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Numerical approaches on shape optimization of elliptic eigenvalue problems and shape study of human brains
Su, Shu

2010, Doctor of Philosophy, Ohio State University, Mathematics.

Shape and topology optimizations have been extensively studied and applied
in the conductivity and elasticity settings. Mathematically, these are infinite-dimensional optimization problems and the closed-form
solutions for most problems are difficult to find. Thus, numerical
approaches are commonly used to solve the problems by using iterative
methods.


In the first part of this thesis, a new efficient numerical approach
is developed and applied to the elliptic eigenvalue problems which
are related to mathematical physics. The study investigates the
minimization and maximization of the k-th eigenvalue and the
maximization of the spectrum ratio of the differential operator.
Physically, the problem is motivated by the question of determining
the optimal vibrating membrane made of two materials with distinct
mass densities such that the k-th frequency or the spectrum ratio of
the resulting membrane is extremized. This approach utilizes the
Rayleigh's Principle of eigenvalues and can handle the topology
changes automatically. It turns out to be more robust and efficient
than the classical level set approach. We further extend the method
to solve principle eigenvalue minimization problem on surfaces.


Another topic we studied is related to morphology of human brains.
Human brains are highly convoluted surfaces with multiple folds. To
characterize the complexity of these folds and their relationship
with neurological and psychiatric conditions, different techniques
have been developed to quantify the folding patterns and gyrification
of the brain. In the second part of this thesis, a new geometric approach is proposed to
measure the gyrification of human brains from magnetic resonance images
(MRI). This approach is based on intrinsic 3D measurements that relate
the local brain surface area to the corresponding area of a tightly
wrapped sheet. Geodesic depth is incorporated into the gyrification
computation as well. These quantities are efficiently and accurately
computed by solving geometric partial differential equations. The
presentation of the geometric framework is complemented with experimental
results for brain complexity in typically developing children and
adolescents. Using this novel approach, evidence of developmental
alterations in brain surface complexity throughout childhood and adolescence is provided.

Chiu-Yen Kao, PhD (Advisor)
Gregory Baker, PhD (Committee Member)
Yuan Lou, PhD (Committee Member)
209 p.

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Su, S. (2010). Numerical approaches on shape optimization of elliptic eigenvalue problems and shape study of human brains. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Su, Shu. "Numerical approaches on shape optimization of elliptic eigenvalue problems and shape study of human brains." Electronic Thesis or Dissertation. Ohio State University, 2010. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Su, Shu "Numerical approaches on shape optimization of elliptic eigenvalue problems and shape study of human brains." Electronic Thesis or Dissertation. Ohio State University, 2010. https://etd.ohiolink.edu/

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