# Research Projects

## Mathematical simulations towards preventing cerebral hemorrhage in premature infants

It is known that half of new born infants with the birth weight less than to 1500 gram develop an intracranial hemorrhage within the first 24 hours of life. Another quarter reveals intracranial bleedings after 24 hours of age. The cause of brain bleeding is always the germinal matrix of the immature brain. The germinal matrix consists of richly vascularized neuroepithelial cells and is especially vulnerable due to the precapillary arteriolar-venular anastomoses and the absence of muscle and collagen layers in its immature blood vessels. The pathogenesis of germinal matrix hemorrhage is multifactorial. The most important factors are fluctuating cerebral blood flow because of impaired cerebral autoregulation; increase in cerebral venous pressure due to e.g. pneumothorax, mechanical or any pressure ventilation; and platelet and coagulation disturbances. Cerebrovascular system of the mature brain is the object of mathematical study during recent 20 years. However, despite of significant progress in understanding of cerebral regulation mechanisms, there is a lack of knowledge of normal and disturbed cerebral circulation in premature newborns. The role of blood pressure and impaired autoregulation is far from being resolved. Therefore, creation of adequate mathematical models of cerebral blood circulation, taking into account special features of the immature brain, is a challenging task. The aim of this project is the development of mathematical models and suitable numerical methods for the simulation of the blood vessel system and blood flow in the germinal matrix. The major attention will be drawn to modeling cerebral autoregulation that plays essential role in maintaining the appropriate level of brain perfusion. The model should take into account the dependence of the cerebral blood flow on the blood pressure which, in turn, depends on the size, length, diameter of blood vessels, CO2 reactivity, and variations of the intracranial pressure. The implementation of these intentions requires the development of new and the enhancement of existing mathematical models. First, a realistic model of the blood vessel network of the germinal matrix will be created. Second, an adequate model of blood flow through this network, accounting for the above mentioned dependencies, will be developed, and the influence of different injuring and control factors will be estimated. In the third stage, the values of dynamic characteristics and parameters (e.g. blood flow velocities and pressures) computed on the second stage will be used in finite element simulations of typical blood vessels of the germinal matrix to compute critical stresses in vessel walls. The results of mathematical modeling will contribute to a better understanding of blood flow processes in the germinal matrix, to the recognition and evaluation of damaging factors leading to bleedings in the premature brain, and to practical recommendations on the reduction of injuring effects. Project funded by Klaus von Tschira Stiftung. In cooperation with Klinikum rechts der Isar (Prof. Dr. Renée Lampe). Project Leader: Prof. Dr. Renée Lampe, Prof. Dr. Dr. h.c. mult. Karl-Heinz Hoffmann, Prof. Dr. Martin Brokate Associate investigator: Dr. Nikolai Botkin (TUM, Centre for Mathematics - M6), Dr. Irina Sidorenko (TUM, Centre for Mathematics - M6), Dr. Andrey Kovtanyuk (Klinikum rechts der Isar), Dr. Varvara Turova (Klinikum rechts der Isar)## Robust dynamic programming approach to aircraft control problems with disturbances

The objective of this project is the development of optimal control techniques and numerical methods for the treatment of various problems of aircraft control. The main attention is drawn to dangerous flight situations where disturbances, severe windshears must be accounted for. Optimal and quasi-optimal aircraft control laws guaranteeing successful withstanding of complicated flight situations are intended to be designed. Furthermore, fast numerical methods for “on-board” solving of optimal control problems with accounting for wind disturbances are being developed. Quasi optimal feedback laws that produce extreme disturbances are being elaborated and implemented in a flight simulator. The theory of differential games, where disturbances are considered as counteractions of an opposite player, is appropriate for constructing control laws that guarantee a certain value of the objective function, as long as the disturbances remain in a given range. The numerical treatment of differential games is based on computing viscosity solutions to Hamilton-Jacobi equations, which is closely related to dynamic programming methods. One of the objectives of this project is the development of numerical methods for solving nonlinear differential games of high dimension and their applications to aircraft trajectory optimization problems. It is very important that the project is worked out in close cooperation between mathematicians and engineers, which allows the applicants to calibrate the models and to verify the efficiency of the designed controls using flight simulators. Project funded by DFG Individual Grants Programme. In cooperation with Institute of Flight System Dynamics, TUM (Prof. Dr.-Ing. Florian Holzapfel). Project Leader: Dr. Varvara Turova (TUM, Klinikum rechts der Isar), Prof. Dr.-Ing. Florian Holzapfel (TUM, Institute of Flight System Dynamics) Main investigators: Dr. Nikolai Botkin (TUM, Centre for Mathematics - M6), M.Sc. Johannes Diepolder (TUM, Institute of Flight System Dynamics)## Solution of high dimensional Hamilton-Jacoby equations with application to aircraft control in the presence of windshear

