Numerical Approaches on Shape Optimization of Elliptic Eigenvalue Problems and Shape Study of Human Brains
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.