermotherapy. Our work in this field
is motivated, e.g., by applications in lithotripsy, where a better understanding and control of
the physical effects via mathematical analysis, numerical simulation, and optimization should
lead to a considerable reduction of lesion and complication risks.
An important prerequisite for reliable and well-founded numerical simulation and optimization
of high intensity ultrasound devices is a mathematical analysis of the underlying partial
differential equation (PDE) models in a general spatially three dimensional geometrical
setting with appropriate boundary and initial conditions. This has so far only been done for
the classical models of nonlinear acoustics. In this project we plan to analyze the qualitative
and quantitative behavior of recently developed models, which is crucial for assessing the
required level of modelling for practically relevant applications. Another important issue which
we plan to investigate is the coupling of nonlinear acoustics to other physical fields
(excitation mechanisms, focusing devices, heat generation, interaction with kidney stones).
Also numerical simulation poses a major challenge due to nonlinearity, coupling to other
physical fields, different spatial and temporal scales resulting from different wavelengths
within the subdomains, and the fact that we deal with open domain problems. Here we are
going to use domain decomposition methods (nonmatching grids, mortar elements) for
coupling and work on operator splitting methods for efficient and robust time integration as
well as on absorbing boundary conditions for simulating unbounded wave propagation using
a truncated computational domain.
The design of high intensity ultrasound devices leads to shape optimization and optimal
control problems in the context of the above mentioned PDE models, with state and control
constraints arising from physical and technical restrictions. Here we plan to derive theory
based first and second order sensitivities for use in efficient mathematical optimization
methods, and investigate multilevel methods based on natural model hierarchies.


Adaptive Discretization Methods for the Regularization of Inverse Problems (DFG, KA1778-5/1)
DescriptionMany complex processes from natural sciences, medicine and technology are described by partial differential equations (PDEs). The resulting systems of PDEs usually contain unknown parameters such as space dependent coefficient functions, source terms, initial- and boundary data whose determination leads to large scale inverse problems.

The numerical effort for solving inverse problems in the context of PDEs is usually a multiple of the one for the numerical simulation of the underlying process with given parameters. Moreover, the inherent instability of inverse problems requires the use of appropriate regularization techniques. A high potential for the development of efficient algorithms for the solution of such inverse problems lies in their adaptive discretization. While the use of adaptive concepts for the discretization for numerical simulation is well-investigated, adaptivity for inverse problems is a very new and timely field.

The goal of this project is to provide generally applicable and analytically well founded methods for the adaptive discretization of inverse problems. Our emphasis is both on efficiency of the developed algorithms and on rigorous convergence analysis.


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