This proposal is related to grid methods for solving Hamilton-Jacobi equations arising from problems of optimization of aircraft trajectories in the presence of windshears. Computed solutions (value functions) yield guaranteeing controls ensuring safe runway acceleration, take-off, cruise flight, landing, abort landing, etc. independently on wind disturbances. Moreover, such solutions allow us to design extreme wind disturbances that can be used in flight simulators for testing autopilots and training pilots in extreme situations. The applicants have developed stable grid methods for the treatment of Hamilton-Jacobi equations arising from conflict control problems with state constraints and have tested them on multiprocessor systems with relative small numbers of processors. The results are very promising bearing in mind restrictive computer resources. Using the full power of the`SuperMUC`

system and sparse representations of grid functions, we expect to treat realistic aircraft models involving more than 6 state variables. This allows us to solve actual engineering problems related to flight operating safety.
Project Leader: Prof. Dr. Martin Brokate, Prof. Dr.-Ing. Florian Holzapfel
Principal investigator:
Dr. Nikolai Botkin (TUM, Centre for Mathematics - M6)
Associate investigator:
M.Sc. Johannes Diepolder (TUM, Institute of Flight System Dynamics)
## Conformal monogenic frames for image analysis

Conformal monogenic signals - in contrast to classical monogenic signals - include a curvature term, which allows for the detection of curve singularities. In the project, we will combine spline frames with the idea of the conformal monogenic signal to make these common analysis families available for higher dimensional image processing with phase information. Project Leader: Prof. Dr. Brigitte Forster-Heinlein A cooperation with Prof. Uwe Kähler, University of Aveiro, Portugal. Project funded by the DAAD.## Optimal control in cryopreservation of cells and tissues

The proposal concerns the application of the theory of partial differential equations and optimal control techniques to the minimization of damaging factors in cryopreservation of living cells and tissues in order to increase the survival rate of frozen and subsequently thawed out cells. Project Leader: Prof. Dr. Dr. h.c. mult. Karl-Heinz Hoffmann Investigators: Dr. Nikolai Botkin, Dr. Varvara Turova## Modelling of CO_{2} sequestration including parameter identification and numerical simulation

This subproject is a part of the Special Partnership between King Abdullah University of Science and Technology (KAUST) and Technische Universität München (TUM).
The subproject deals with the simulation of different scenarios of CO_{2}sequestration in order to predict possible leakage sources of CO

_{2}, estimate expected storage capacities of CO

_{2}repositories, and optimize the injection process. The development and numerical implementation of multiphase flow models including such phenomena as phase changes, hysteresis effects, chemical reactions, etc is the content of the investigation. Project Leader: Prof. Dr. Dr. h.c. mult. Karl-Heinz Hoffmann Investigator: Dr. Nikolai Botkin

## Hysteretic aspects of CO_{2} sequestration modelling

This subproject is a part of the Special Partnership between King Abdullah University of Science and Technology (KAUST) and Technische Universität München (TUM).
The multiphase nature of the flow in porous media is characterized by hysteretic effects on the macroscopic level. These effects have a significant influence on the behavior of the whole system and therefore have to be taken into account. The aim of this subproject is to develop and implement models describing the hysteresis in the context of the CO_{2}sequestration process. Project Leader: Prof. Dr. Martin Brokate Investigators: Dr. Oleg Pykhteev, Dr. Varvara Turova