Parametric Shape-Based Methods for Inverse Problems.
Abstract: This work
presents a so called parametric level set method (PaLS) for reconstruction of obstacles
in general inverse problems. For the proposed parametric technique, general evolution
equations for the reconstruction of unknown obstacles are derived in terms of the
underlying parameters. The main feature of this technique is efficiently reducing the
dimensionality of the problem by ... read moremeans of an appropriate parametrization. This approach
bypasses many difficulties with conventional inverse problems, such as ill-posedness and
regularization challenges and many implementation complexities with traditional shape
based and level set methods such as re-initialization, use of signed distance function
and difficulties in using Newton type methods. As a proof of concept, the performance of
proposed approach is examined in three different inverse problems, i.e., electrical
resistance tomography, X-ray computed tomography and diffuse optical tomography.
However, the ultimate and driving application of this work is the characterization of
water contaminant source zones as an extremely challenging real-world inverse problem.
Remediation of sites polluted by hazardous contaminants such as dense non-aqueous phase
liquids (DNAPLs) represents an important environmental problem due to the large scale
threat these releases pose to water supplied throughout the world. Successful cleanup
and restoration relies on methods to characterize the source zone structure prior to
remediation and afterwards to monitor the corresponding chemical levels. We apply the
PaLS technology to DNAPL source zone characterization based on the joint inversion of
hydrological and geophysical data. The hydraulic modality corresponds to interaction of
groundwater with the contaminant source zone, for which a fully 3-D flow and transport
model is used to provide the downstream contaminant concentration associated to the
source zone saturation distribution. As the geophysical modality we utilize electrical
resistance tomography (ERT) where electric potential measurements related to the
electrical properties of the medium are obtained cross gradient to the water flow
direction. The joint inversion technique presented is able to extract the profiles of
the low and high saturation regions in a DNAPL source zone. Through some challenging
simulations we show that the proposed algorithm is a markedly suitable and
computationally tractable choice for the complex inversion problem
Thesis (Ph.D.)--Tufts University, 2012.
Submitted to the Dept. of Electrical Engineering.
Advisor: Eric Miller.
Committee: Eric Miller, Linda Abriola, Shuchin Aeron, and Aria Abubakar.
Keywords: Electrical engineering, Environmental engineering, and Mathematics.read less