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1. Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

   https://doi.org/10.1017/apr.2021.20  

   Budhiraja, Amarjit; Fraiman, Nicolas; Waterbury, Adam (March 2022, Advances in Applied Probability)

   Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD. 

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2. A Swing of Beauty: Pendulums, Fluids, Forces, and Computers

   https://doi.org/10.3390/fluids5020048  

   Mongelli, Michael; Battista, Nicholas A. (June 2020, Fluids)

   While pendulums have been around for millennia and have even managed to swing their way into undergraduate curricula, they still offer a breadth of complex dynamics to which some has still yet to have been untapped. To probe into the dynamics, we developed a computational fluid dynamics (CFD) model of a pendulum using the open-source fluid-structure interaction (FSI) software, IB2d. Beyond analyzing the angular displacements, speeds, and forces attained from the FSI model alone, we compared its dynamics to the canonical damped pendulum ordinary differential equation (ODE) model that is familiar to students. We only observed qualitative agreement after a few oscillation cycles, suggesting that there is enhanced fluid drag during our setup’s initial swing, not captured by the ODE’s linearly-proportional-velocity damping term, which arises from the Stokes Drag Law. Moreover, we were also able to investigate what otherwise could not have been explored using the ODE model, that is, the fluid’s response to a swinging pendulum—the system’s underlying fluid dynamics. 

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3. Exit Time Analysis for Approximations of Gradient Descent Trajectories Around Saddle Points

   https://doi.org/10.1093/imaiai/iaac025  

   Dixit, Rishabh; Gürbüzbalaban, Mert; Bajwa, Waheed U (February 2023, Information and Inference: A Journal of the IMA)

   This paper considers the problem of understanding the exit time for trajectories of gradient-related first-order methods from saddle neighborhoods under some initial boundary conditions. Given the ‘flat’ geometry around saddle points, first-order methods can struggle to escape these regions in a fast manner due to the small magnitudes of gradients encountered. In particular, while it is known that gradient-related first-order methods escape strict-saddle neighborhoods, existing analytic techniques do not explicitly leverage the local geometry around saddle points in order to control behavior of gradient trajectories. It is in this context that this paper puts forth a rigorous geometric analysis of the gradient-descent method around strict-saddle neighborhoods using matrix perturbation theory. In doing so, it provides a key result that can be used to generate an approximate gradient trajectory for any given initial conditions. In addition, the analysis leads to a linear exit-time solution for gradient-descent method under certain necessary initial conditions, which explicitly bring out the dependence on problem dimension, conditioning of the saddle neighborhood, and more, for a class of strict-saddle functions. 

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4. High-Resolution Modeling of the Fastest First-Order Optimization Method for Strongly Convex Functions

   https://doi.org/10.1109/CDC42340.2020.9304444  

   Sun, Boya; George, Jemin; Kia, Solmaz (December 2020, 2020 59th IEEE Conference on Decision and Control (CDC))

   null (Ed.)

   Motivated by the fact that the gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs), here we derive an ODE representation of the accelerated triple momentum (TM) algorithm. For unconstrained optimization problems with strongly convex cost, the TM algorithm has a proven faster convergence rate than the Nesterov's accelerated gradient (NAG) method but with the same computational complexity. We show that similar to the NAG method, in order to accurately capture the characteristics of the TM method, we need to use a high-resolution modeling to obtain the ODE representation of the TM algorithm. We propose a Lyapunov analysis to investigate the stability and convergence behavior of the proposed high-resolution ODE representation of the TM algorithm. We compare the rate of the ODE representation of the TM method with that of the NAG method to confirm its faster convergence. Our study also leads to a tighter bound on the worst rate of convergence for the ODE model of the NAG method. In this paper, we also discuss the use of the integral quadratic constraint (IQC) method to establish an estimate on the rate of convergence of the TM algorithm. A numerical example verifies our results. 

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5. ESTIMATION OF ARTERIAL PARAMETERS FROM NONINVASIVELY MEASURED ARTERIAL PULSE SIGNALS: VIBRATION-MODEL-BASED ANALYSIS

   Md Mahfuzur Rahman, Najmin Ara (November 2020, IMECE2020)

   null (Ed.)

   With the goal of achieving consistence in interpretation of an arterial pulse signal between its vibration model and its hemodynamic relations and improving its physiological implications in our previous study, this paper presents an improved vibration-model-based analysis for estimation of arterial parameters: elasticity (E), viscosity (), and radius (r0) at diastolic blood pressure (DBP) of the arterial wall, from a noninvasively measured arterial pulse signal. The arterial wall is modeled as a unit-mass vibration model, and its spring stiffness (K) and damping coefficient (D) are related to arterial parameters. Key features of a measured pulse signal and its first-order and second-order derivatives are utilized to estimate the values of K and D. These key features are then utilized in hemodynamic relations, where their interpretation is consistent with the vibration model, to estimate the value of r0 from K and D. Consequently, E, , and pulse wave velocity (PWV) are also estimated from K and D. The improved vibration-model-based analysis was conducted on pulse signals of a few healthy subjects measured under two conditions: at-rest and immediately post-exercise. With E, r0, and PWV at-rest as baseline, their changes immediately post-exercise were found to be consistent with the related findings in the literature. Thus, this improved vibration-model-based analysis is validated and contributes to estimation of arterial parameters with better physiological implications, as compared with its previous counterpart. 

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