Georgia Institute of Technology College of Sciences

We propose a Generalized Dantzig Selector (GDS) for linear models, in which any norm encoding the parameter structure can be leveraged for estimation. We investigate both computational and statistical aspects of the GDS. Based on conjugate proximal operator, a flexible inexact ADMM framework is designed for solving GDS, and non-asymptotic high-probability bounds are established on the estimation error, which rely on Gaussian width of unit norm ball and suitable set encompassing estimation error. Further, we consider a non-trivial example of the GDS using k-support norm. We derive an efficient method to compute the proximal operator for k-support norm since existing methods are inapplicable in this setting. For statistical analysis, we provide upper bounds for the Gaussian widths needed in the GDS analysis, yielding the first statistical recovery guarantee for estimation with the k-support norm. The experimental results confirm our theoretical analysis.

We give a brief analysis of the oscillations of the earth and then extract the system of equations describing acousto-elastic, seismic waves. Processes in Earth's interior are encoded in the coefficients of this system, which also parametrize its structure and material properties. We introduce the seismic inverse problem with its different aspects including a dual time-frequency point of view. Central in the analysis is the formulation as an inverse boundary value problem with the Dirichlet-to-Neumann map or Neumann-to-Dirichlet map as the data. We discuss various conditional Lipschitz stability estimates for this problem for coefficients containing discontinuities, and with partial boundary data, which involves the introduction of an unstructured tetrahedral mesh. Quantitative estimates of the stability constants play acritical role in analyzing convergence for iterative reconstruction schemes, making use of Hausdorff warping and leading to a multilevel approach requiring hierarchical, multi-scale compression. We present computational experiments on the regional and geophysical exploration scales. We conclude with some results pertaining to the high-frequency inverse boundary value or geometric inverse problems, again, in the presence of discontinuities.

Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. In this talk, I will advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input one. Our motivating examples will be plane elliptic cubics and genus two hyperelliptic curves. Based on joint work with Hannah Markwig (arXiv:1409.7430) and ongoing work in progress with Hannah Markwig and Ralph Morrison.