In this dissertation, we apply the level set method and the operator-splitting method to solve geometry related problems. In the first part we use the level set method for dimension reduction on a Riemannian manifold. We design an energy based on the distance from the solution to the data set and the principal direction generated from the data set. The functional is minimized by gradient decent. This algorithm is implemented using level set functions where the solution is represented implicitly. So we do not need any a priori knowledge of the structure of the solution. Our algorithm is very robust to noise and outliers. In the second part we apply operator-splitting method to solve some Monge-Ampère equation related problems. The Monge-Ampère equation originates from the Mi...[ Read more ]

In this dissertation, we apply the level set method and the operator-splitting method to solve geometry related problems. In the first part we use the level set method for dimension reduction on a Riemannian manifold. We design an energy based on the distance from the solution to the data set and the principal direction generated from the data set. The functional is minimized by gradient decent. This algorithm is implemented using level set functions where the solution is represented implicitly. So we do not need any a priori knowledge of the structure of the solution. Our algorithm is very robust to noise and outliers. In the second part we apply operator-splitting method to solve some Monge-Ampère equation related problems. The Monge-Ampère equation originates from the Minkowski problem in differential geometry which asks to reconstruct a convex surface from a prescribed Gaussian curvature. In these problem, the solution is constructed by solving the Monge-Ampère equation. Operator-splitting method for two dimensional Monge-Ampère equation with the Dirichlet problem is first discussed. We rewrite the equation in an elliptic divergence form and then solve an initial value problem to steady state. For problem with classical solution, a two-stage strategy is introduced to accelerate the convergence rate. Then the algorithm is modified to solve various related problems.