Abstract |
Viscosity solutions are a class of generalized solutions of nonlinear partial differential
equations. The great value of this kind of weak solutions is the fact that very general
existence, uniqueness and stability results hold for them in many problems of various
fields of application. They are used in many kinds of applications, two of which are analyzed in this work.
The Shallow Lake Problem, a non-standard optimal control problem derived from the
combination of agricultural activities and ecological services that a shallow lake provides,
is the first application studied. The optimal dynamics of the problem, necessary conditions
of which are provided by the Pontryagin Maximum Principle, are studied in the beginning
for a range of values of the discount factor. The number and the type of the equilibrium
points are also investigated. We then prove that the value function of the Shallow Lake
Problem is a viscosity solution of an Optimal Hamilton-Jacobi-Bellman (OHJB) equation,
a basic for the rest of this work result. To derive this, we study the control problem on a
compact control space and we prove monotonicity, semiconvexity and other, related to the
subdifferential, properties for the corresponding value function, considered as a viscosity
solution of a modified HJB equation. We extract many regularity results for the value
(welfare) function, using this result. Furthermore, three different numerical schemes are
presented, the "forward", the "backward" and the "upwind" schemes, for the approximation
of the viscosity solution, based on a finite difference space discretization. Their convergence is proved using fixed point arguments. For validation of the numerical results,
we compare them with the results obtained from the Simple Shooting Method, which we
use as the "gold standard". The small mean relative error for all cases (different number of
saddle points, different spatial step) proves the accuracy of the numerical approximations.
The second application analyzed is connected with the geometric representation of the
Abdominal Aortic Aneurysm (AAA), which is a localized dilatation of the aortic wall. A
reliable estimate of AAA rupture risk demands accurate measurements of its geometric
characteristics. So, our main objective in this part of our work is the extraction of the
thrombus and outer wall boundaries from cross sections of a 3D CTA AAA image data set,
using the level set framework and new geometrical methods to address the basic problem
of no sufficient intensity contrast between thrombus and surrounding tissue. Tools like the
inversion mapping and the convex hull of a closed curve are used to trace and reconstruct
these boundaries, exploiting the presence of calcifications, which are detected by combining
these tools with a thresholding technique. We also introduce three novel stopping criteria
to address the leakage problem that Level Set Methods (LSM’s) present and a method
for detecting the leakage regions. A Fast Marching Method (FMM) is initially used to
resolve another problem of the LSM’s, namely speed, with a proper modification for images
difficult to segment. In regions with few or no calcifications, an interpolation distance
technique may be used to obtain the two boundaries, if required.
Artificial images which simulate the real cases are then presented to test the versatility
of the methods. Sensitivity to the parameter settings and reproducibility are also analyzed
and segmentation time is presented. A manual segmentation, created by a medical expert,
was performed in the slices of ten patient data sets (450 slices) to compare with our results.
Mean distance, area overlap and relative volume error are three of the quantities used to
evaluate outer wall segmentation error. For the validation of the thrombus results a
method for approximating the mean wall thickness of the AAA is introduced, utilizing
another tool, namely the division of a curve into sectors using its centroid.
The results for the outer wall and for the mean wall thickness are comparable with
previous values reported in literature, in which, however, there is no segmentation of the
thrombus boundary, at least using the level set framework. These results indicate that geometrically accurate 3D reconstructions of AAA anatomy can be produced through LSM based segmentation of image data obtained from currently available imaging technology.
This information is quite important for finally obtaining a reliable patient specific measure
of AAA rupture risk.
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