Mathematical optimization has significantly expanded its scope during the last decades. This is, in part, due to the fact that it is increasingly often considered in a function space frame-work which allows for differential equations or variational problems as constraints. It provides the natural setting for parameter estimation and optimal control problems, as well as for the variational formulations of image processing, non-invasive testing, and modeling of complex growth processes as they arise in biosciences.
Mathematicians in Graz have contributed significantly to the development of optimization with partial differential equations as constraints. In the future these activities will be expanded including new directions in scientific computing. The recent hiring of two new professors in numerical analysis and scientific computing at TUG and KFU offers a new perspective for collaboration between the fields of optimization in the context of partial differential equations and scientific computing in Graz.
The groups on optimization and on numerical methods will be joined by a third group of researchers with expertise in selected branches of the biomedical sciences.
The central scheme of the proposed SFB is continuous optimization in the context of differential equations and variational inequalities as well as the development of associated numerical methods. The proposed research includes topics on optimal control based on model reduction techniques, semi-smooth Newton methods, optimization in the context of free boundaries and interfaces, inherent optimizing properties of multigrid cycles, and efficient and robust numerical strategies for solving large-scale optimality systems. These are timely problems within optimization and control per se. But they are also triggered by the applications which are investigated in the biomedical sciences group. The topics treated there include magnetic resonance imaging, special near-field techniques for biomedical imaging, computer models for the heart, the cardio-vascular and the insulin-glucose systems. Advancing the solution strategies for the proposed bio-engineering optimization problems will provide new insight, and has the potential of improving diagnostic methods and tools.
The combination of expertise in optimization and control for biomedical sciences involving mathematicians at the KFU and TUG as well as biomedical engineering partners at TUG and MUG makes this group of researchers unique.
Fast Finite and Boundary Element Methods for Optimality Systems (Direction: Gundolf Haase, Olaf Steinbach, 1.5.2007 - 30.4.2018)
The aim of this subproject is to formulate, analyse, implement and provide an efficient simulation tool based on finite and boundary element methods for the approximate solution of optimality problems. This should cover potential, electromagnetic, Helmholtz problems, linear elastostatics and their coupling which originate from medical applications.
Special Near Field Techniques for Biomedical Imaging (Direction: Hermann Scharfetter, Olaf Steinbach, 1.4.2007 - 31.12.2010)
Several biomedical imaging methods rely on the interaction between tissue and the near field of electromagnetic sources. For the solution of the underlying partial differential equations and their adjoints efficient numerical solvers are mandatory. In particular we will focus on fast finite and boundary element methods for inverse problems for the Maxwell system.
Quantification of Functional and Biophysical Information in Magnetic Resonance Imaging (Direction: Stollberger, Teilnehmer: Michael Hofer, Florian Knoll, 1.4.2007 - 30.4.2018)
Magnetic resonance imaging (MRI) is now developing into an important functional imaging technique. The focus of this project is the development and implementation of methods for the improved quantification of functional and physiological parameters describing properties of the micro-vessel system, tissue perfusion and diffusion of water molecules in the microenvironment of tumours. The quantification of functional information in MRI is a multi step procedure which includes in general optimization of data acquisition and appropriate image analysis. In the whole field, many of the mathematical challenges refer to inverse or optimization problems. The two main goals in the short term perspective of this proposal are the improvement of data acquisition with optimized parallel imaging and the implementation of non-rigid registration techniques for functional assessment of organs effected by physiological motion. Further goals, treated by additional staff of the working group and in cooperation with different groups are the correction of inhomogeneities of the transmit and receive field and noise related topics in functional imaging. Principal and independent component analysis will be explored for model reduction of multidimensional data sets. New targeted MR-tracers ("molecular imaging") play an important role in the long-term perspectives. These molecules will be captured by active processes, therefore new models for biophysical interpretation of tracer distribution and strategies to solve the specific inverse problems will be necessary. Data deconvolution as spatiotemporal process instead of pixel by pixel should also be investigated in the long term perspective. The further development of robust non-rigid registration algorithms and minimal encoding strategies in MR-imaging is in the long term range, too.