Abstract-Model reduction methods from diverse fieldsincluding control, statistical mechanics and economics-aimed at systems that can be represented by Markov chains, are discussed in terms of their general properties and common features. These methods include decomposability, optimal prediction techniques, and Mori-Zwanzig representations. Our objective in this paper is to present a survey of and highlight connections between the approaches pursued in different fields, and demonstrate application of the methods on a set of wellknown examples. I. INTRODUCTIONThere has been a long and ubiquitous need for model reduction methods, with a history of research in fields as varied as economics [23], speech and signal processing [15], Internet analysis [24], and statistical mechanics [2], [3], to name a few. The common goal in all of this research is to find a simple mathematical model that adequately represents the behavior of a given complex system. Any particular reduction algorithm is then judged upon the level of complexity reduction achieved, how closely the reduced model captures the given system behavior, and the computational complexity of implementing the algorithm. In this paper we focus on model reduction from a controls and dynamical systems perspective, with our specific goal being the reduction of large scale Markov chain representations for complex systems. We begin with a brief overview of research in this area.One of the most standard approaches to model reduction of Markov chains has been to aggregate states into metastates based on the concept of completely decomposable and nearly completely decomposable systems, first introduced in [23]. In this setting, the aggregated states captured by each of the meta-states have dynamics which evolve along a similar short-run time scale, whereas the interactions between the meta-states evolve on a long-run time scale. This approach provides the basis for singular perturbation methods such as those proposed in [20] and discussed more recently in [27]. Similar state-aggregation approaches for Markov chains have been established which are directly related to the property of lumpability [13], [26], [21], where lumpability of a Markov chain refers to the partitioning of the states of the chain into aggregated sets which exhibit similar dynamics and observation statistics. An alternate approach for analyzing
Chaotic mixing strategies produce high mixing rates in microfluidic channels and other applications. In prior numerical and experimental work the variance of a tracer field in a chaotic mixer has been observed to decay rapidly after an initial slower transient. We relate this to the cutoff phenomenon observed in finite Markov chains and provide numerical evidence to suggest that chaotic mixing indeed exhibits cutoff. We provide results for a herringbone passive microfluidic mixer and the Standard Map. INTRODUCTIONThe question of how chaotic advection mixes a passive scalar function has attracted much research effort in recent years [1]. The main issues in this field are: how to measure the thoroughness of the mixing, how the mixing process changes qualitatively and quantitatively when the diffusion is close to zero, and how to enhance the overall mixing process by designing the map which produces chaotic advection. Unfortunately, we have only partial understanding for most of these topics. In spite of the fact that the detailed mechanism of mixing is unclear, nontrivial mixing processes have been observed in experiments [2] and can be simulated by large-scale computations [3].A widely observed phenomenon in the chaotic mixing process when small diffusion exists is the two or three-stage transition [4][5][6]. The map does not mix the scalar function with a constant rate in general. When the variance of the scalar function is measured during the mixing process, one can in general observe a relatively flat decay initially, followed by a super-exponential change, and then finally it tends to an exponential decay. We are interested in when these transitions happen, why they happen, and how to predict the slope of the exponential region. A good review and physical interpretation can be found in [7].
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A systematic study of photon and neutron radiation doses generated in high-intensity laser-solid interactions is underway at SLAC National Accelerator Laboratory. These laser-solid experiments are being performed using a 25 TW (up to 1 J in 40 fs) femtosecond pulsed Ti:sapphire laser at the Linac Coherent Light Source's (LCLS) Matter in Extreme Conditions (MEC) facility. Radiation measurements were performed with passive and active detectors deployed at various locations inside and outside the target chamber. Results from radiation dose measurements for laser-solid experiments at SLAC MEC in 2014 with peak intensity between 10 and 7.1 × 10 W cm are presented.
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