THESIS
2018
xxx, 229 pages : illustrations (chiefly color) ; 30 cm
Abstract
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
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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.
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