arXiv subjects

Muruhan Rathinam

Publications and source records attributed to Muruhan Rathinam.

12 recordsLinked to original sources

arXiv2026

Data Assimilation for Chemical Reaction Networks and Population Models via a Tunable Observer

We consider the problem of state reconstruction for a nonlinear dynamical system from observations of a linear function of the state. We present a design method for a tunable observer and provide a general theorem which under certain conditions guarantees exponential convergence of the observer regardless of initial error. Additional results are provided that apply this theorem to chemical reaction network models. Moreover, these results are illustrated via examples of mass action form of chemical reaction networks where a subset of the species concentrations are observed. Numerical results are provided to show the efficacy of our proposed observer. Numerical results are also shown for the case of noisy observations and our observer is compared favorably with the particle filter when the observation noise is small.

math.DS↗

arXiv2024

Stochastic Filtering of Reaction Networks Partially Observed in Time Snapshots

Stochastic reaction network models arise in intracellular chemical reactions, epidemiological models and other population process models, and are a class of continuous time Markov chains which have the nonnegative integer lattice as state space. We consider the problem of estimating the conditional probability distribution of a stochastic reaction network given exact partial state observations in time snapshots. We propose a particle filtering method called the targeting method. Our approach takes into account that the reaction counts in between two observation snapshots satisfy linear constraints and also uses inhomogeneous Poisson processes as proposals for the reaction counts to facilitate exact interpolation. We provide rigorous analysis as well as numerical examples to illustrate our method and compare it with other alternatives.

math.NA↗

arXiv2022

A Stochastic Binary Opinion Model: Opinion Dominance vs. Balance

We propose and study a stochastic binary opinion model where agents in a group are considered to hold an opinion of 0 or 1 at each moment. An agent in the group updates his/her opinion based on the group's opinion configuration and his/her \emph{personality}. Considering the number of agents with opinion 1 as a continuous time Markov process, we analyze the long-term probabilities for large population size in relation to the personalities of the group. In particular, we focus on the question of ``balance'' where both opinions are present in nearly equal numbers as opposed to ``dominance'' where one opinion is present in a greater number.

physics.soc-ph↗

arXiv2022

A generalized model of flocking with steering

We introduce and analyze a model for the dynamics of flocking and steering of a finite number of agents. In this model, each agent's acceleration consists of flocking and steering components. The flocking component is a generalization of many of the existing models and allows for the incorporation of many real world features such as acceleration bounds, partial masking effects and orientation bias. The steering component is also integral to capture real world phenomena. We provide rigorous sufficient conditions under which the agents flock and steer together. We also provide a formal singular perturbation study of the situation where flocking happens much faster than steering. We end our work by providing some numerical simulations to illustrate our theoretical results.

math.DS↗

arXiv2020

State and parameter estimation from exact partial state observation in stochastic reaction networks

We consider chemical reaction networks modeled by a discrete state and continuous in time Markov process for the vector copy number of the species and provide a novel particle filter method for state and parameter estimation based on exact observation of some of the species in continuous time. The conditional probability distribution of the unobserved states is shown to satisfy a system of differential equations with jumps. We provide a method of simulating a process that is a proxy for the vector copy number of the unobserved species along with a weight. The resulting weighted Monte Carlo simulation is then used to compute the conditional probability distribution of the unobserved species. We also show how our algorithm can be adapted for a Bayesian estimation of parameters and for the estimation of a past state value based on observations up to a future time.

q-bio.MN↗

arXiv2020

On the Validity of the Girsanov Transformation Method for Sensitivity Analysis of Stochastic Chemical Reaction Networks

We investigate the validity of the Girsanov Transformation (GT) method for parametric sensitivity analysis of stochastic models of chemical reaction networks. The validity depends on the likelihood ratio process being a martingale and the commutation of a certain derivative with expectation. We derive some exponential integrability conditions which imply both these requirements. We provide further conditions in terms of a reaction network that imply these exponential integrability conditions.

math.PR↗

arXiv2019

Homogenization of a Random Walk on a Graph in $\mathbb{R}^d$: An approach to predict macroscale diffusivity in media with finescale obstructions and interactions

We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in ${\mathbb R}^d$ and the scaling limit of the associated continuous-time random walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process $\varepsilon^2 Z(t/\varepsilon^2)$ converges to a linear drift $\bar{U}t$ and the case of interest to us is that of null drift $\bar{U}=0$. In this case, we show that $\varepsilon Z(t/\varepsilon^2)$ converges weakly to a Brownian motion. The diffusivity of the limiting Brownian motion can be computed by solving a set of linear algebra problems. As we allow for jump rates to be irreversible, our framework allows for the modeling of very general forms of interactions such as attraction, repulsion, and bonding. We provide some sufficient conditions for null drift that include certain symmetries of the graph. We also provide a formal asymptotic derivation of the effective diffusivity in analogy with homogenization theory for PDEs. For the case of reversible jump rates, we derive an equivalent variational formulation. This derivation involves developing notions of gradient for functions on the graph's nodes, divergence for ${\mathbb R}^d$-valued functions on the graph's edges, and a divergence theorem.

