Applied Mathematics

One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model. A new approach to data assimilation known as the Azouani-Olson-Titi algorithm (AOT) introduced a feedback control term to the 2D incompressible Navier-Stokes equations (NSE) in order to incorporate sparse measurements. The solution to the AOT algorithm applied to the 2D NSE was proven to converge exponentially to the true solution of the 2D NSE with respect to the given initial data. In this talk, we present our tests on the robustness, improvements, and implementation of the AOT algorithm, as well as generate new ideas based off of these investigations. First, we discuss the application of the AOT algorithm to the 2D NSE with an incorrect parameter and prove it still converges to the correct solution up to an error determined by the error in the parameters. This led to the development of a simple parameter recovery algorithm, whose convergence we recently proved in the setting of the Lorenz equations. The implementation of this algorithm led us to provide rigorous proofs that solutions to the corresponding sensitivity equations are in fact the Fréchet derivative of the solutions to the original equations. Next, we present a proof of the convergence of a nonlinear version of the AOT algorithm in the setting of the 2D NSE, where for a portion of time the convergence rate is proven to be double exponential. Finally, we implement the AOT algorithm in the large scale Model for Prediction Across Scales - Ocean model, a real-world climate model, and investigate the effectiveness of the AOT algorithm in recovering subgrid scale properties.

Speaker Biography

Elizabeth Carlson, is a homeschooler turned math PhD! She grew up in Helena, MT, USA, where she also graduated from Carroll College with a Bachelor's in mathematics and minor in physics. She became interested in fluid dynamics as an undergraduate, and followed this interest through her graduate work at the University of Nebraska - Lincoln in Lincoln, NE, USA, where she just earned my PhD in May 2021. Her research focus is in fluid dynamics, focusing on the well-posedness of systems of partial differential equations and numerical computations and analysis in fluid dynamics. In her free time, she enjoys hiking, playing piano, reading, and martial arts.

Read more about Elizabeth Carlson on our PIMS Medium blog here.

The development of quantum electrodynamics is one of the major achievements of theoretical physics and mathematics of the 20th century, called the "Jewel of physics" by Richard Feynman. This talk is not about that. Instead, I explain two of its basic ingredients - Feynman diagrams, and Spinor bundles - and then describe how these can be adapted to "electron-like" strings. This will lead us naturally to the Spinor bundle on loop space, which I will describe in some detail. An element of loop space, i.e. a smooth function from the circle into some fixed manifold, is supposed to represent a string at a fixed moment in time. I will then explain the notion of a fusion product (on this bundle), and argue that this is a manifestation of the principle of locality, which is ubiquitous in physics. If time permits, I will discuss some ongoing work, in collaboration with Matthias Ludewig, Darvin Mertsch, and Konrad Waldorf, where we describe how this fusive spinor bundle on loop space fits beautifully in the higher categorical framework of 2-vector bundles.

I will present nonlinear dynamics for distributed decision-making that derive from principles of symmetry and bifurcation. Inspired by studies of animal groups, including house-hunting honeybees and schooling fish, the nonlinear dynamics describe a group of interacting agents that can manage flexibility as well as stability in response to a changing environment.

Biography:

Naomi Ehrich Leonard is Edwin S. Wilsey Professor of Mechanical and Aerospace Engineering and associated faculty in Applied and Computational Mathematics at Princeton University. She is a MacArthur Fellow, and Fellow of the American Academy of Arts and Sciences, SIAM, IEEE, IFAC, and ASME. She received her BSE in Mechanical Engineering from Princeton University and her PhD in Electrical Engineering from the University of Maryland. Her research is in control and dynamics with application to multi-agent systems, mobile sensor networks, collective animal behavior, and human decision dynamics.

Moderated Questions

1. What was the first job you had after graduation and how did you get it?
2. What do you like most/least about your current work?
3. If you could go back in time and change one thing about your career choices what would you do?
4. What advice do you have for the students in the audience looking for their first job?

Speaker Bios

Bernard Chan is currently a data scientist at BuildDirect.com (BD), an e-commerce platform in flooring, tiles and other home improvement products. At BD, Bernard is part of the analytics team and he specializes in logistics related data problems such as freight rate and route planning. Prior to working at BD, Bernard was a applied mathematics researcher in dynamical systems and bifurcation theory.

Soyean Kim is a professional statistician (P.STAT) who is passionate about ethical use of data and algorithms to contribute to the betterment of society. She currently leads a team of data scientists at Technical Safety BC, a safety regulator in Canada. Her key contribution includes advancing ethics roadmap in predictive system and deployment of AI and machine learning to help safety inspection process. Her previous leadership roles include her tenure at PricewaterhouseCoopers and Fortis as a rate design manager. She is an advocate for “Data for Good” and a speaker on the topic of real world applications of AI. Her latest speaking engagement includes PAPIs in London, UK which is a series of international AI conferences, and BC Tech Summit in Vancouver.

Michael Reid received a Bachelor’s in Mathematics from UMBC before starting work as junior web developer for a US federal government consulting agency. After moving to Vancouver, he’s worked in software engineering at companies ranging from small consulting firms to Amazon Web Services. He recently co-founded Nautilus Technologies, a machine learning and data privacy startup in Vancouver.

Parin Shah is a Data Scientist at KPMG focused on solving machine learning and data engineering problems in the space of mining, gaming, insurance and social media. Previously, he spent 2.5 years helping develop the digital analytics practice for an e-commerce firm, Natural Wellbeing, where he worked on setting up data infrastructure, building consumer analytics models and website experimentation. Parin was a fellow at a UBC machine learning workshop and has an undergraduate degree from the University of British Columbia (UBC) with a coursework concentrated in economics with statistics and computer science electives.

Dr. Aanchan Mohan is a machine learning scientist and software engineer at Synaptitude Brain Health. He is currently working on software and machine learning methods to encourage circadian regulation with the goal of improving an individual’s brain health. His current research interests include problems in natural language processing. Dr. Mohan has worked on Bayesian and deep learning methods applied to time series signals across multiple domains. He holds a PhD from McGill University where he focused on transfer learning and parameter sharing in acoustic models for speech recognition. He supervises students and actively publishes in the area of speech processing. He is a named co-inventor on two issued patents in the area of speech processing, and one filed patent in the area of wearable devices. He is a co-organizer of the AI in Production, and Natural Language Processing meetups in Vancouver.

There have been several recent outbreaks of cholera (for example, in Haiti and Yemen), which is a bacterial disease caused by the bacterium Vibrio cholerae. It can be transmitted to humans directly by person-to-person contact or indirectly via contaminated water. Random mixing cholera models from the literature are first formulated and briefly analyzed. Heterogeneities in person-to-person contact are introduced, by means of a multigroup model, and then by means of a contact network model. Utilizing an interplay of analysis and linear algebra, various control strategies for cholera are suggested by these models.

Pauline van den Driessche is a Professor Emeritus in the Department of Mathematics and Statistics at the University of Victoria. Her research focuses on aspects of stability in biological models and matrix analysis. Current research projects include disease transmission models that are appropriate for influenza, cholera and Zika. Most models include control strategies (e.g., vaccination for influenza) and aim to address questions relevant for public health. Sign pattern matrices occur in these models, and the possible inertias of such patterns is a current interest.