Reduction of Markov Chains using a Value-of-Information-Based Approach

In this paper, we propose an approach to obtain reduced-order models of Markov chains. Our approach is composed of two information-theoretic processes. The first is a means of comparing pairs of stationary chains on different state spaces, which is done via the negative Kullback-Leibler divergence defined on a model joint space. Model reduction is achieved by solving a value-of-information criterion with respect to this divergence. Optimizing the criterion leads to a probabilistic partitioning of the states in the high-order Markov chain. A single free parameter that emerges through the optimization process dictates both the partition uncertainty and the number of state groups. We provide a data-driven means of choosing the `optimal' value of this free parameter, which sidesteps needing to a priori know the number of state groups in an arbitrary chain.Comment: Submitted to Entrop

Approximations of countably-infinite linear programs over bounded measure spaces

We study a class of countably-infinite-dimensional linear programs (CILPs) whose feasible sets are bounded subsets of appropriately defined weighted spaces of measures. We show how to approximate the optimal value, optimal points, and minimal points of these CILPs by solving finite-dimensional linear programs. The errors of our approximations converge to zero as the size of the finite-dimensional program approaches that of the original problem and are easy to bound in practice. We discuss the use of our methods in the computation of the stationary distributions, occupation measures, and exit distributions of Markov~chains

A tiered modelling approach for condition-based maintenance of industrial assets with load sharing interaction and fault propagation

This paper considers the problem of condition-based maintenance optimization for complex industrial assets involving load sharing interaction and fault propagation among their components using a two-tiered approach. The upper layer represents the deterioration and failure at the asset level and uses a continuous-time Markov chain to calculate the optimal inspection interval and maintenance threshold. The model is mathematics tractable. We illustrate a numerical example with closed-form solutions for a simple case. The lower layer models the load sharing interaction and fault propagation, and in turn calculates the deterioration rates and failure rates to be used in the lower layer. We derive a partitioning rule to transform the lower layer model into the layer model’s format. Hence we are able to reduce the size of state space and circumvent the state space explosion problem.This is the author accepted manuscript. The final version can be found on the publisher's website at: http://imaman.oxfordjournals.org/content/early/2014/06/23/imaman.dpu013.abstract © The authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserve

Performance Analysis of Online Social Platforms

We introduce an original mathematical model to analyze the diffusion of posts within a generic online social platform. Each user of such a platform has his own Wall and Newsfeed, as well as his own self-posting and re-posting activity. As a main result, using our developed model, we derive in closed form the probabilities that posts originating from a given user are found on the Wall and Newsfeed of any other. These probabilities are the solution of a linear system of equations. Conditions of existence of the solution are provided, and two ways of solving the system are proposed, one using matrix inversion and another using fixed-point iteration. Comparisons with simulations show the accuracy of our model and its robustness with respect to the modeling assumptions. Hence, this article introduces a novel measure which allows to rank users by their influence on the social platform, by taking into account not only the social graph structure, but also the platform design, user activity (self- and re-posting), as well as competition among posts.Comment: Preliminary version of accepted paper at INFOCOM 2019 (Paris, France

TRIPLET POPULATION DYNAMICS AND EXCITED STATE RELAXATION IN CHALCOGENOPHENE POLYMERS

The conventional understanding of intersystem crossing in multichromophoric conju- gated polymers is usually depicted via a pure electronic model, neglecting contributions of vibrations or conformational order. Obtaining accurate structure-function correlations on spin-conversion processes involving photogenerated singlet excitons to triplet excitons and the excited state dynamics requires sensitivity to the subtle conformational ordering within conjugated polymers. This dissertation seeks to understand the kinetics of multi-exciton singlet-triplet interactions and the excited state relaxation of chalcogen containing (S, Se) conjugated polymers. Utilizing single molecule modulation spectroscopy allows determi- nation of triplet formation of individual conjugated polymer chains and aggregates. This technique resolves triplet-induced fluorescence quenching to ascertain the dynamics of the triplet population. In parallel, we have utilized the solutions to the probabilistic master equation describing the time-dependent kinetics of triplet formation. Finally, investigat- ing the excited state relaxation of strongly aggregating, non-emissive poly(3-decylthieneyl- enevinylene) (P3DTV)anditsheavyatomanalogpoly(3-decyl-seleneylenevinylene) (P3DSV) we demonstrate an alternative hypothesis for the observed ultrafast excited state dynamics