math.PR↗

arXiv2018

Estimation of parameter sensitivities for stochastic reaction networks using tau-leap simulations

We consider the important problem of estimating parameter sensitivities for stochastic models of reaction networks that describe the dynamics as a continuous-time Markov process over a discrete lattice. These sensitivity values are useful for understanding network properties, validating their design and identifying the pivotal model parameters. Many methods for sensitivity estimation have been developed, but their computational feasibility suffers from the critical bottleneck of requiring time-consuming Monte Carlo simulations of the exact reaction dynamics. To circumvent this problem one needs to devise methods that speed up the computations while suffering acceptable and quantifiable loss of accuracy. We develop such a method by first deriving a novel integral representation of parameter sensitivity and then demonstrating that this integral may be approximated by any convergent tau-leap method. Our method is easy to implement, works with any tau-leap simulation scheme and its accuracy is proved to be similar to that of the underlying tau-leap scheme. We demonstrate the efficiency of our methods through numerical examples. We also compare our method with the tau-leap versions of certain finite-difference schemes that are commonly used for sensitivity estimations.

math.PR↗

arXiv2017

Convergence, stability and robustness of multidimensional opinion dynamics in continuous time

We analyze a continuous time multidimensional opinion model where agents have heterogeneous but symmetric and compactly supported interaction functions. We consider Filippov solutions of the resulting dynamics and show strong Lyapunov stability of all equilibria in the relative interior of the set of equilibria. We investigate robustness of equilibria when a new agent with arbitrarily small weight is introduced to the system in equilibrium. Assuming the interaction functions to be indicators, we provide a necessary condition and a sufficient condition for robustness of the equilibria. Our necessary condition coincides with the necessary and sufficient condition obtained by Blondel et al. for one dimensional opinions.

math.DS↗

arXiv2016

Efficiency of the Girsanov transformation approach for parametric sensitivity analysis of stochastic chemical kinetics

Most common Monte Carlo methods for sensitivity analysis of stochastic reaction networks are the finite difference (FD), the Girsanov transformation (GT) and the regularized pathwise derivative (RPD) methods. It has been numerically observed in the literature, that the biased FD and RPD methods tend to have lower variance than the unbiased GT method and that centering the GT method (CGT) reduces its variance. We provide a theoretical justification for these observations in terms of system size asymptotic analysis under what is known as the classical scaling. Our analysis applies to GT, CGT and FD, and shows that the standard deviations of their estimators when normalized by the actual sensitivity, scale as $\mathcal{O}(N^{1/2}), \mathcal{O}(1)$ and $\mathcal{O}(N^{-1/2})$ respectively, as system size $N \to \infty$. In the case of the FD methods, the $N \to \infty$ asymptotics are obtained keeping the finite difference perturbation $h$ fixed. Our numerical examples verify that our order estimates are sharp and that the variance of the RPD method scales similarly to the FD methods. We combine our large $N$ asymptotics with previously known small $h$ asymptotics to obtain the best choice of $h$ in terms of $N$, and estimate the number $N_s$ of simulations required to achieve a prescribed relative $\mathcal{L}_2$ error $δ$. This shows that $N_s$ depends on $δ$ and $N$ as $δ^{-2 - \frac{γ_2}{γ_1}} N^{-1}, δ^{-2}$ and $N δ^{-2}$, for FD, CGT and GT respectively. Here $γ_1 >0, γ_2>0$ depend on the type of FD method used.

math.NA↗

arXiv2015

Convergence of moments of tau leaping schemes for unbounded Markov processes on integer lattices

Tau leap schemes were originally designed for the efficient time stepping of discrete state and continuous in time Markov processes arising in stochastic chemical kinetics. Previous convergence results on tau leaping schemes have been restricted to systems that remain in a bounded subdomain (which may depend on the initial condition) or satisfy global Lipschitz conditions on propensities. This paper extends the convergence results to fairly general tau leap schemes applied to unbounded systems that possess certain moment growth bounds. Specifically, we prove a weak convergence result, which shows order $q$ convergence of all moments under certain form of moment growth bound assumptions on the stochastic chemical system and the tau leap method, as well as polynomial bound assumption on the propensity functions. The results are stated for a general class of Markov processes with $\integ^N$ as their state space.

math.NA↗

arXiv2015

Moment growth bounds on continuous time Markov processes on non-negative integer lattices

We consider Markov processes in continuous time with state space $\posint^N$ and provide two sufficient conditions and one necessary condition for the existence of moments $E(\|X(t)\|^r)$ of all orders $r \in \nat$ for all $t \geq 0$. The sufficient conditions also guarantee an exponential in time growth bound for the moments. The class of processes studied have finitely many state independent jumpsize vectors $ν_1,\dots,ν_M$. This class of processes arise naturally in many applications such as stochastic models of chemical kinetics, population dynamics and queueing theory for example. We also provide a necessary and sufficient condition for stochiometric boundedness of species in terms of $ν_j$.

math.PR↗