Applied Math

Many MCMC methods either rely on gradients – such as the No-U-Turn Sampler (NUTS) – or struggle with multi-scale distributions, where different regions require vastly different exploration strategies.  NURS is a new  locally adaptive MCMC method that overcomes these…

The cochlea is the main organ of the inner ear. It is responsible for converting the incoming mechanical sound signals from the middle ear into neural signals in the auditory nerve. This transformation involves a fascinating sequence of mechanical and hydrodynamical components, responsible for…

We consider conditional McKean-Vlasov processes that arise in the study of hydrodynamic limits of interacting diffusions on random regular graphs. We establish an H-theorem that characterizes the long-time behavior of these processes. Specifically, we show that a certain function related to the…

The simulation of all-atom molecular dynamics is limited in both length and time scales. The same difficulty applies to simulating the unitary dynamics of large, closed quantum systems. Therefore, we turn to modeling coarse-grained molecular dynamics and open quantum dynamics, using approximate…

Subspace-based signal processing techniques, such as the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular methods for spectral estimation. These algorithms can achieve the so-called super-resolution scaling under low noise conditions, surpassing…

Operator learning has undergone rapid development in recent years and gradually emerged as a transformative machine-learning paradigm. In this talk, I will provide an overview of the operator-learning framework and explore its applications to computational problems, with a particular focus on…

A longstanding challenge in data science is to effectively quantify systems of interest by integrating information from heterogeneous datasets, a problem known as multiview learning. In this talk, I will present recent advancements in this direction, focusing on novel algorithms based on…

Manifold optimization has found wide applications across various scientific and engineering domains. In this talk, I will present our recently developed algorithms for large-scale decentralized and federated manifold optimization. In addition, I will present a retraction-free and penalty…