Group Meeting

Synopsis:

Describing incompressible fluids necessitates incorporating pressure, however, it is introduced in the equations with a lower differential order compared to velocity. This discrepancy complicates numerical schemes, often requiring additional boundary conditions or different discretization approaches for pressure and velocity.

In this talk we introduce a new formulation for the periodic channel, presenting an unexpectedly simple yet effective solution to resolve the mismatch between pressure and velocity in incompressible fluids. By employing this seemingly unconventional approach, we craft a well-conditioned system that successfully eliminates the traditional mismatch problem. Notably, this method's versatility extends beyond incompressible fluids, applicable to various differential-algebraic equations, such as inelastic fibers and curl-free fluids

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Synopsis:

Charles will give an introduction of algorithmic differentiation, followed by a research talk from Ella.

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In this talk, I will give an overview of PNNL and working at the national laboratory. I will go over our strategic mission at a high level and focus on the work being done in AI/ML through the Mathematics in Artificial Reasoning in Sciences (MARS) laboratory initiative. I will showcase some of the research that my team has done under the initiative.

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New Courant faculty working on applied and computational mathematics will introduce themselves and their research to the group.

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Prof. Leif Ristroph will introduce his research group and current direction. After, there will be a series of short 5-10 minute demos of useful graphic design tools for figure-making, including a Tikz demo from Prof. Georg Stadler and an Adobe Illustrator demo from Mariya Savinov

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Praharsh will give a Julia tutorial with demos to demonstrate its use and benefits. Then, Ed and Shafer will give short 15-20 min talks introducing their research groups and current direction.

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This is a minifestival of short, 10-15 minute presentations by current 4th years working in applied and computational mathematics at Courant.

Mariya will discuss their work with Prof. Alex Mogilner on the modeling and simulation of various cell skeleton structures and their dynamics. They will outline two of their published works: the first on size-dependent periodic contractile dynamics of cell extracts in water droplets, and the second on how friction can robustly control in vitro actomyosin contraction. Finally, Mariya will introduce ongoing research in which they utilize the immersed boundary method to investigate the dynamics of stress fibers. 

Olivia will talk about her work with Leif Ristroph on the existence and stability of equilibria of free-falling thin flat plates. She will first prove the existence of a unique equilibrium gliding mode for any choice of physical properties of the plate, and then explore the stability of such equilibria using linear stability analysis. She will conclude by proposing a new flight mode in addition to those characterized in previous literature, and show some preliminary experimental demonstrations of its existence.

Ryan will show simulations of a balanced model of the ocean. At the scale of O(1-10 km), the ocean flow exhibits statistical asymmetries because of the abundance of fronts at that scale. The balanced model is a reduced-order model from the full fluids equation that can capture these statistical asymmetries and the growth of fronts.

Paul will discuss ongoing work with Mike O’Neil on numerical methods for elliptic PDEs with random boundary data. In the case where the boundary data is a Gaussian process, we will construct boundary integral equation representations of the covariance of the solution, and apply these methods to build statistical models of experimental data in acoustic and electromagnetic scattering problems.

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A subset of MSG faculty will each introduce their research group and current direction. These short talks are aimed to be broadly accessible to those interested in applied and computational mathematics, showcasing current research being done at Courant.

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Environmental variability greatly influences how the size and composition of a population evolve. In microbial communities, variations of the composition and size of the population, i.e. their eco-evolutionary dynamics, are key to understand the mechanisms of antimicrobial resistance, and may lead to population bottlenecks, where new colonies consisting of few individuals are prone to fluctuations. How the composition and size of these communities evolve is often interdependent, and demographic fluctuations are generally coupled to environmental variability, often resulting in feedback loops and cooperative behavior.

In this seminar, I will focus on a class of simple and insightful models of populations of fluctuating size whose growth is limited by a binary carrying capacity that endlessly switches between values corresponding to abundant and scarce resources. In these models, the population consists of two strains, one growing faster than the other, that compete under various scenarios. In the basic model, the competition is only for the same resources, and one classically expects that the faster strain always prevails (exclusion principle). Using analytical tools and computational means, I will show how the population size distribution and the strains fixation/extinction probability are dramatically influenced by the coupling of demographic fluctuations and environmental variability. In addition to the basic model with random binary switching, the cases of periodic switching and different forms of environmental noise (of discrete and continuous range) will be mentioned. I will then discuss how these ideas and techniques are being generalized to study the eco-evolutionary dynamics of cooperative antimicrobial resistance, as well as the role of twofold environmental variability (toxin concentration and nutrients abundance) on the long-lived coexistence of the strains.

References:

Phys. Rev. Lett.119, 158301 (2017); J. Roy. Soc. Interface 15, 20180343 (2018); Phys. Rev. Lett. 25, 048105 (2020); J. Theor. Biol. 491, 110135 (2020); J. Roy. Soc. Interface  18, 20210613 (2021); Phys. Rev. Research 5, L022004 (2023); bioRxiv 2023.07.06.547929; arXiv:2307.06314