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.

Time: December 8, 2023 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Paula Vasquez

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]]> 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.

Time: November 17, 2023 2:30pm-3:30pm

Location: LeConte 440

Host: Changhui Tan

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]]> 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.

Time: October 27, 2023 2:30pm-3:30pm

Location: LeConte 440

Host: Changhui Tan

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]]>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.

Time: December 1, 2023 3:40pm-4:40pm

Location: LeConte 440

Host: Wuchen Li

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]]> 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.

Time: November 3, 2023 2:30pm-3:30pm

Location: LeConte 440

Host: Yi Sun

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]]> 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).

Time: October 6, 2023 3:40pm-4:40pm

Location: LeConte 440

Host: Wuchen Li

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]]> 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.

Time: September 29, 2023 3:40pm-4:40pm

Location: LeConte 440

Host: Wuchen Li

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]]>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 U. South Carolina.

Time: September 22, 2023 3:40pm-4:40pm

Location: LeConte 440

Host: Wuchen Li

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]]>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.

Time: September 1, 2023 2:30pm-3:30pm

Location: LeConte 440

Host: Wuchen Li

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]]>The Euler Alignment system is a hydrodynamic PDE version of the celebrated Cucker-Smale ODE's of collective behavior. Together with Changhui Tan, we developed a theory of weak solutions in 1D, which provide a uniquely determined way to evolve the dynamics after a blowup. Inspired by Brenier and Grenier's work on the pressureless Euler equations, we show that the dynamics of our system are captured by a nonlocal scalar balance law. We generate the unique entropy solution of a discretization of this balance law by introducing the "sticky particle Cucker-Smale" system to track the shock locations. Our approximation scheme for the density converges in the Wasserstein metric; it does so with a quantifiable rate as long as the initial velocity is at least Holder continuous. In this talk, we will discuss the limiting configurations, or "flocking states," that arise in this system, and how to predict them from the initial data.

Time: April 21, 2023 2:30pm-3:30pm

Location: LeConte 440

Host: Changhui Tan

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