Math Colloquium - Fall 2022

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September 9

Helen Wilson, University College London
Hosted by: Linda Cummings

*Please note, this seminar will be held virtually via WebEx only. Please contact math@njit.edu for access information*

Mathematical Modelling of Suspensions

Materials made from a mixture of liquid and solid are, instinctively, very obviously complex. From dilatancy (the reason wet sand becomes dry when you step on it) to extreme shear-thinning (quicksand) or shear-thickening (cornflour oobleck) there is a wide range of behaviours to explain and predict. I'll discuss the seemingly simple case of solid spheres suspended in a Newtonian fluid, which still has plenty of surprises up its sleeve.

September 16

Wooyoung Choi, NJIT
Hosted by: Anand Oza

New Directions and Challenges in Modeling Nonlinear Waves in Shallow Water

Nonlinear waves in shallow water have been often described by weakly nonlinear and weakly dispersive long wave models. Considering that the number of extreme events increases, strongly nonlinear long wave models have recently attracted much attention. In this talk, a high-order strongly nonlinear long wave model will be introduced and its well-posedness and solitary wave solution will be discussed. To solve numerically the high-order long wave model, it is found that finite difference schemes could be unstable due to discretization errors in approximating high-order derivatives and, therefore, a more accurate numerical scheme is required. Here a pseudo-spectral method is adopted and  some numerical solutions  of the high-order model will be presented. Additional challenges in practical applications of the high-order long wave model will be also discussed. 

September 23

Jean-Luc Thiffeault, University of Wisconsin Madison
Hosted by: Michael Booty

Shake Your Hips: An Active Particle with a Fluctuating Propulsion Force

The active Brownian particle (ABP) model describes a microswimmer, synthetic or living, whose direction of swimming is a Brownian motion. The swimming is due to a propulsion force, and the fluctuations are often assumed thermal in origin.  We present a 2D model where the fluctuations arise from nonthermal noise in a propelling force acting at a single point, such as that due to a flagellum.  We carefully take the overdamped limit and find several modifications to the traditional ABP model.  Since the fluctuating force causes a fluctuating torque, the diffusion tensor describing the process has a coupling between translational and rotational degrees of freedom.  An anisotropic particle also exhibits a noise-induced drift.  We show that these effects have measurable consequences for the long-time diffusivity of active particles, in particular adding a contribution that is independent of where the force acts.  This is joint work with Jiajia Guo.

September 30

Saleh Tanveer, Ohio State University
Hosted by: Wooyoung Choi and Mike Siegel

Study of a Reduced Model for Two Fluid Shear Flow

Mathematical reduction of complicated equations describing any phenomenon is common in Applied Mathematics. Instead of ad-hoc modeling, it is desirable to systematically reduce complicated models to simpler more tractable equations using asymptotic arguments, particularly when there are many parameters involved. Such reductions elucidate parameter dependence of phenomena in far greater detail than possible for the original model. 

In this vein, we consider a nonlocal thin film model Kaligirou et al (2016) model for two fluid shear flows in a channel. Shear flows involving two fluids are quite common in nature, whether they occur in the context of flow in a coated pipe, wind-water interaction or other geo or astrophysical context.  We use tools from analysis, numerical calculations and asymptotics in this study. We present results on whether or not the 2+1 model equations allow smooth solutions for all time, or if there is singularity in finite time. Furthermore, we discuss existence of different branches of steady traveling wave solutions in one dimension that initially bifurcate from a flat state and present stability and bifurcation properties, including Hopf-bifurcation to time periodic states. We also highlight surprising singular effects of small slip on an otherwise stably stratified flow. 

*The work presented is in collaboration with D. Papageorgiou, Imperial.

October 7

Shuwen Lou, Loyola
Hosted by: Eliza Michalopoulou

A Model of Distorted Brownian Motion with Varying Dimension

We will begin with an introduction to standard Brownian motion and its applications in STEM fields since early 1900s. Then we will introduce what a "distorted Brownian motion" is, and what we mean by "varying dimension". The last 1/3 of the talk will be focused on our main results: heat kernel estimates of such a model. The term "heat kernel" comes from the link between Brownian motion and heat equations in PDE. No background in probability is required to understand the first half of the talk. 

