Réunion d'hiver SMC 2024

Vancouver/Richmond, 29 novembre - 2 decembre 2024

Optimization, control, dynamics and stochastics: interplay and applications
Org: Eric Foxall (University of British Columbia), Jinniao Qiu (University of Calgary) et Zhongwei Shen (UA)
[PDF]

ERIC FOXALL, UBC Okanagan Optimal control of ribosome population for gene expression under periodic nutrient intake [PDF]: Ribosomes are molecular machines that build proteins out of available amino acid resources, and are largely made up of those resources. There is evidence that ribosomes are actively degraded when resources are scarce, and then reassembled once resources become more plentiful. In order to understand why, we formulate a model of protein production that allows for varying resource input and control over the ribosome population, and pose the following optimization problem: subject to periodically varying resource input, find the (time-dependent) rates of ribosome degradation and assembly that yield the highest, constant (with respect to time) rate of protein production. Using a quasi-static approximation that we justify analytically, we find that in optimal solutions, the ribosome population varies in response to the input, suggesting that the intense regulation observed in experiments occurs in order to maximize protein production. Joint work with Luca Ciandrini, Khanh Dao Duc and Clément Soubrier.
AMY HURFORD, Memorial University Optimal control strategies for community and traveler isolation under resource constraints [PDF]: Health authorities allocate limited resources to support the isolation of infected community members and travelers to reduce infectious disease spread. We consider an epidemic model and characterize the optimal controls. When resources are not limiting, if the maximum daily isolation rate is high, the optimal control corresponds to an elimination strategy, which results in a small outbreak of short duration. However, if the maximum daily isolation rate is low, the optimal control corresponds to a mitigation strategy, which results in a large outbreak of short duration. When resources are limiting, the optimal control is any strategy that uses all available resources, including circuit breaker strategies of this type, which results in a large outbreak of a duration ranging from short to long. We recommend implementing control measures at the start of an outbreak, as this action is always optimal, and is consistent with the precautionary principle, which recommends action even when important information, such whether resources will be limiting, is unknown. The elimination strategy results in substantially smaller outbreaks of short duration, and increasing the maximum daily isolation rate, or increasing the total resources available so as to achieve elimination, is likely optimal in some circumstances. Our modelling could be reformulated to consider multiple outbreaks over a fixed period of time, and would then serve as a suitable framework to further explore the conditions for when travel measures are an optimal control.
TYLER MEADOWS, Queen's University Optimizing biomass production in bioreactors [PDF]: A chemostat is a simple bioreactor used to study microorganisms under controlled conditions. Similar bioreactors are used to mass produce microorganisms and harvest important metabolites, such as biofuels and antibiotics. In this talk, we consider a control problem in the chemostat where the flow rate is used to maximize the amount of biomass harvested from the fermentation vessel.
NHU NGUYEN, University of Rhode Island Stochastic Approximation and Applications [PDF]: This work develops new results for stochastic approximation algorithms. The emphases are on treating algorithms and limits with discontinuities. The main ingredients include the use of differential inclusions, set-valued analysis, and non-smooth analysis, and stochastic differential inclusions. Under broad conditions, it is shown that a suitably scaled sequence of the iterates has a differential inclusion limit. In addition, it is shown for the first time that a centered and scaled sequence of the iterates converges weakly to a stochastic differential inclusion limit. The results are then used to treat several application examples including Markov decision process, Lasso algorithms, Pegasos algorithms, support vector machine classification, and learning. Some numerical demonstrations are also provided.
JINIAO QIU, University of Calgary A particle consensus approach to solving nonconvex-nonconcave min-max problems [PDF]: A zero-order optimization method is introduced for sequential min-max problems based interacting particles. The systems are coupled so that one population aims to solve the inner maximization problem, while the other aims to solve the outer minimization problem. The dynamics are characterized by a consensus-type interaction with additional stochasticity to promote exploration of the objective landscape. Without relying on convexity or concavity assumptions, theoretical convergence guarantees of the algorithm are established via a suitable mean-field approximation of the particle systems. Numerical experiments illustrate the validity of the proposed approach. In particular, the algorithm is able to identify a global min-max solution, in contrast to gradient-based methods, which typically converge to possibly suboptimal stationary points. This talk is based on joint work with Giacomo Borghi and Hui Huang.
