arXiv subjects

Carolyn L. Beck

Publications and source records attributed to Carolyn L. Beck.

At least 19 recordsLinked to original sources

arXiv2025

Online Model Order Reduction of Linear Systems via $(γ,δ)$-Similarity

Model order reduction aims to determine a low-order approximation of high-order models with least possible approximation errors. For application to physical systems, it is crucial that the reduced order model (ROM) is robust to any disturbance that acts on the full order model (FOM) -- in the sense that the output of the ROM remains a good approximation of that of the FOM, even in the presence of such disturbances. In this work, we present a framework for online model order reduction for a class of continuous-time linear systems that ensures this property for any $\mathcal{L}_2$ disturbance. Apart from robustness to disturbances in this sense, the proposed framework also displays other desirable properties for model order reduction: (1) a provable bound on the error defined as the $L_2$ norm of the difference between the output of the ROM and FOM, (2) preservation of stability, (3) compositionality properties and a provable error bound for arbitrary interconnected systems, (4) a provable bound on the output of the FOM when the controller designed for the ROM is used with the FOM, and finally, (5) compatibility with existing approaches such as balanced truncation and moment matching. Property (4) does not require computation of any gap metric and property (5) is beneficial as existing approaches can also be equipped with some of the preceding properties. The theoretical results are corroborated on numerical case studies, including on a building model.

eess.SY↗

arXiv2024

Modeling and Analysis of a Coupled SIS Bi-Virus Model

The paper deals with the setting where two viruses (say virus 1 and virus 2) coexist in a population, and they are not necessarily mutually exclusive, in the sense that infection due to one virus does not preclude the possibility of simultaneous infection due to the other. We develop a coupled bi-virus susceptible-infected-susceptible (SIS) model from a 4n-state Markov chain model, where n is the number of agents (i.e., individuals or subpopulation) in the population. We identify a sufficient condition for both viruses to eventually die out, and a sufficient condition for the existence, uniqueness and asymptotic stability of the endemic equilibrium of each virus. We establish a sufficient condition and multiple necessary conditions for local exponential convergence to the boundary equilibrium (i.e., one virus persists, the other one dies out) of each virus. Under mild assumptions on the healing rate, we show that there cannot exist a coexisting equilibrium where for each node there is a nonzero fraction infected only by virus 1; a nonzero fraction infected only by virus 2; but no fraction that is infected by both viruses 1 and 2. Likewise, assuming that healing rates are strictly positive, a coexisting equilibrium where for each node there is a nonzero fraction infected by both viruses 1 and 2, but no fraction is infected only by virus 1 (resp. virus 2) does not exist. Further, we provide a necessary condition for the existence of certain other kinds of coexisting equilibria. We show that, unlike the competitive bivirus model, the coupled bivirus model is not monotone. Finally, we illustrate our theoretical findings using an extensive set of in-depth simulations.

q-bio.PE↗

arXiv2023

Distributed Dual Subgradient Methods with Averaging and Applications to Grid Optimization

We study finite-time performance of a recently proposed distributed dual subgradient (DDSG) method for convex constrained multi-agent optimization problems. The algorithm enjoys performance guarantees on the last primal iterate, as opposed to those derived for ergodic means for vanilla DDSG algorithms. Our work improves the recently published convergence rate of $\Ocal(\log T/\sqrt{T})$ with decaying step-sizes to $\Ocal(1/\sqrt{T})$ with constant step-size on a metric that combines suboptimality and constraint violation. We then numerically evaluate the algorithm on three grid optimization problems. Namely, these are tie-line scheduling in multi-area power systems, coordination of distributed energy resources in radial distribution networks, and joint dispatch of transmission and distribution assets. The DDSG algorithm applies to each problem with various relaxations and linearizations of the power flow equations. The numerical experiments illustrate various properties of the DDSG algorithm--comparison with vanilla DDSG, impact of the number of agents, and why Nesterov-style acceleration fails in DDSG settings.