October 14

Yue Yu, Lehigh
Hosted by: Roy Goodman

Learning Nonlocal Operators for Heterogeneous Material Modeling

Constitutive modeling based on the continuum mechanics theory has been a classical approach for modeling the mechanical responses of materials. However, when constitutive laws are unknown or when defects and/or high degrees of heterogeneity present, these classical models may become inaccurate. In this talk we propose to use data-driven modeling which directly utilizes high-fidelity simulation and experimental measurements on displacement fields, to predict a material's response. Two nonlocal operator regression approaches will be presented, to obtain homogenized and heterogeneous surrogates for material modeling, respectively.

In the first approach, we propose to consider nonlocal models as coarse-grained, homogenized models to accurately capture the system's global behavior, and learn optimal kernel functions directly from data. By combining machine learning identifiability theory, known physics, and nonlocal theory, this combination guarantees that the resulting model is mathematically well-posed, physically consistent, and converging as the data resolution increases. As an application, we employ this ``linear operator regression’’ technique to model wave propagation in metamaterials.

In the second approach, we further propose to leverage the linear operator regression to nonlinear models by combining it with neural networks, and model the heterogeneous material response as mapping between loading conditions and its resultant displacement fields. To this end, we develop deep integral neural operator architectures, which is resolution-independent and naturally embeds the material micromechanical properties and defects in the integrand. As an application, we learn soft tissue models directly from digital image correlation (DIC) displacement tracking measurements, and show that the learnt model substantially outperforms conventional constitutive models.

October 21

Marcus Roper, UCLA
Hosted by: Anand Oza

Dead Spots and Short Circuits: Optimization in the Microcirculation

A decade-old revolution in imaging of microvessels has opened new windows on the exquisite hydraulics of the microvessels, little bigger than red blood cells themselves, that perfuse all tissues with blood. The availability of these new data allow us to develop new hypotheses about the organizing principles for the flows and the networks that sculpt them, but also require new mathematical tools to assimilate large and noisy data sets. In my talk, I will tell two stories of finding optimality principles within the microcirculation: First we look at the mammalian sensory cortex, which must be kept continuously nourished by a dense web of capillaries. I will discuss how a clear picture of the network’s constraints and capabilities comes from upscaled models in which the capillary web is replaced by a porous medium coupled to a network model of features above the finest and noisiest length scales. Our model reveals how the topology of dead spots sets the optimal ratio of arterioles and venules seen across species. Second, we turn to the developing zebrafish microvasculatre in which are previous work has revealed that vessels must have precisely tuned radii to prevent short circuiting of blood flows. But how do vessels find their right radii? We show that shear stresses alone, which have long been dismissed as carrying insufficient information to robustly guide networks, can direct vessels to avoid short circuits.

October 28

Gerard Ben Arous, Courant Institute
Hosted by: James Maclaurin

High-Dimensional Limit Theorems for Stochastic Gradient Descent: Effective Dynamics and Critical Scaling

This is a joint work with Reza Gheissari (Northwestern) and Aukosh Jagannath (Waterloo), to appear in NeurIPS 2022 (Arxiv2206.04030)

We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. Interestingly, we find a critical scaling regime for the step-size below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub-optimal solutions with probability bounded away from zero from random (e.g.,Gaussian) initializations. 

November 4

Amin Doostmohammadi, Niels Bohr Institute
Hosted by: Yuan-Nan Young

*Please note, this seminar will be held virtually via WebEx only. Please contact math@njit.edu for access information*

Active Matter: Flow, Topology, and Control

The spontaneous emergence of collective flows is a generic property of active fluids and often leads to chaotic flow patterns characterized by swirls, jets, and topological disclinations in their orientation field [1]. I will first discuss two examples of these collective features helping us understand biological processes:

(i) to explain the tortoise & hare story in bacterial competition: how motility of Pseudomonas aeruginosa bacteria leads to a slower invasion of bacteria colonies, which are individually faster [2], and

(ii) how self-propelled defects lead to finding an unanticipated mechanism for cell death [3,4].

