### 2024 – 2025 Academic Year

**Organized by: **Changhui Tan (tan@math.sc.edu) & Siming He (siming@mailbox.sc.edu)

Unless otherwise noted, the seminar will be held on Fridays from 2:30pm to 3:30pm in LeConte 440.

This page will be update as new seminar information becomes available. Check back often for the most up to date information!

**When: **October 11^{th} 2024 from 2:30 – 3:30 p.m.

**Where:** LeConte 440

**Speaker:** Xiaoqian Xu, (Duke Kunshan University)

**Abstract: **In the study of incompressible fluid, one fundamental phenomenon that arises in a
wide variety of applications is dissipation enhancement by so-called mixing flow.
In this talk, I will give a brief introduction to the idea of mixing flow and the
role it plays in the field of advection-diffusion-reaction equation. I will also discuss
the examples of such flows in this talk.

**When: **August 30^{th} 2024 from 3:40 – 4:30 p.m.

**Where:** LeConte 440

**Speaker:** Viktor Stein (TU Berlin)

**Abstract:** We give a comprehensive description of Wasserstein gradient flows of maximum mean
discrepancy (MMD) functionals \(F_\nu := \text{MMD}_K^2(\cdot, \nu)\) towards given
target measures \(\nu\) on the real line, where we focus on the negative distance
kernel \(K(x,y) := -|x-y|\). In one dimension, the Wasserstein-2 space can be isometrically
embedded into the cone \(C(0,1) \subset L_2(0,1)\) of quantile functions leading to
a characterization of Wasserstein gradient flows via the solution of an associated
Cauchy problem on \(L_2(0,1)\). Based on the construction of an appropriate counterpart
of \(F_\nu\) on \(L_2(0,1)\) and its subdifferential, we provide a solution of the
Cauchy problem. For discrete target measures \(\nu\), this results in a piecewise
linear solution formula. We prove invariance and smoothing properties of the flow
on subsets of \(C(0,1)\). For certain \(F_\nu\)-flows this implies that initial point
measures instantly become absolutely continuous, and stay so over time. Finally, we
illustrate the behavior of the flow by various numerical examples using an implicit
Euler scheme and demonstrate differences to the explicit Euler scheme, which is easier
to compute, but comes with limited convergence guarantees. This is joint work with
Richard Duong (TU Berlin), Robert Beinert (TU Berlin), Johannes Hertrich (UCL) and
Gabriele Steidl (TU Berlin).

*Joint Seminar with the RTG Seminars on Data Science*

### Previous Years

**Organized by: **Changhui Tan (tan@math.sc.edu) & Siming He (siming@mailbox.sc.edu)

Unless otherwise noted, the seminar will be held on Fridays from 2:30pm to 3:30pm in LeConte 440.

This page will be update as new seminar information becomes available. Check back often for the most up to date information!

**When: **April 19^{th} 2024 from 2:30 p.m. to 3:30 p.m.

**Where:** LeConte 440

**Speaker:** Weinan Wang (University of Oklahoma)

**Abstract:** In this talk, I will talk about some recent well-posedness and stability results
for three incompressible fluid equations. More precisely, I will first discuss a global
well-posedness result for the 2D Boussinesq equations with fractional dissipation
and the long-time behavior of solutions. For the Oldroyd-B model, we show that small
smooth data lead to global and stable solutions. When the Navier-Stokes is coupled
with the magnetic field in the magneto-hydrodynamics (MHD) system, solutions near
a background magnetic field are shown to be always global in time. The magnetic field
stabilizes the fluid. In the examples for Oldroyd-B and MHD, the systems governing
the perturbations can be converted to damped wave equations, which reveal the smoothing
and stabilizing effect. If time permits, I will discuss some open problems.

**When:** April 12^{th} 2024 from 2:30 p.m. to 3:30 p.m.

**Where:** LeConte 440

**Speaker:** Feng Bao (Florida State University)

**Abstract:** Generative machine learning models, including variational auto-encoders (VAE), normalizing
flows (NF), generative adversarial networks (GANs), diffusion models, have dramatically
improved the quality and realism of generated content, whether it's images, text,
or audio. In science and engineering, generative models can be used as powerful tools
for probability density estimation or high-dimensional sampling that critical capabilities
in uncertainty quantification (UQ), e.g., Bayesian inference for parameter estimation.
Studies on generative models for image/audio synthesis focus on improving the quality
of individual sample, which often make the generative models complicated and difficult
to train. On the other hand, UQ tasks usually focus on accurate approximation of statistics
of interest without worrying about the quality of any individual sample, so direct
application of existing generative models to UQ tasks may lead to inaccurate approximation
or unstable training process. To alleviate those challenges, we developed several
new generative diffusion models for various UQ tasks, including diffusion-model-assisted
supervised learning of generative models, a score-based nonlinear filter for recursive
Bayesian inference, and a training-free ensemble score filter for tracking high dimensional
stochastic dynamical systems. We will demonstrate the effectiveness of those methods
in various UQ tasks including density estimation and data assimilation problems.

