Department of Mathematics, University of California Los Angeles
Title
Geometric Scattering on Measure Spaces
Abstract
Geometric Deep Learning is an emerging field of research that aims to extend the success of convolutional neural networks (CNNs) to data with non-Euclidean geometric structure. Despite being in its relative infancy, this field has already found great success in many applications such as recommender systems, computer graphics, and traffic navigation. In order to improve our understanding of the networks used in this new field, several works have proposed novel versions of the scattering transform, a wavelet-based model of CNNs for graphs, manifolds, and more general measure spaces. In a similar spirit to the original Euclidean scattering transform, these geometric scattering transforms provide a mathematically rigorous framework for understanding the stability and invariance of the networks used in geometric deep learning.Additionally, they also have many interesting applications such as drug discovery, solving combinatorial optimization problems, and predicting patient outcomes from single-cell data. In particular, motivated by these applications to single-cell data, I will also discuss recent work proposing a diffusion maps style algorithm with quantitative convergence guarantees for implementing the manifold scattering transform from finitely many samples of an unknown manifold.
Department of Applied Mathematics, University of Colorado at Boulder
Title
Direct parallel solvers for a variable coefficient Helmholtz equation applied to 3D wave propagation problems.
Abstract
We present an efficient parallel solver for the linear system that arises from the Hierarchical Poincare-Steklov (HPS)
discretization of three dimensional variable coefficient Helmholtz problems. We tackle problems of 100 wavelengths in
each direction that require more than a billion unknowns to achieve approximately 4 digits of accuracy in less than 20 minutes.
Additionally, we show results on a parallel Direct solver for the same problem and explore the potential of both techniques
for inverse problems. This work was funded by Total Energies and NSF.