math.OC↗

arXiv2021

On the Role of Asymptomatic Carriers in Epidemic Spread Processes

We present an epidemiological compartment model, SAIR(S), that explicitly captures the dynamics of asymptomatic infected individuals in an epidemic spread process. We first present a group model and then discuss networked versions. We provide an investigation of equilibria and stability properties for these models, and present simulation results illustrating the effects of asymptomatic-infected individuals on the spread of the disease. We also discuss local isolation effects on the epidemic dynamics in terms of the networked models. Finally, we provide initial parameter estimation results based on simple least-squares approaches and local test-site data.\par Keywords: Epidemic dynamics, networks, data-informed modeling, stability analysis, parameter estimation

q-bio.PE↗

arXiv2021

On Concentration Inequalities for Vector-Valued Lipschitz Functions

We derive two upper bounds for the probability of deviation of a vector-valued Lipschitz function of a collection of random variables from its expected value. The resulting upper bounds can be tighter than bounds obtained by a direct application of a classical theorem due to Bobkov and Götze.

math.PR↗

arXiv2020

On the Consistency of Maximum Likelihood Estimators for Causal Network Identification

We consider the problem of identifying parameters of a particular class of Markov chains, called Bernoulli Autoregressive (BAR) processes. The structure of any BAR model is encoded by a directed graph. Incoming edges to a node in the graph indicate that the state of the node at a particular time instant is influenced by the states of the corresponding parental nodes in the previous time instant. The associated edge weights determine the corresponding level of influence from each parental node. In the simplest setup, the Bernoulli parameter of a particular node's state variable is a convex combination of the parental node states in the previous time instant and an additional Bernoulli noise random variable. This paper focuses on the problem of edge weight identification using Maximum Likelihood (ML) estimation and proves that the ML estimator is strongly consistent for two variants of the BAR model. We additionally derive closed-form estimators for the aforementioned two variants and prove their strong consistency.

math.ST↗

arXiv2020

Controlling a Networked SIS Model via a Single Input over Undirected Graphs

This paper formulates and studies the problem of controlling a networked SIS model using a single input in which the network structure is described by a connected undirected graph. A necessary and sufficient condition on the values of curing and infection rates for the healthy state to be exponentially stable is obtained via the analysis of signed Laplacians when the control input is the curing budget of a single agent. In the case when the healthy state is stabilizable, an explicit expression for the minimum curing budget is provided. The utility of the algorithm is demonstrated using a simulation over a network of cities in the northeastern United States.

math.OC↗

arXiv2020

Analysis, Identification, and Validation of Discrete-Time Epidemic Processes

Models of spread processes over non-trivial networks are commonly motivated by modeling and analysis of biological networks, computer networks, and human contact networks. However, identification of such models has not yet been explored in detail, and the models have not been validated by real data. In this paper, we present several different spread models from the literature and explore their relationships to each other; for one of these processes, we present a sufficient condition for asymptotic stability of the healthy equilibrium, show that the condition is necessary and sufficient for uniqueness of the healthy equilibrium, and present necessary and sufficient conditions for learning the spread parameters. Finally, we employ two real datasets, one from John Snow's seminal work on cholera epidemics in London in the 1850's and the other one from the United States Department of Agriculture, to validate an approximation of a well-studied network-dependent susceptible-infected-susceptible (SIS) model.

math.OC↗

arXiv2019

On the convergence rate of distributed gradient methods for finite-sum optimization under communication delays

Motivated by applications in machine learning and statistics, we study distributed optimization problems over a network of processors, where the goal is to optimize a global objective composed of a sum of local functions. In these problems, due to the large scale of the data sets, the data and computation must be distributed over processors resulting in the need for distributed algorithms. In this paper, we consider a popular distributed gradient-based consensus algorithm, which only requires local computation and communication. An important problem in this area is to analyze the convergence rate of such algorithms in the presence of communication delays that are inevitable in distributed systems. We prove the convergence of the gradient-based consensus algorithm in the presence of uniform, but possibly arbitrarily large, communication delays between the processors. Moreover, we obtain an upper bound on the rate of convergence of the algorithm as a function of the network size, topology, and the inter-processor communication delays.

