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1. Driving Markov Chains to Desired Equilibria via Linear Programming

   https://doi.org/10.1109/IEEECONF44664.2019.9048874  

   Bhat, Harish S. ; Huang, Li-Hsuan ; Rodriguez, Sebastian ( November 2019 , Proceedings of the 53rd Asilomar Conference on Signals, Systems, and Computers)

   In this paper, we develop methods to find the most sparse perturbation to a given Markov chain (either discrete- or continuous-time) such that the perturbed Markov chain achieves a desired equilibrium. 

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2. Numerical Solutions for Detecting Contingency in Modern Power Systems

   https://doi.org/10.1109/ICTC57116.2023.10154790  

   Ma, Xiaohang ; Qian, Hongjiang ; Wang, Le Yi ; Nazari, Masoud H. ; Yin, George ( May 2023 , Proc. 2023 4th IEEE Inform. Comm. Tech. Conference)

   This paper is devoted to the detection of contingencies in modern power systems. Because the systems we consider are under the framework of cyber-physical systems, it is necessary to take into consideration of the information processing aspect and communication networks. A consequence is that noise and random disturbances are unavoidable. The detection problem then becomes one known as quickest detection. In contrast to running the detection problem in a discretetime setting leading to a sequence of detection problems, this work focuses on the problem in a continuous-time setup. We treat stochastic differential equation models. One of the distinct features is that the systems are hybrid involving both continuous states and discrete events that coexist and interact. The discrete event process is modeled by a continuous-time Markov chain representing random environments that are not resented by a continuous sample path. The quickest detection then can be written as an optimal stopping problem. This paper is devoted to finding numerical solutions to the underlying problem. We use a Markov chain approximation method to construct the numerical algorithms. Numerical examples are used to demonstrate the performance. 

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3. 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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4. Efficient parameter generation for constrained models using MCMC

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   Kravtsova, Natalia ; Chamberlin, Helen_M ; Dawes, Adriana_T ( September 2023 , Scientific Reports)

   Mathematical models of complex systems rely on parameter values to produce a desired behavior. As mathematical and computational models increase in complexity, it becomes correspondingly difficult to find parameter values that satisfy system constraints. We propose a Markov Chain Monte Carlo (MCMC) approach for the problem of constrained model parameter generation by designing a Markov chain that efficiently explores a model’s parameter space. We demonstrate the use of our proposed methodology to analyze responses of a newly constructed bistability-constrained model of protein phosphorylation to perturbations in the underlying protein network. Our results suggest that parameter generation for constrained models using MCMC provides powerful tools for modeling-aided analysis of complex natural processes.

    

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5. Saddle point least squares discretization for convection-diffusion

   https://doi.org/10.1080/00036811.2023.2291511  

   Bacuta, Constantin ; Hayes, Daniel ; O'Grady, Tyler ( August 2024 , Applicable Analysis)

   We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models. 

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