Mathematics

Solving dynamic user equilibrium by mean field routing game with explicit congestion dynamics

Speaker: 
Theophile Cabannes
Date: 
Wed, Oct 27, 2021
Location: 
Online
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

This work introduces a new N-player dynamic routing game that extend current Markovian traffic static assignment model.
It extends the N-player dynamic routing game to the corresponding mean field routing game, which models congestion in its dynamics.
Therefore, this new mean field routing game does not need to model congestion in the player cost function as done in the existing literature.
Both games are implemented in the open source library OpenSpiel.
The mean field game is used to solve the N-player dynamic game which leads to efficient computation of a approximate dynamic user equilibrium of the dynamic routing game.

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Networked Mean Field Games with Elements of Robustness and Learning

Speaker: 
Tamer Başar
Date: 
Wed, Oct 27, 2021
Location: 
Online
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

This talk will be a fairly high-level one, addressing various current issues in mean field games (MFGs), the underlying challenges primarily with regard to robustness, learning, and incentivization, and paths toward their resolution. Among these are: (i) use of multi-agent reinforcement learning for the computation of mean-field equilibrium (MFE) with state samples drawn from an unmixed Markov chain, and studying the performance of the associated (actor-critic) algorithms; (ii) adversarial MFGs on multi-graphs where agents interact with their neighbors, with such interactions propagating from neighborhoods to the entire network, and with an adversary counteracting the consensus formation process among the agents; and (iii) MFGs with a decision hierarchy, where the agent at the top of the hierarchy (leader) aims at designing incentive strategies (as in mechanism design) to induce a high population of agents at the lower level (followers) to act rationally toward a globally optimal solution in spite of their non-cooperative behavior. The talk will also identify several fruitful directions of research in this domain.

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2021 PIMS-UBC Math Job Forum for Postdoctoral Fellows & Graduate Students

Speaker: 
Moderator: Stephanie van Willigenburg
Date: 
Fri, Oct 22, 2021
Location: 
Online
Abstract: 

The PIMS-UBC Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in Mathematics and related areas with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion.

Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?

Panelists:

Dan Coombs, Head of Mathematics, UBC

Pamela Harris, Associate Professor, Williams College, and Faculty Fellow of the Office of Institutional Diversity, Equity and Inclusion

Eugene Li, Chair of Mathematics and Statistics, Langara

Luis Serrano, Quantum AI Research Scientist at Zapata Computing, and ex-Google, ex-Apple

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Local dynamics for large sparse networks of interacting diffusions

Speaker: 
Daniel Lacker
Date: 
Tue, Oct 26, 2021
Location: 
Online
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

This talk is an overview of a recent and ongoing line of work on large sparse networks of interacting diffusion processes. Each process is associated with a vertex in a graph and interacts only with its neighbors. When the graph is complete and the size grows to infinity, the system is well-approximated by its mean field limit, which describes the behavior of one typical process. For general graphs, however, the mean field approximation can fail, most dramatically when the graph is sparse. Nevertheless, if the underlying graph is locally tree-like (as is the case for many canonical sparse random graph models), we show that a single process and its nearest neighbors are characterized by an autonomous evolution which we call the "local dynamics." This can be viewed as a sparse counterpart of the usual McKean-Vlasov equation. The structure of the local dynamics depend heavily on the symmetries of the underlying graph and the conditional independence structure of the solution process. In the time-stationary case, the local dynamics take a particular tractable form. Based on joint works with Kavita Ramanan, Ruoyu Wu, and Jiacheng Zhang.

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Long time limits and concentration bounds for graphon mean field systems

Speaker: 
Ruoyu Wu
Date: 
Tue, Oct 26, 2021
Location: 
Online
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable convexity/dissipativity assumptions, we show the exponential ergodicity for both systems, establish a uniform-in-time law of large numbers for the empirical measure of particle states, and introduce the uniform-in-time Euler approximation. The precise rate of convergence of the Euler approximation is provided. We also provide uniform-in-time exponential concentration bounds for the rate of the LLN convergence under additional integrability conditions. Based on joint works with Erhan Bayraktar and Suman Chakraborty.

