Regularity of invariant densities for 1D systems with random switching

© 2015 IOP Publishing Ltd & London Mathematical Society.This is a detailed analysis of invariant measures for one-dimensional dynamical systems with random switching. In particular, we prove the smoothness of the invariant densities away from critical points and describe the asymptotics of the invariant densities at critical points.

Stochastic switching in infinite dimensions with applications to random parabolic PDE

© 2015 Society for Industrial and Applied Mathematics.We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.

Sensor Planning for Bayesian Nonparametric Target Modeling

Bayesian nonparametric models, such as the Gaussian process and the Dirichlet process, have been extensively applied for target kinematics modeling in various applications including environmental monitoring, traffic planning, endangered species tracking, dynamic scene analysis, autonomous robot navigation, and human motion modeling. As shown by these successful applications, Bayesian nonparametric models are able to adjust their complexities adaptively from data as necessary, and are resistant to overfitting or underfitting. However, most existing works assume that the sensor measurements used to learn the Bayesian nonparametric target kinematics models are obtained a priori or that the target kinematics can be measured by the sensor at any given time throughout the task. Little work has been done for controlling the sensor with bounded field of view to obtain measurements of mobile targets that are most informative for reducing the uncertainty of the Bayesian nonparametric models. To present the systematic sensor planning approach to leaning Bayesian nonparametric models, the Gaussian process target kinematics model is introduced at first, which is capable of describing time-invariant spatial phenomena, such as ocean currents, temperature distributions and wind velocity fields. The Dirichlet process-Gaussian process target kinematics model is subsequently discussed for modeling mixture of mobile targets, such as pedestrian motion patterns.

Novel information theoretic functions are developed for these introduced Bayesian nonparametric target kinematics models to represent the expected utility of measurements as a function of sensor control inputs and random environmental variables. A Gaussian process expected Kullback Leibler divergence is developed as the expectation of the KL divergence between the current (prior) and posterior Gaussian process target kinematics models with respect to the future measurements. Then, this approach is extended to develop a new information value function that can be used to estimate target kinematics described by a Dirichlet process-Gaussian process mixture model. A theorem is proposed that shows the novel information theoretic functions are bounded. Based on this theorem, efficient estimators of the new information theoretic functions are designed, which are proved to be unbiased with the variance of the resultant approximation error decreasing linearly as the number of samples increases. Computational complexities for optimizing the novel information theoretic functions under sensor dynamics constraints are studied, and are proved to be NP-hard. A cumulative lower bound is then proposed to reduce the computational complexity to polynomial time.

Three sensor planning algorithms are developed according to the assumptions on the target kinematics and the sensor dynamics. For problems where the control space of the sensor is discrete, a greedy algorithm is proposed. The efficiency of the greedy algorithm is demonstrated by a numerical experiment with data of ocean currents obtained by moored buoys. A sweep line algorithm is developed for applications where the sensor control space is continuous and unconstrained. Synthetic simulations as well as physical experiments with ground robots and a surveillance camera are conducted to evaluate the performance of the sweep line algorithm. Moreover, a lexicographic algorithm is designed based on the cumulative lower bound of the novel information theoretic functions, for the scenario where the sensor dynamics are constrained. Numerical experiments with real data collected from indoor pedestrians by a commercial pan-tilt camera are performed to examine the lexicographic algorithm. Results from both the numerical simulations and the physical experiments show that the three sensor planning algorithms proposed in this dissertation based on the novel information theoretic functions are superior at learning the target kinematics with

little or no prior knowledge

Resonance Energy Transfer-Based Molecular Switch Designed Using a Systematic Design Process Based on Monte Carlo Methods and Markov Chains

A RET network consists of a network of photo-active molecules called chromophores that can participate in inter-molecular energy transfer called resonance energy transfer (RET). RET networks are used in a variety of applications including cryptographic devices, storage systems, light harvesting complexes, biological sensors, and molecular rulers. In this dissertation, we focus on creating a RET device called closed-diffusive exciton valve (C-DEV) in which the input to output transfer function is controlled by an external energy source, similar to a semiconductor transistor like the MOSFET. Due to their biocompatibility, molecular devices like the C-DEVs can be used to introduce computing power in biological, organic, and aqueous environments such as living cells. Furthermore, the underlying physics in RET devices are stochastic in nature, making them suitable for stochastic computing in which true random distribution generation is critical.

In order to determine a valid configuration of chromophores for the C-DEV, we developed a systematic process based on user-guided design space pruning techniques and built-in simulation tools. We show that our C-DEV is 15x better than C-DEVs designed using ad hoc methods that rely on limited data from prior experiments. We also show ways in which the C-DEV can be improved further and how different varieties of C-DEVs can be combined to form more complex logic circuits. Moreover, the systematic design process can be used to search for valid chromophore network configurations for a variety of RET applications.

We also describe a feasibility study for a technique used to control the orientation of chromophores attached to DNA. Being able to control the orientation can expand the design space for RET networks because it provides another parameter to tune their collective behavior. While results showed limited control over orientation, the analysis required the development of a mathematical model that can be used to determine the distribution of dipoles in a given sample of chromophore constructs. The model can be used to evaluate the feasibility of other potential orientation control techniques.

A Molecular-scale Programmable Stochastic Process Based On Resonance Energy Transfer Networks: Modeling And Applications

While molecular and cellular processes are often modeled as stochastic processes, such as Brownian motion, chemical reaction networks and gene regulatory networks, there are few attempts to program a molecular-scale process to physically implement stochastic processes. DNA has been used as a substrate for programming molecular interactions, but its applications are restricted to deterministic functions and unfavorable properties such as slow processing, thermal annealing, aqueous solvents and difficult readout limit them to proof-of-concept purposes. To date, whether there exists a molecular process that can be programmed to implement stochastic processes for practical applications remains unknown.

In this dissertation, a fully specified Resonance Energy Transfer (RET) network between chromophores is accurately fabricated via DNA self-assembly, and the exciton dynamics in the RET network physically implement a stochastic process, specifically a continuous-time Markov chain (CTMC), which has a direct mapping to the physical geometry of the chromophore network. Excited by a light source, a RET network generates random samples in the temporal domain in the form of fluorescence photons which can be detected by a photon detector. The intrinsic sampling distribution of a RET network is derived as a phase-type distribution configured by its CTMC model. The conclusion is that the exciton dynamics in a RET network implement a general and important class of stochastic processes that can be directly and accurately programmed and used for practical applications of photonics and optoelectronics. Different approaches to using RET networks exist with vast potential applications. As an entropy source that can directly generate samples from virtually arbitrary distributions, RET networks can benefit applications that rely on generating random samples such as 1) fluorescent taggants and 2) stochastic computing.

By using RET networks between chromophores to implement fluorescent taggants with temporally coded signatures, the taggant design is not constrained by resolvable dyes and has a significantly larger coding capacity than spectrally or lifetime coded fluorescent taggants. Meanwhile, the taggant detection process becomes highly efficient, and the Maximum Likelihood Estimation (MLE) based taggant identification guarantees high accuracy even with only a few hundred detected photons.

Meanwhile, RET-based sampling units (RSU) can be constructed to accelerate probabilistic algorithms for wide applications in machine learning and data analytics. Because probabilistic algorithms often rely on iteratively sampling from parameterized distributions, they can be inefficient in practice on the deterministic hardware traditional computers use, especially for high-dimensional and complex problems. As an efficient universal sampling unit, the proposed RSU can be integrated into a processor / GPU as specialized functional units or organized as a discrete accelerator to bring substantial speedups and power savings.