**When:** March 1^{st} 2024 from 2:30 p.m. to 3:30 p.m.

**Where:** LeConte 440

**Speaker:** Ruiwen Shu (University of Georgia)

**Abstract:** I will discuss the behavior of interaction energy minimizers on bounded domains.
When the interaction potential is more singular than Newtonian, the mass does not
tend to concentrate on the boundary; when it is Newtonian or less singular, the mass
necessarily concentrates on the boundary for purely repulsive potentials. We also
draw a connection between bounded-domain minimizers and whole-space minimizes.

**When: **February 28^{th} 2024 from 2:30 p.m. to 3:30 p.m.

**Where:** Virtual via Zoom

**Speaker:** David Pardo (University of the Basque Country, Spain)

**Abstract: **Download here [PDF]

Join Zoom Meeting**Meeting ID:** 982 8541 4873**Passcode:** 839056

**When: **December 8^{th} 2023 from 2:30pm to 3:30pm

**Where: **Virtual via Zoom

**Speaker: **Yuan-Nan Young (New Jersey Institute of Technology)

**Host: **Paula Vasquez

**Abstract:** The Stoichiometric Model for the interaction of centrosomes with cortically anchored
pulling motors, through their associated microtubules (MTs), has been applied to study
key steps in the cell division such as spindle positioning and elongation. In this
work we extend the original Stoichiometric Model to incorporate (1) overlap in the
cortical motors, and (2) the dependence of velocity in the detachment rate of MTs
from the cortical motors. We examine the effects of motor overlap and velocity-dependent
detachment rate on the centrosome dynamics, such as the radial oscillation around
the geometric center of the cell, the nonlinear nature (supercritical and subcritical
Hopf bifurcation) of such oscillation, and the nonlinear orbital motions previously
found for a centrosome. We explore biologically feasible parameter regimes where these
effects may lead to significantly different centrosome/nucleus dynamics. Furthermore
we use this extended Stoichiometric Model to study the migration of a nucleus being
positioned by a centrosome. This is joint work with Justin Maramuthal, Reza Farhadifar
and Michael Shelley.

**Meeting ID:** 942 9769 4178 **Passcode:** 488494

**When: **December 1^{st} 2023 from 3:40pm to 4:40pm

**Where: **LeConte 440

**Speaker: **Yuehaw Khoo (University of Chicago)

**Host: **Wuchen Li (Joint RTG Seminar)

**Abstract:** Tensor-network ansatz has long been employed to solve the high-dimensional Schrödinger
equation, demonstrating linear complexity scaling with respect to dimensionality.
Recently, this ansatz has found applications in various machine learning scenarios,
including supervised learning and generative modeling, where the data originates from
a random process. In this talk, we present a new perspective on randomized linear
algebra, showcasing its usage in estimating a density as a tensor-network from i.i.d.
samples of a distribution, without the curse of dimensionality, and without the use
of optimization techniques. Moreover, we illustrate how this concept can combine the
strengths of particle and tensor-network methods for solving high-dimensional PDEs,
resulting in enhanced flexibility for both approaches.

**When: **November 17^{th}, 2023 from 2:30pm to 3:30pm

**Where: **LeConte 440

**Speaker: **Quyuan Lin (Clemson University)

**Host: **Changhui Tan

**Abstract:** Large scale dynamics of the ocean and the atmosphere are governed by the primitive
equations (PE). In this presentation, I will first review the derivation of the PE
and some well-known results for this model, including well-posedness of the viscous
PE and ill-posedness of the inviscid PE. The focus will then shift to discussing singularity
formation and the stability of singularities for the inviscid PE, as well as the effect
of fast rotation (Coriolis force) on the lifespan of the analytic solutions. Finally,
I will talk about a machine learning algorithm, the physics-informed neural networks
(PINNs), for solving the viscous PE, and its rigorous error estimate.

**When: **November 3^{rd}, 2023 from 2:30pm to 3:30pm

**Where: **LeConte 440

**Speaker: **Xiantao Li (Penn State University)

**Host: **Yi Sun

**Abstract:** Quantum computing has recently emerged as a potential tool for large-scale scientific
computing. In sharp contrast to their classical counterparts, quantum computers use
qubits that can exist in superposition, potentially offering exponential speedup for
many computational problems. Current quantum devices are noisy and error-prone, and
in near term, a hybrid approach is more appropriate. I will discuss this hybrid framework
using three examples: quantum machine learning, quantum algorithms for density-functional
theory and quantum optimal control. In particular, this talk will outline how quantum
algorithms can be interfaced with a classical method, the convergence properties and
the overall complexity.

**When:** October 27^{nd}, 2023 from 2:30pm to 3:30pm

**Where:** LeConte 440

**Speaker:** Adrian Tudorascu (West Virginia University)

**Host: **Changhui Tan

**Abstract:** We study Zeldovich's Sticky-Particles system when the evolution is confined to arbitrary
closed subsets of the real line. Only the sticky boundary condition leads to a rigorous
formulation of the initial value problem, whose well-posedness is proved under the
Oleinik and initial strong continuity of energy conditions. For solutions confined
to compact sets a long-time asymptotic limit is shown to exist.