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Describing interacting particle systems via partial differential equations and graphons

Speaker: 
Fabio Coppini
Date: 
Tue, Oct 26, 2021
Location: 
Onlin
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

The study of large populations of interacting agents connected via a non-trivial network of connections represents a field of growing interest in Applied Mathematics. The relatively recent theory of graphons turns out to be well-adapted to model the emergence of complex networks and has been applied in several contexts by now, including mean-field systems. In this talk, we discuss how some of the key properties of graphon objects, e.g., exchangeability and labelling, are related to the study of interacting particle systems. Hopefully, this will shed some light on the behavior of systems described by partial differential equations and graphons, as the Graphon Mean-Field Game equations.

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Graphon mean field systems: large population and long time limits

Speaker: 
Erhan Bayraktar
Date: 
Tue, Oct 26, 2021
Location: 
Online
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. A law of large numbers result is established as the system size increases and the underlying graphons converge. Under suitable additional assumptions, we show the exponential ergodicity for the system, establish the uniform in time law of large numbers, and introduce the uniform in time Euler approximation. The precise rate of convergence of the Euler approximation is provided.

Based on joint works with Suman Chakraborty and Ruoyu Wu.

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Epidemic Model-Based Benchmark for Optimal Control on Networks

Speaker: 
Yaroslav V. Salii
Date: 
Tue, Oct 26, 2021
Location: 
Online
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

Network dynamical systems add an additional challenge of scale to optimal control schemes. There are many options of overcoming it, such as approximations and heuristics based on mean field games, neural networks, or reinforcement learning, or the actual structure of the networks, each with its own advantages and tradeoffs.

Metapopulation epidemic models, where each population is an entity on a map, such as a city or a district, are a convenient option for benchmarking varying optimal control schemes: these can be designed with varying number of nodes (dimension), have a natural per-node optimal control, e.g. the “lockdown level,” and a straightforward visualization option of choropleth maps.

In this talk, we will describe a procedure for generating plausible instances of such models with from 1 to circa 64,000 nodes based on publicly available census data for the contiguous U.S., each with the network of short-range travel (commute) and long-range travel (airplane), the latter derived from publicly available passenger flight statistics---along with a formal aggregation routine enabling a view of the same geography at different resolutions.

As a showcase, we designed a “baseline” optimal control scheme for three instances covering Oregon and Washington states: a 2-node instance on state level, a 75-node on county level, and a 2,072-node instance made of “atomic” population units, the census tracts, which are put through a metapopulation SIR model with per-node “lockdown level” optimal control on a 180-day time horizon, with the objective of minimizing the cumulative number of infections and the square of this lockdown control; the results are compared with the “no-lockdown” model.
The optimal control was derived through the Pontryagin Maximum Principle and numerically computed by the forward-backward sweep method, which converges within 5 seconds on the 2- and 75-node instances and within 40 seconds on the 2,072-node one.

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Controlling Human Microbiota

Speaker: 
Yang-Yu Liu
Date: 
Tue, Oct 26, 2021
Location: 
Online
Conference: 
Workshop on Mean Field Games on Networks
Abstract: 

We coexist with a vast number of microbes—our microbiota—that live in and on our bodies, and play an important role in human physiology and diseases. Many scientific advances have been made through the work of large-scale, consortium-driven metagenomic projects. Despite these advances, there are still many fundamental questions regarding the dynamics and control of microbiota to be addressed. Indeed, it is well established that human-associated microbes form a very complex and dynamic ecosystem, which can be altered by drastic diet change, medical interventions, and many other factors. The alterability of our microbiome offers opportunities for practical microbiome-based therapies, e.g., fecal microbiota transplantation and probiotic administration, to restore or maintain our healthy microbiota. Yet, the complex structure and dynamics of the underlying ecosystem render the quantitative study of microbiome-based therapies extremely difficult. In this talk, I will discuss our recent theoretical progress on controlling human microbiota from network science, dynamical systems, and control theory perspectives.

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Differential Equations and Algebraic Geometry - 1

Speaker: 
Hossein Movasati
Date: 
Wed, Oct 27, 2021
Location: 
PIMS, University of Alberta
Zoom
Conference: 
PIMS Network Courses
Differential Equations and Algebraic Geometry
Abstract: 

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

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