math.OC↗

arXiv2019

Analysis and Control of a Continuous-Time Bi-Virus Model

This paper studies a distributed continuous-time bi-virus model in which two competing viruses spread over a network consisting of multiple groups of individuals. Limiting behaviors of the network are characterized by analyzing the equilibria of the system and their stability. Specifically, when the two viruses spread over possibly different directed infection graphs, the system may have (1) a unique equilibrium, the healthy state, which is globally stable, implying that both viruses will eventually be eradicated, (2) two equilibria including the healthy state and a dominant virus state, which is almost globally stable, implying that one virus will pervade the entire network causing a single-virus epidemic while the other virus will be eradicated, or (3) at least three equilibria including the healthy state and two dominant virus states, depending on certain conditions on the healing and infection rates. When the two viruses spread over the same directed infection graph, the system may have zero or infinitely many coexisting epidemic equilibria, which represents the pervasion of the two viruses. Sensitivity properties of some nontrivial equilibria are investigated in the context of a decentralized control technique, and an impossibility result is given for a certain type of distributed feedback controller.

math.OC↗

arXiv2018

Distributed Resource Allocation Over Dynamic Networks with Uncertainty

Motivated by broad applications in various fields of engineering, we study a network resource allocation problem where the goal is to optimally allocate a fixed quantity of resources over a network of nodes. We consider large scale networks with complex interconnection structures, thus any solution must be implemented in parallel and based only on local data resulting in a need for distributed algorithms. In this paper, we study a distributed Lagrangian method for such problems. By utilizing the so-called distributed subgradient methods to solve the dual problem, our approach eliminates the need for central coordination in updating the dual variables, which is often required in classic Lagrangian methods. Our focus is to understand the performance of this distributed algorithm when the number of resources is unknown and may be time-varying. In particular, we obtain an upper bound on the convergence rate of the algorithm to the optimal value, in expectation, as a function of the size and the topology of the underlying network. The effectiveness of the proposed method is demonstrated by its application to the economic dispatch problem in power systems, with simulations completed on the benchmark IEEE-14 and IEEE-118 bus test systems.

math.OC↗

arXiv2018

Convergence of the Iterates in Mirror Descent Methods

We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simulations comparing entropic mirror descent and standard subgradient methods for the robust regression problem.

math.OC↗

arXiv2017

Distributed Lagrangian Methods for Network Resource Allocation

Motivated by a variety of applications in control engineering and information sciences, we study network resource allocation problems where the goal is to optimally allocate a fixed amount of resource over a network of nodes. In these problems, due to the large scale of the network and complicated inter-connections between nodes, any solution must be implemented in parallel and based only on local data resulting in a need for distributed algorithms. In this paper, we propose a novel distributed Lagrangian method, which requires only local computation and communication. Our focus is to understand the performance of this algorithm on the underlying network topology. Specifically, we obtain an upper bound on the rate of convergence of the algorithm as a function of the size and the topology of the underlying network. The effectiveness and applicability of the proposed method is demonstrated by its use in solving the important economic dispatch problem in power systems, specifically on the benchmark IEEE-14 and IEEE-118 bus systems.

math.OC↗

arXiv2017

Multi-Competitive Viruses over Static and Time--Varying Networks

Epidemic processes are used commonly for modeling and analysis of biological networks, computer networks, and human contact networks. The idea of competing viruses has been explored recently, motivated by the spread of different ideas along different social networks. Previous studies of competitive viruses have focused only on two viruses and on static graph structures. In this paper, we consider multiple competing viruses over static and dynamic graph structures, and investigate the eradication and propagation of diseases in these systems. Stability analysis for the class of models we consider is performed and an antidote control technique is proposed.

