Dynamic behavior of stochastic gene expression models in the presence of bursting

This paper considers the behavior of discrete and continuous mathematical models for gene expression in the presence of transcriptional/translational bursting. We treat this problem in generality with respect to the distribution of the burst size as well as the frequency of bursting, and our results are applicable to both inducible and repressible expression patterns in prokaryotes and eukaryotes. We have given numerous examples of the applicability of our results, especially in the experimentally observed situation that burst size is geometrically or exponentially distributed.Comment: 22 page

Modeling and Estimation of Thermal Flows Based on Transport and Balance Equations

Heat transfer in counterflow heat exchangers is modeled by using transport and balance equations with the temperatures of cold fluid, hot fluid, and metal pipe as state variables distributed along the entire pipe length. Using such models, boundary value problems can be solved to estimate the temperatures over all the length by means of measurements taken only at the boundaries. Conditions for the stability of the estimation error given by the difference between the temperatures and their estimates are established by using a Lyapunov approach. Toward this end, a method to construct nonlinear Lyapunov functionals is addressed by relying on a polynomial diagonal structure. This stability analysis is extended in case of the presence of bounded modeling uncertainty. The theoretical findings are illustrated with numerical results, which show the effectiveness of the proposed approach

On multi-dimensional hypocoercive BGK models

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the L1 distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.Comment: 55 pages, 2 figure

Efficient Sequential Monte-Carlo Samplers for Bayesian Inference

In many problems, complex non-Gaussian and/or nonlinear models are required to accurately describe a physical system of interest. In such cases, Monte Carlo algorithms are remarkably flexible and extremely powerful approaches to solve such inference problems. However, in the presence of a high-dimensional and/or multimodal posterior distribution, it is widely documented that standard Monte-Carlo techniques could lead to poor performance. In this paper, the study is focused on a Sequential Monte-Carlo (SMC) sampler framework, a more robust and efficient Monte Carlo algorithm. Although this approach presents many advantages over traditional Monte-Carlo methods, the potential of this emergent technique is however largely underexploited in signal processing. In this work, we aim at proposing some novel strategies that will improve the efficiency and facilitate practical implementation of the SMC sampler specifically for signal processing applications. Firstly, we propose an automatic and adaptive strategy that selects the sequence of distributions within the SMC sampler that minimizes the asymptotic variance of the estimator of the posterior normalization constant. This is critical for performing model selection in modelling applications in Bayesian signal processing. The second original contribution we present improves the global efficiency of the SMC sampler by introducing a novel correction mechanism that allows the use of the particles generated through all the iterations of the algorithm (instead of only particles from the last iteration). This is a significant contribution as it removes the need to discard a large portion of the samples obtained, as is standard in standard SMC methods. This will improve estimation performance in practical settings where computational budget is important to consider.Comment: arXiv admin note: text overlap with arXiv:1303.3123 by other author

Reaction Networks and Population Dynamics

Reaction systems and population dynamics constitute two highly developed areas of research that build on well-defined model classes, both in terms of dynamical systems and stochastic processes. Despite a significant core of common structures, the two fields have largely led separate lives. The workshop brought the communities together and emphasised concepts, methods and results that have, so far, appeared in one area but are potentially useful in the other as well

Formal synthesis of partially-observable cyber-physical systems

This dissertation is motivated by the challenges arising in the synthesis of controllers for partially-observable cyber-physical systems (PO-CPSs). In the past decade, CPSs have become ubiquitous and an integral part of our daily lives. Examples of such systems range from autonomous vehicles, drones, and aircraft to robots and advanced manufacturing. In many applications, these systems are expected to do complex logic tasks. Such tasks can usually be expressed using temporal logic formulae or as (in)finite strings over finite automata. In the past few years, abstraction-based techniques have been very promising for the formal synthesis of controllers. Since these techniques are based on the discretization of state and input sets, when dealing with large-scale systems, unfortunately, they suffer severely from the curse of dimensionality (i.e., the computational complexity grows exponentially with the dimension of the state set). In order to overcome the large computa- tional burden, a discretization-free approach based on control barrier functions has shown great potential to solve formal synthesis problems. In this thesis, we provide a systematic approach to synthesize a hybrid control policy for partially-observable (stochastic) control systems without discretizing the state sets. In many real-life applications, full-state information is not always available (due to the cost of sensing or the unavailability of the measurements). Therefore, in this thesis, we consider partially-observable (stochastic) control systems. Given proper state estimators, our goal is to utilize a notion of control barrier functions to synthesize control policies that provide (and potentially maximize) a lower bound on the probability that the trajectories of the partially-observable (stochastic) control system satisfy complex logic specifications such as safety and those that can be expressed as deterministic finite automata (DFA). Two main approaches are presented in this thesis to construct control barrier functions. In the first approach, no prior knowledge of estimation accuracy is needed. The second approach utilizes a (probability) bound on the estimation accuracy. Though the synthesis procedure for lower-dimensional systems is challenging itself, the task is much more computationally expensive (if not impossible) for large-scale interconnected systems. To overcome the challenges encountered with large-scale systems, we develop approaches to reduce the computational complexity. In particular, by considering a large-scale partially-observable control system as an interconnection of lower-dimensional subsystems, we compute so-called local control barrier functions for subsystems along with the corresponding local controllers. By assuming some small-gain type conditions, we then utilize local control barrier functions of subsystems to compositionally construct an overall control barrier function for the interconnected system. Finally, since closed-form mathematical models of many physical systems are either unavailable or too complicated to be of any use, we also extend our work to the synthesis of safety controllers for partially-observable systems with unknown dynamics. To tackle this problem, we utilize a data-driven approach and construct control barrier functions and their corresponding controllers via sets of data collected from the output trajectories of the systems and the trajectories of the estimators. To demonstrate the effectiveness of the proposed results in the thesis, we consider various case studies, such as a DC motor, an adaptive cruise control (ACC) system consisting of vehicles in a platoon, and a Moore-Greitzer jet engine model