I will then discuss various strategies to tame, otherwise chaotic, active flows, showing how hydrodynamic screening of active flows can act as a robust way of controlling and guiding active particles into dynamically ordered coherent structures [5]. I will also explain how combining hydrodynamics with topological constraints can lead to further control of exotic morphologies of active shells [6].

[1] A. Amiri, R. Mueller, and A. Doostmohammadi, J. Phys. A. (2021). [2] O. J. Meacock et al., Nat. Phys. (2021). [3] T. N. Saw et al., Nature. (2017). [4] R. Mueller, J. M. Yeomans, and A. Doostmohammadi, Phys. Rev. Lett. (2019). [5] A. Doostmohammadi et al., Nat. Comm. (2018). [6] L. Metselaar, J. M. Yeomans, and A. Doostmohammadi, Phys. Rev. Lett. (2019).

November 11

Javier Gomez Serrano, Brown University
Hosted by: Michael Siegel

Self-Similar Blow up Profiles for Fluids via Physics-Informed Neural Networks

In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.

November 18

James Kelly, U.S. Naval Research Lab
Hosted by: Simone Marras (Mechanical Engineering)

Fractional Partial Differential Equations: Boundary Conditions and Duality

Fractional partial differential equations (FPDEs) provide a powerful framework for modeling nonlocal behavior in many areas of physics and earth science. This talk gives a quick introduction to space- and time-fractional FPDEs and two recent contributions. First, we develop appropriate boundary conditions for the two-sided fractional diffusion equation, where the usual second derivative in space is replaced by a weighted average of positive and negative fractional derivatives. Mass preserving, reflecting boundary conditions for two-sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary. Stable, consistent explicit and implicit Euler methods are detailed, and steady state solutions are derived. Second, we develop a duality relationship between time-fractional PDEs and space- fractional PDEs. We show how a space-fractional model is mathematically equivalent to the corresponding time-fractional model, which has applications to hydrology and anomalous diffusion.

November 25

No Colloquium - Thanksgiving Break

December 2

Javier Diez, CIFICEN-CONICET-CICPBA, Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la Provincia de Buenos Aires
Hosted by: Lou Kondic

Dependence of Surface Tension and Hamaker Constant on Concentration

We study the instability of a thin film composed of two miscible liquids (binary fluid) on top of a solid planar surface including the fact that both the free surface and wetting energies are dependent on the mixture concentration. By assuming a linear relationship between these energies and both the bulk and surface concentrations, we analyze their effect on the eventual phase separation of the constituent fluids. The problem is treated within the gradient dynamics formulation applied to the thin film limit of the Cahn-Hilliard Navier-Stokes equations. The dependence of the free surface energy on concentration leads to a Marangoni type of effect, while the wetting energy resulting from van der Waals type of interaction between the film and the substrate is described by a concentrations dependant Hamaker constant. The linear stability analysis uncovers that both real and imaginary growth rates are possible, suggesting monotonic or oscillatory evolution of any small perturbation of either the film thickness or the concentration field. While our problem formulation applies to any binary mixture that can be consistently modeled via the presented approach, a particular interpretation of the results is provided for the case of liquid metal alloy films on nanoscale.

December 9

Stephanie Chaillat-Loseille, ENSTA Paris - UMA Laboratoire POems
Hosted by: Travis Askham

Fast Boundary Element Methods to Simulate Underwater Explosions and their Interactions with Submarines 

Assessing the impact of underwater explosions on submerged structures (submarines) is an important naval engineering problem. An underwater explosion mainly induces two distinct phenomena: a "shock wave" followed by an oscillating bubble of gas. Our goal is to create an efficient numerical method that accounts for the effects of both phenomena on submerged structures. Due to the unbounded nature of the ocean and the complex mechanical behavior of the submarine we want to take into account, it is natural to consider a Boundary Element Method/Finite Element Method (BEM/FEM) coupling. I will present how we can take advantage of fast BEMs in the frequency domain to model this time-domain problem, all the necessary improvements to be able to consider realistic configurations and the consequences on the convergence of the FEM/BEM coupling.

Updated: December 6, 2022