Fall 2023

Date Speaker Topic
M Sep 11 Organizational Meeting  
M Sep 18 Anca Andrei (Tufts) Nonlinear Methods for Shape Optimization Problems in Liquid Crystal Tactoids



Abstract: An emerging theme across many science and engineering domains is modeling materials that can change shape. With Morpho, domain scientists gain a powerful new simulation tool to tackle larger and more complex shape evolution problems than presently possible. One example involves modeling the evolution of nematic liquid crystals with free boundaries, known as nematic tactoids, that are in contact with an isotropic fluid. In this talk, we discuss results from applying a class of classical nonlinear numerical methods to this model and compare them with previously used gradient-descent methods. Moreover, by wrapping the algorithms in a multilevel nested iteration approach, we see significant improvements in the efficiency of the simulations with a variety of initial guesses.

M Sep 25 Julie C Blackwood (Williams College) Transboundary management of ecological systems with applications to diseases


The development of public health policy is inextricably linked with governance structure. In our increasingly globalized world, human migration and infectious diseases often span multiple administrative jurisdictions that might have different systems of government and divergent management objectives. In this talk, I’ll introduce two examples in which spatial coordination may be critical for disease control. The first is in the management of rabies in vampire bats, where we demonstrate that spatial interactions likely play a key role in driving disease persistence. The second will describe a more general infectious disease in humans. We show that successful management may depend on both the actions of multiple managers and their desired objectives.

M Oct 2

Kevin O’Neill (Yale) Two Techniques Involving High-Dimensional Data



Abstract:  Data in high dimensional Euclidean space may often be described with fewer parameters than the ambient space, perhaps lying on or near a submanifold of lower intrinsic dimension. In part one of this talk, we will discuss a new method for estimating this intrinsic dimension from the data: a version of local PCA which is calibrated on quadratic embeddings which may better approximate a manifold at larger scales. The second part of the talk will focus on a technique of random embeddings to reduce ambient dimension. The topics of this talk are joint work with Anna Gilbert.

M Oct 9   Indigenous Peoples’ Day
M Oct 16   No seminar
M Oct 23 Sigal Gottlieb (UMass Dartmouth) New developments in strong stability preserving time-stepping



In this talk I discuss strong stability preserving (SSP) methods. After telling about the results of the last 30 years, I present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge–Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge–Kutta schemes where the time-step restriction is independent of the stiff term, as well as new SSP additive implicit Runge–Kutta schemes with downwinding that are unconditionally SSP. These methods depend on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation.

M Oct 30 Andrea Arnold (Worcester Polytechnic Institute) Bayesian Filtering Methods for Dynamic Parameter Estimation



Abstract: Estimating and quantifying uncertainty in unknown system parameters from partial, noisy system measurements remains a challenging inverse problem. In addition to constant parameters, a variety of systems stemming from real-world applications include unobservable parameters that change with time but have unknown evolution models. In this talk, we present several approaches using Bayesian filtering techniques to estimate time-varying parameters in deterministic dynamical systems governed by differential equations.

M Nov 6 Martin Molina-Fructuoso (Brandeis) Hamilton-Jacobi equations and data depth



Statistical depths extend the concepts of quantiles and medians to multidimensional settings. In the context of data science, one of the applications of the concept of statistical depth is to establish a ranking order within data clusters. The Tukey depth is one classical geometric construction of a highly robust statistical depth that has deep connections with convex geometry. However, calculating the Tukey depth for general measures is a computationally expensive problem, particularly for high dimensional problems.
In our recent work (a collaboration with Ryan Murray) we have shown a link between the Tukey depth of measures with a certain degree of regularity and a partial differential equation of the Hamilton-Jacobi type. In this talk I will detail this connection as well as a new computational approach to calculate the Tukey depth based on its geometrical characterizations. Additionally, I will describe a new concept of depth based on the eikonal equation, which can be seen as a local variant of the partial differential equation associated with the Tukey depth.
M Nov 13 Lidia Mrad (Mt Holyoke College) Order Reconstruction in Microfluidic Channels



Nematic liquid crystals are classified as partially-ordered materials due to the tendency of their molecules to align along preferred directions. This partial alignment, combined with their fluidic properties, render these materials useful in a range of applications, including their widespread use in optical displays. When confined to thin channels, the fluid flow extends the applications of nematics further to optofluidic devices and guided micro-cargo transport. We examine the intrinsic coupling between the fluid motion and the orientation of nematic molecules by considering steady unidirectional flows in thin channels. We work within a reduced Beris-Edwards framework which leads to a system of nonlinear and coupled differential equations. Our analytical and numerical results highlight the universality of order reconstruction solutions, where distinct sub-domains arise, separated by so-called domain walls.

M Nov 20   Thanksgiving week
M Nov 27 Xiaoqian Gong (Amherst College) On the Mathematical Properties of Some Multi-Scale Traffic Models
In this talk, we will present the mathematical properties of some microscopic, mesoscopic and macroscopic descriptions of traffic flow models. From the microscopic perspective, we will discuss the limitations and improvements of the Intelligent Driver Model (IDM), as well as the well-posedness of the “Bando-Follow-the-Leader” (Bando-FtL) Model and a time-delayed version of the Bando-FtL. As one of the applications of the microscopic car-following models, we will talk about optimal cruise control for traffic smoothing. From the mesoscopic perspective, we will derive rigorously the mean-field limit of a finite-dimensional hybrid system describing multi-lane and multi-class traffic flow in presence of human-driven and autonomous vehicles. From the mesoscopic perspective, we will briefly talk about the well-posedness of a non-local LWR model with memory. Numerical simulations and field experiment results will also be presented.
M Dec 4

Frederike Dumbgen (U of Toronto)


Toward plug-and-play global optimality for robotics



Recent years have seen promising developments in so-called certifiably optimal estimation, showing that many state estimation and planning problems in robotics, although highly non-convex, may be solved to global optimality. In particular, the quest for global optimality has led to 1) the development of solvers that are more likely to converge to the global optimum, 2) efficient certificates to verify if they did, and 3) a deeper understanding of the theory of when they do.

In this talk, I aim to introduce the audience to this exciting research field and to our recent efforts to make certifiably optimal solvers – for the field of state estimation in particular – not only faster, but also more accessible. Among those efforts, I will present efficient optimality certificates as a low-cost add-on to off-the-shelf local solvers, which apply to a variety of problems including range-only, stereo-camera and, more generally, matrix-weighted localization. Then, I will present our most recent solutions to automatically certify almost any state estimation problem, using a sampling-based method to automatically find a sufficient subset of so-called redundant constraints. I end with an outlook on promising research directions in state estimation and beyond.