Jason Bramburger

Title:Deep learning of conjugate mappings

Rosemary Renault

Title:Examining the Feasibility of Computational Methods for Inverting Gravity and Magnetic Data: Numerical Methods and Joint Inversion

Chrysoula Tsogka

Title:Fast Signal Recovery from Quadratic Measurements.

Abstract:

We present a novel approach for recovering a sparse signal from quadratic measurements corresponding to a rank-one tensorization of the data vector. Such quadratic measurements, referred to as interferometric or cross-correlated data, naturally arise in many fields such as remote sensing, spectroscopy, holography and seismology. Compared to the sparse signal recovery problem that uses linear measurements, the unknown in this case is a matrix, $X=\rho \rho^*$, formed by the cross correlations of $\rho \in C^K$. This creates a bottleneck for the inversion since the number of unknowns grows quadratically in $K$. The main idea of the proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal elements of the unknown matrix, $| \rho_i|^2$, $i=1,\ldots,K$.  The contribution of the off-diagonal terms $\rho_i \rho_j^*$ for $i \neq j$ to the data is treated as noise and is absorbed using the Noise Collector approach introduced in [Moscoso et al, The noise collector for sparse recovery in high dimensions, PNAS 117 (2020)]. With this strategy, we recover the unknown  by solving a convex linear problem whose cost is similar to the one that uses linear measurements. The proposed approach provides exact support recovery when the data is not too noisy, and there are no false positives for any level of noise.

Todd Arbogast (with Chieh-Sen Huang, Ming-Hsien Kuo, and Xikai Zhao)

Abstract: We analyze weighted essentially nonoscillatory (WENO) reconstructions of functions from finite volume average values, basing the approximations on either polynomials or radial basis functions (RBFs). In the polynomial case, we present new multi-level WENO reconstructions with adaptive order (WENO-AO). We give conditions under which the polynomial reconstructions achieve optimal order accuracy for both smooth solutions and solutions with discontinuities. The theory of existence of finite volume RBF approximations is developed, and numerical evidence is given to show that the RBF approximation converges to the same order as a polynomial approximation when the RBF is infinitely differentiable. Specific multiquadric RBFs on stencils of 2 and 3 mesh cells are proven to have this convergence property. These WENO-AO and RBF-WENO-AO reconstructions are applied to develop finite volume schemes for hyperbolic conservation laws on nonuniform meshes over multiple space dimensions. Numerical examples show that the schemes maintain proper accuracy and achieve the essentially non-oscillatory property when solving hyperbolic conservation laws.

Rachidi B. Salako

Abstract: We study some multi-strain reaction-diffusion epidemic model dynamics with spatially heterogeneous infection and recovery rates. The nonexistence of coexistence endemic equilibrium (EE) solutions and the final extinction of one or multiple strains are proved under some proper assumptions on the model’s parameters. However, when some of these assumptions fail to hold for the case of the two strains model, we show that both strains of the disease may coexist, and solutions of the initial value problem stabilize at the unique coexistence endemic equilibrium. Finally, the asymptotic behaviors of the coexistence endemic equilibrium solutions, as the diffusion rate of the populations approaches zero, are also investigated, where the spatial segregation of multiple strains is found and determined.

Wenjing Liao

Abstract: In the past decade, deep learning has made astonishing breakthroughs in various real-world applications. It is a common   belief that deep neural networks are good at learning various geometric structures hidden in data sets, such as rich local regularities,   global symmetries, or repetitive patterns. One of the central interests in deep learning theory is to understand why deep neural   networks are successful, and how they utilize low-dimensional data structures. In this talk, I will present some statistical learning   theory for deep ReLU networks where data exhibit low-dimensional structures, such as lying on a low-dimensional manifold. The   learning tasks include regression, classification and learning operators between Hilbert spaces. When data are sampled on a low-   dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient   dimension of the data. These results demonstrate that deep neural networks are adaptive to low-dimensional geometric structures of   data sets.