math.OC↗

arXiv2016

Mixing Times and Structural Inference for Bernoulli Autoregressive Processes

We introduce a novel multivariate random process producing Bernoulli outputs per dimension, that can possibly formalize binary interactions in various graphical structures and can be used to model opinion dynamics, epidemics, financial and biological time series data, etc. We call this a Bernoulli Autoregressive Process (BAR). A BAR process models a discrete-time vector random sequence of $p$ scalar Bernoulli processes with autoregressive dynamics and corresponds to a particular Markov Chain. The benefit from the autoregressive dynamics is the description of a $2^p\times 2^p$ transition matrix by at most $pd$ effective parameters for some $d\ll p$ or by two sparse matrices of dimensions $p\times p^2$ and $p\times p$, respectively, parameterizing the transitions. Additionally, we show that the BAR process mixes rapidly, by proving that the mixing time is $O(\log p)$. The hidden constant in the previous mixing time bound depends explicitly on the values of the chain parameters and implicitly on the maximum allowed in-degree of a node in the corresponding graph. For a network with $p$ nodes, where each node has in-degree at most $d$ and corresponds to a scalar Bernoulli process generated by a BAR, we provide a greedy algorithm that can efficiently learn the structure of the underlying directed graph with a sample complexity proportional to the mixing time of the BAR process. The sample complexity of the proposed algorithm is nearly order-optimal as it is only a $\log p$ factor away from an information-theoretic lower bound. We present simulation results illustrating the performance of our algorithm in various setups, including a model for a biological signaling network.

stat.ML↗

arXiv2016

Epidemic Processes over Time-Varying Networks

The spread of viruses in biological networks, computer networks, and human contact networks can have devastating effects; developing and analyzing mathematical models of these systems can be insightful and lead to societal benefits. Prior research has focused mainly on network models with static graph structures, however the systems being modeled typically have dynamic graph structures. Therefore to better understand and analyze virus spread, further study is required. In this paper, we consider virus spread models over networks with dynamic graph structures, and investigate the behavior of diseases in these systems. A stability analysis of epidemic processes over time-varying networks is performed, examining conditions for the disease free equilibrium, in both the deterministic and stochastic cases. We present simulation results, propose a number of corollaries based on these simulations, and discuss quarantine control via simulation.

math.OC↗

arXiv2016

Vehicle Routing Problem with Time Windows: A Deterministic Annealing approach

The Vehicle Routing Problem with Time-Windows (VRPTW) is an important problem in allocating resources on networks in time and space. We present in this paper a Deterministic Annealing (DA)-based approach to solving the VRPTW with its aspects of routing and scheduling, as well as to model additional constraints of heterogeneous vehicles and shipments. This is the first time, to our knowledge, that a DA approach has been used for problems in the class of the VRPTW. We describe how the DA approach can be adapted to generate an effective heuristic approach to the VRPTW. Our DA approach is also designed to not get trapped in local minima, and demonstrates less sensitivity to initial solutions. The algorithm trades off routing and scheduling in an n-dimensional space using a tunable parameter that allows us to generate qualitatively good solutions. These solutions differ in the degree of intersection of the routes, making the case for transfer points where shipments can be exchanged. Simulation results on randomly generated instances show that the constraints are respected and demonstrate near optimal results (when verifiable) in terms of schedules and tour length of individual tours in each solution.

math.OC↗

arXiv2016

On the Analysis of a Continuous-Time Bi-Virus Model

Motivated by the spread of opinions on different social networks, we study a distributed continuous-time bi-virus model for a system of groups of individuals. An in-depth stability analysis is performed for more general models than have been previously considered, for the healthy and epidemic states. In addition, we investigate sensitivity properties of some nontrivial equilibria and obtain an impossibility result for distributed feedback control.

math.OC↗