POURIA RAMAZI, Brock University Towards Optimizing Vaccine Uptake Through Tailored Communication Strategies [PDF]: In this talk, I will explore how decision-making strategies influence vaccine uptake, focusing on two key groups: evidence-based decision-makers, who prioritize immediate personal benefits, and social-based decision-makers, who rely on the experiences and behaviors of others. The proportions of these two types within a population are critical in determining vaccine uptake, a well-established theoretical insight. I will demonstrate that these proportions are both theoretically identifiable and practically estimable. By presenting fitting results from jurisdictions across the USA and Canada, I will show that these proportions can vary significantly. These findings pave the way for developing tailored communication strategies to influence each group's decisions, ultimately optimizing public health efforts and enhancing vaccine promotion effectiveness.
SIDDHARTH SABHARWAL, Texas A&M University Population Size in Stochastic Ecological Dynamics [PDF]: We study how environmental stochasticity influences the long-term population size in certain one- and two-species models. The difficulty is that even when one can prove that there is coexistence, it is usually impossible to say anything about the invariant probability measure which describes the coexisting species. We are able to circumvent this problem for some important ecological models by noticing that the per-capita growth rates at stationarity are zero, something which can sometimes yield information about the invariant probability measure. For more complicated models we use a recent result by Cuello to explore how small noise influences the population size. We are able to showcase that sometimes environmental fluctuations lead to an increase in the population sizes, contrary to the Cushing-Henson conjecture. Further we look at the interaction of dispersal and environmental stochasticity in an $n$-patch model. We are able to prove persistence and extinction results even in the setting when the the dispersal rates are stochastic.
ZHONGWEI SHEN, University of Alberta WKB Approximation of Quasi-stationary Distributions with Applications [PDF]: Quasi-stationary distribution (QSD) is a powerful tool in characterizing the local dynamics of a dynamical system under noise perturbations. Its WKB approximation can be used to extract essential dissipative and conservative structures, thus aiding in gaining a clearer understanding of the local dynamics under noise perturbations. This talk is dedicated to discussing recent mathematical advancements surrounding the WKB approximation of QSDs and their applications to the potential-landscape and flux framework and Helmholtz decomposition.
XIONG WANG, Johns Hopkins University
YANG YANG, University of Calgary Infinite dimensional optimal control differential systems with randomness and path-dependence [PDF]: This talk is devoted to the stochastic optimal control problem of infinite-dimensional differential systems allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases studied by Bayraktar and Keller [J. Funct. Anal. 275 (2018), 2096--2161], the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.
KEXUE ZHANG, Queen's University Impulsive Synchronization of Complex Networks: an Event-Triggered Pinning Algorithm [PDF]: Complex networks (CNs) consist of an extensive collection of nodes, which are usually modelled by dynamical systems, and these nodes are connected according to specific topological structures. As a typical collective behavior, the synchronization of CNs has been investigated extensively due to its wide applications in various scientific fields ranging from biology and engineering to physics and sociology. As a particular type of feedback control, impulsive control uses impulses, which are state abrupt changes or jumps at a sequence of discrete times, to achieve network synchronization. The impulsive control paradigm has proven robust and efficient in network synchronization.
In this talk, we discuss the synchronization problem for a class of CNs with a pinning impulsive control approach. We propose a novel event-triggering algorithm to determine the impulse times and then introduce sufficient conditions on the network topology, impulsive control gains, and parameters in the event-triggering conditions to guarantee network synchronization. Next, we introduce an adaptive tuning method on the network coupling strength to allow arbitrary pinning schemes for the event-triggered impulsive controller. With the adaptive coupling strength, the synchronization of CNs can be realized via the proposed control method with an arbitrary selection of the pinning nodes.

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