Nearest neighbours for solving PDEs, random sampling, and classifying point clouds
When these searches are performed for each point on a grid, we obtain a discretization which captures the geometry of the object. The first half of the presentation reviews “closest point methods”, which use such a grid to numerically solve PDEs on curved surfaces or other general domains, using simple finite differences and interpolation.
The second half of the talk covers work-in-progress on a sampling framework whereby random one-dimensional rays are projected onto an object using nearest-neighbour searches. One possible application is the classification of point clouds: the sampled data can be fed into a simple neural network. This is already giving promising results.
This is joint work with Lewis Liu, Louis Ly, and Richard Tsai.