Browsing by Author "Ullrich, Marcel"

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Monotonicity-based methods for inverse parameter identification problems in partial differential equations
(2015) Ullrich, Marcel; von Harrach, Bastian (Prof. Dr.)
This work is concerned with monotonicity-based methods for inverse parameter reconstruction problems in partial differential equations. The first three chapters address the anomaly detection problem of electrical impedance tomography. While electrical impedance tomography aims on reconstructing the interior conductivity distribution of a conductive subject from boundary data, the goal of the specific anomaly detection problem is the reconstruction of areas inside a conductive subject where the conductivity differs from an expected reference conductivity. The considered boundary data can be understood as an operator that describes current-voltage measurements. In the final chapter we prove a novel uniqueness result for the inverse potential problem of the Schrödinger equation with partial data. For the development of anomaly detection methods, both known and novel variants of a monotonicity relation are used. Roughly speaking, these monotonicity relations particularly show that a pointwise decrease of the conductivity leads to larger boundary data (in sense of operator definiteness). At first glance, it is not obvious at all whether the converse of this implication holds also true, i.e., it is not clear whether larger boundary data could also result from a local decrease of the conductivity in some parts and a local increase in other parts. Assuming a local definiteness condition for the conductivity change we prove a partial converse of the monotonicity implication that holds for the case in which the measurements are modeled with the idealized continuum model. In the first chapter we develop novel anomaly detection methods for measurement data modeled with the continuum model. Moreover, fast linearized variants are presented that only require the computation of reference measurements for one homogeneous reference conductivity. We prove that all presented methods are capable of reconstructing the exact outer shape of conductivity anomalies. In realistic electrical impedance tomography settings in which measurement data is collected on a finite number of electrodes, the reconstruction of the exact outer shape of anomalies cannot be guaranteed anymore. On top of that, systematic errors resulting from imprecise knowledge of the setting parameters as well as additional random measurement errors need to be taken into account. In the second chapter we show that nevertheless certain resolution guarantees are principally possible for such settings. In the third chapter we develop a novel hybrid method that does not require the simulation of reference data. We apply an idealized model for ultrasound modulation that alters the conductivity uniformly in a test region and we develop a test criterion to check whether the test region is located inside an anomaly. The test criterion consists of a monotonicity-based comparison of ultrasound modulated and weighted frequency-difference measurements. Finally, in the fourth chapter, a local uniqueness result for the inverse potential problem of the Schrödinger equation on a bounded Lipschitz domain with partial boundary data is shown. More precisely, we show that positive-valued bounded potentials that do not completely coincide in a neighborhood of a potentially arbitrarily small part of the boundary can be distinguished from Cauchy data on this boundary part provided that a local definiteness condition is fulfilled.