Fall 2023
Schedule
Time | Location | Speaker | Title |
---|---|---|---|
Sep 22, 11am | Zoom | Paolo Piersanti | Grounded shallow ice sheets melting as an obstacle problem |
Sep, 29, 11am | GP346 | Yuanzhen Shao | On critical spaces of the Navier-Stokes equation over manifolds with boundary |
Oct 13, 11am | GP346 | Matthias Dogbatsey | Assessing the impact of intervention programs in gang dynamics: A mathematical modeling approach |
Oct 20, 11am | GP346 | Ryan Murray | Variational approaches to statistical learning |
Nov 10, 11am | GP346 | Chongze Hu | Accelerating phase-field predictions via recurrent neural networks for microstructural evolution |
Nov 17, 11am | Zoom | Ziyao Xu | A high-order well-balanced alternative finite difference WENO (A-WENO) method with the exact conservation property for systems of hyperbolic balance laws |
Dec 1, 11am | GP346 | Keisha Cook | Complexities of the cytoskeleton and applications of single particle tracking |
Abstracts
- Sep 22, 2023
Paolo Piersanti
Title: Grounded shallow ice sheets melting as an obstacle problem
Abstract: In this talk I will show you how to formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative. This renders the problem under consideration an obstacle problem. A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method. - Sep 29, 2023
Yuanzhen Shao
Title: On critical spaces of the Navier-Stokes equation over manifolds with boundary
Abstract: In this talk, we consider the motion of an incompressible viscous fluid on a compact Riemannian manifold (M,g) with boundary. The motion is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to the Navier boundary condition. We establish the existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. In case M is two-dimensional, solutions with divergence free initial condition in L2 exist globally and converge to an equilibrium. Moreover, we show that solutions that start close to a Killing vector field that satisfy the Navier boundary condition exist globally and converge exponentially fast to a (possibly different) Killing vector field. I will make the talk friendly to graduate students. - Oct 13, 2023
Matthias Dogbatsey
Title: Assessing the impact of intervention programs in gang dynamics: A mathematical modeling approach
Abstract: In this talk, we will explore the role of reformed gang members in the spread of gangs in the population. We develop a compartmental deterministic model of nonlinear ordinary differential equations to study gang dynamics. We will look at some basic properties of the model and then show that the model has a global asymptotically stable gang-free equilibrium whenever a certain criminological threshold, known as the effective reproduction number R0, is less than unity. We will conclude by looking at some bifurcation scenarios. - Oct 20, 2023
Ryan Murray
Title: Variational approaches to statistical learning
Abstract: Traditionally mathematicians have utilized variational methods and partial differential equations to broaden our understanding of problems from physics and engineering. However, recently there has been a growing need for a deeper understanding of statistical learning algorithms that come from data science and machine learning. It turns out that many of the algorithms used in those contexts actually have similar structure to the problems mathematicians studied in the 19th and 20th centuries in the context of physics and engineering. This talk will survey a number of recent advances where tools from the calculus of variations and partial differential equations have been used to provide a deeper understanding of statistical learning algorithms. This talk will not assume specialist knowledge of either statistics or PDE techniques, instead aiming to provide an introduction to the area. - Nov 10, 2023
Chongze Hu
Title: Accelerating phase-field predictions via recurrent neural networks for microstructural evolution
Abstract: Phase-field modeling is a highly powerful computational tool which can simulate the dynamics of microstructures and their physical properties for various engineering materials at the mesoscale. However, phase-field simulations are computationally expensive because these simulations need to solve a system of partial differential equations for various continuous variables that evolve both in space and time. In this talk, we will present an accelerated phase-field approach which uses a recurrent neural network (RNN) to learn the microstructural evolution in latent space. We will discuss a comprehensive analysis of the combination of different dimensionality-reduction methods and types of recurrent units in RNNs, focusing on a classical spinodal decomposition problem. Our comparison reveals that using an autocorrelation-based principal component analysis (PCA) method is the most efficient methods to reduce microstructural image in latent space. In addition, we find that the implementations of LSTM and GRU models provide comparable accuracy with respect to the high-fidelity phase-field predictions, but with a considerable computational speedup relative to the full simulation. This study not only enhances our understanding of the performance of dimensionality reduction on the microstructure evolution, but it also provides insights on strategies for accelerating phase-field modeling through machine learning techniques. - Nov 17, 2023
Ziyao Xu
Title: A high-order well-balanced alternative finite difference WENO (A-WENO) method with the exact conservation property for systems of hyperbolic balance laws
Abstract: In this work, we develop a high-order well-balanced alternative finite difference weighted essentially non-oscillatory (A-WENO) method with the exact conservation property and high efficiency for a class of hyperbolic balance laws whose steady states are characterized by constant equilibrium variables. In particular, the method preserves the non-hydrostatic equilibria of the shallow water equations with non- flat bottom topography and the Euler equations in gravitational fields. Our method comprises three essential gradients. First, we adopt the finite difference framework to discretize the equations, thus we approximate the value of source terms at grid points rather than their averages on cells. Then, we rewrite the source terms in flux-gradient forms at local reference equilibrium states and discretize them using the same approach as the true flux gradient to achieve the well-balanced property. Most importantly, the exact conservation property and high efficiency are achieved through the interpolation of equilibrium variables in the A-WENO framework, which is different from the more widely used finite difference framework based on the reconstruction of fluxes. Since the equilibrium variables are constants at equilibria in the equations we study, the WENO interpolation becomes trivial for the local reference equilibrium states in the flux-gradient formulation of source terms, which is the key to efficiency and preservation of conservation property. - Dec 1, 2023
Keisha Cook
Title: Complexities of the Cytoskeleton and Applications of Single Particle Tracking
Abstract: Biological systems are traditionally studied as isolated processes (e.g. regulatory pathways, motor protein dynamics, transport of organelles, etc.). Although more recent approaches have been developed to study whole cell dynamics, integrating knowledge across biological levels remains largely unexplored. In experimental processes, we assume that the state of the system is unknown until we sample it. Many scales are necessary to quantify the dynamics of different processes. These may include a magnitude of measurements, multiple detection intensities, or variation in the magnitude of observations. The interconnection between scales, where events happening at one scale are directly influencing events occurring at other scales, can be accomplished using mathematical tools for integration to connect and predict complex biological outcomes. In this work, we focus on building inference methods to study the complexity of the cytoskeleton from one scale to another. We rely on single particle tracking techniques based on stochastic models and explore long-term dynamics of the systems.