**When:** October 27^{nd}, 2023 from 3:40pm to 4:40pm

**Where:** LeConte 440

**Speaker:** Jiajia Yu (Duke University)

**Host: **Wuchen Li (Joint RTG Seminar)

**Abstract:** Mean-field games study the Nash Equilibrium in a non-cooperative game with infinitely
many agents. Most existing works study solving the Nash Equilibrium with given cost
functions. However, it is not always straightforward to obtain these cost functions.
On the contrary, it is often possible to observe the Nash Equilibrium in real-world
scenarios. In this talk, I will discuss a bilevel optimization approach for solving
inverse mean-field game problems, i.e., identifying the cost functions that drive
the observed Nash Equilibrium. With the bilevel formulation, we retain the essential
characteristics of convex objective and linear constraint in the forward problem.
This formulation permits us to solve the problem using a gradient-based optimization
algorithm with a nice convergence guarantee. We focus on inverse mean-field games
with unknown obstacles and unknown metrics and establish the numerical stability of
these two inverse problems. In addition, we prove and numerically verify the unique
identifiability for the inverse problem with unknown obstacles. This is a joint work
with Quan Xiao (RPI), Rongjie Lai (Purdue) and Tianyi Chen (RPI).

**When: **September 29^{th}, 2023 from 3:40pm to 4:40pm

**Where: **LeConte 440

**Speaker: **Qi Feng (Florida State University)

**Host:** Wuchen Li (Joint RTG Seminar)

**Abstract:** In this talk, I will discuss long-time dynamical behaviors of Langevin dynamics,
including Langevin dynamics on Lie groups and mean-field underdamped Langevin dynamics.
We provide unified Hessian matrix conditions for different drift and diffusion coefficients.
This matrix condition is derived from the dissipation of a selected Lyapunov functional,
namely the auxiliary Fisher information functional. We verify the proposed matrix
conditions in various examples. I will also talk about the application in distribution
sampling and optimization. This talk is based on several joint works with Erhan Bayraktar
and Wuchen Li.

**When: **September 22^{nd}, 2023 from 3:40pm-4:40pm

**Where: **LeConte 440

**Speaker: **Guosheng Fu (University of Notre Dame)

**Host:** Wuchen Li (Joint RTG Seminar)

**Abstract: **We design and compute first-order implicit-in-time variational schemes with high-order
spatial discretization for initial value gradient flows in generalized optimal transport
metric spaces. We first review some examples of gradient flows in generalized optimal
transport spaces from the Onsager principle. We then use a one-step time relaxation
optimization problem for time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto
schemes. Their minimizing systems satisfy implicit-in-time schemes for initial value
gradient flows with first-order time accuracy. We adopt the first-order optimization
scheme ALG2 (Augmented Lagrangian method) and high-order finite element methods in
spatial discretization to compute the one-step optimization problem. This allows us
to derive the implicit-in-time update of initial value gradient flows iteratively.
We remark that the iteration in ALG2 has a simple-to-implement point-wise update based
on optimal transport and Onsager's activation functions. The proposed method is unconditionally
stable for convex cases. Numerical examples are presented to demonstrate the effectiveness
of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher--Kolmogorov-Petrovskii-Piskunov
equation, and two and four species reversible reaction-diffusion systems. This is
a joint work with Stanley Osher from UCLA and Wuchen Li from University South Carolina.

**When: **September 1^{st}, 2023 from 2:30pm-3:30pm

**Where:** Leconte 440

**Speaker:** Tianyi Lin (Massachusetts Institute of Technology)

**Host:** Wuchen Li (Joint RTG Seminar)

**Abstract: **Reliable and multi-agent machine learning has seen tremendous achievements in recent
years; yet, the translation from minimization models to min-max optimization models
and/or variational inequality models --- two of the basic formulations for reliable
and multi-agent machine learning --- is not straightforward. In fact, finding an optimal
solution of either nonconvex-nonconcave min-max optimization models or nonmonotone
variational inequality models is computationally intractable in general. Fortunately,
there exist special structures in many application problems, allowing us to define
reasonable optimality criterion and develop simple and provably efficient algorithmic
schemes. In this talk, I will present the results on structure-driven algorithm design
in reliable and multi-agent machine learning. More specifically, I explain why the
nonconvex-concave min-max formulations make sense for reliable machine learning and
show how to analyze the simple and widely used two-timescale gradient descent ascent
by exploiting such special structure. I also show how a simple and intuitive adaptive
scheme leads to a class of optimal second-order variational inequality methods. Finally,
I discuss two future research directions for reliable and multi-agent machine learning
with potential for significant practical impacts: reliable multi-agent learning and
reliable topic modeling.

Previous seminar information can be found on Dr. Changhui Tan's website