Georgia Institute of Technology College of Sciences

The goal of this lecture is to explain and motivate the connection between Aubry-Mather theory (Dynamical Systems), and viscosity solutions of the Hamilton-Jacobi equations (PDE). The connection is the content of weak KAM Theory. The talk should be accessible to the ''generic" mathematician. No a priori knowledge of any of the two subjects is assumed.

Low-rank structures are common in modern data analysis and signal processing, and they usually play essential roles in various estimation and detection problems. It is challenging to recover the underlying low-rank structures reliably from corrupted or undersampled measurements. In this talk, we will introduce convex and nonconvex optimization methods for low-rank recovery by two examples. The first example is community detection in network data analysis. In the literature, it has been formulated as a low-rank recovery problem, and then SDP relaxation methods can be naturally applied. However, the statistical advantages of convex optimization approaches over other competitive methods, such as spectral clustering, were not clear. We show in this talk that the methodology of SDP is robust against arbitrary outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small fraction of outliers. We also demonstrate that a degree-corrected version of SDP works well for a real-world network dataset with a heterogeneous distribution of degrees. Although SDP methods are provably effective and robust, the computational complexity is usually high and there is an issue of storage. For the problem of phase retrieval, which has various applications and can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by nonconvex optimization. We prove that our method converges reliably to the original signal. It requires far less storage and has much higher rate of convergence compared to convex methods.

Mathematical models of physical phenomena often include parameters that are hard or impossible to measure directly or are subject to variability, and are thus considered uncertain. Different aspects of modeling under uncertainty include forward uncertainty propagation, statistical inver- sion of uncertain parameters, optimal design of experiments, and optimization under uncertainty. I will focus on recent advances in numerical methods for infinite-dimensional Bayesian inverse problems and optimal experimental de- sign. I will also discuss the problem of risk-averse optimization under uncertainty with applications to control of PDEs with uncertain parameters. The driving applications are systems governed by PDEs with uncertain parameter fields, such as ow in the subsurface with an uncertain permeability field, or the diffusive transport of a contaminant with an uncertain initial condition. Such problems are computationally challenging due to expensive forward PDE solves and infinite-dimensional (high-dimensional when discretized) parameter spaces.

The stochastic block model is a random graph model that was originally introduced 30 years ago to model community structure in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.