Markov Approximation Research Articles

Calculation of the Photorecombination Spectrum when Atoms Irradiated by a Strong Laser Field Based on the Markov Approximation

Calculation of the Photorecombination Spectrum when Atoms Irradiated by a Strong Laser Field Based on the Markov Approximation

Adaptive Configuration Selection and Bandwidth Allocation for Edge-Based Video Analytics

Major cities worldwide have millions of cameras deployed for surveillance, business intelligence, traffic control, crime prevention, etc. Real-time analytics on video data demands intensive computation resources and high energy consumption. Traditional cloud-based video analytics relies on large centralized clusters to ingest video streams. With edge computing, we can offload compute-intensive analysis tasks to nearby servers, thus mitigating long latency incurred by data transmission via wide area networks. When offloading video frames from the front-end device to an edge server, the application configuration (i.e., frame sampling rate and frame resolution) will impact several metrics, such as energy consumption, analytics accuracy and user-perceived latency. In this paper, we study the configuration selection and bandwidth allocation for multiple video streams, which are connected to the same edge node sharing an upload link. We propose an efficient online algorithm, called JCAB, which jointly optimizes configuration adaption and bandwidth allocation to address a number of key challenges in edge-based video analytics systems, including edge capacity limitation, unknown network variation, intrusive dynamics of video contents. Our algorithm is developed based on Lyapunov optimization and Markov approximation, works online without requiring future information, and achieves a provable performance bound. We also extend the proposed algorithms to the multi-edge scenario in which each user or video stream has an additional choice about which edge server to connect. Extensive evaluation results show that the proposed solutions can effectively balance the analytics accuracy and energy consumption while keeping low system latency in a variety of settings.

Wander of a Gaussian-Beam Wave Propagating through Kolmogorov and Non-Kolmogorov Turbulence along Laser-Satellite Communication Uplink

It is accepted that there exists two kinds of atmospheric turbulence in the Earth’s aerosphere—Kolmogorov and non-Kolmogorov turbulence; therefore, it is important to research their combined impacts on laser-satellite communications. In this paper, the exponential power spectra of refractive-index fluctuations for non-Kolmogorov turbulence in the free troposphere and stratosphere are proposed, respectively. Based on these two spectra, using the Markov approximation, beam wander displacement variances of a Gaussian-beam wave are derived, respectively, which are valid under weak turbulent fluctuations condition. On this basis, using a three-layer altitude-dependent turbulent spectrum model for vertical/slant path, the combined influence of a three-layer atmospheric turbulence on wander of a Gaussian-beam wave as the carrier wave in laser-satellite communication is studied. This three-layer spectrum is more accurate than a two-layer model. Moreover, the variations of beam wander displacement with beam radius, zenith angles, and nominal value of the refractive-index structure parameter on the ground are estimated. The theory of optical wave propagation through non-Kolmogorov atmospheric turbulence is further enriched and a theoretical model of a three-layer atmospheric turbulence beam wander for a satellite-ground laser communication uplink is established.

Near-Optimal Energy-Efficient Algorithm for Virtual Network Function Placement

To accommodate heterogeneous and sophisticated network services, Network Function Virtualization (NFV) is invented as a hopeful networking technology. The most distinct feature of NFV is that it separates network functions from physical hardware. In the NFV architecture, various types of Virtual Network Functions (VNFs) are placed on specific software-based middleboxes by telecom providers. Traffic traverses through a sequence of Virtual Network Functions (VNFs) in pre-defined order, which is named as Service Function Chain (SFC). However, how to effectively place VNFs at different locations and steer SFC requests while minimizing energy consumption is still an open problem. Accordingly, we investigate on the joint optimization of VNF placement and traffic steering for energy efficiency in telecom networks. We first present the power consumption model in NFV-enabled telecom networks, and then formulate the studied problem as an Integer Linear Programming (ILP) model. Since the problem is proved as NP-hard, we design a polynomial algorithm that can achieve near-optimal performances based on the Markov approximation technique. In addition, our algorithm can be extended to an online version to serve dynamic arriving SFC requests. The online algorithm achieves a near-optimal long-term averaged performance. Extensive simulation results show that compared with the benchmark algorithms, in the offline and online scenario, our algorithm can reduce up to 14.08 and 13.72 percent power consumption in telecom networks, respectively.

Feature Selection Method for Nonintrusive Load Monitoring With Balanced Redundancy and Relevancy

Nonintrusive load monitoring (NILM) has become a key technology in the power Internet of Things as well as an important information source for load characteristics analysis. Whether the selected features are appropriate determines the effectiveness of the load identification in NILM. In order to reduce the redundancy and improve the relevancy of the selected features, a feature selection method that balances redundancy and relevancy is proposed. Based on the approximate Markov blanket decision, this article puts forward the basis of feature set redundancy elimination. This is used to determine the priority of feature selection by taking the number of features preselection as a measure. The mutual information method is combined with the CRITIC weight to implement redundancy elimination for the initial feature set of the load, and the feature correlation ranking algorithm is established based on the Relief-F method. Based on the idea of balancing redundancy and relevancy, a feature selection strategy is established for the initial feature set, in order to obtain the optimal feature subset with minimum redundancy and maximum relevancy. Finally, the <i>K</i>-nearest neighbor identification algorithm after <i>k</i>-value optimization is used to simulate the proposed method. The results show that compared with the other feature selection methods, the proposed method is promising in terms of robustness and generalization and shows an active effect on improving identification accuracy.

Transport Monte Carlo: High-Accuracy Posterior Approximation via Random Transport

In Bayesian applications, there is a huge interest in rapid and accurate estimation of the posterior distribution, particularly for high dimensional or hierarchical models. In this article, we propose to use optimization to solve for a joint distribution (random transport plan) between two random variables, θ from the posterior distribution and β from the simple multivariate uniform. Specifically, we obtain an approximate estimate of the conditional distribution as an infinite mixture of simple location-scale changes; applying the Bayes’ theorem, can be sampled as one of the reversed transforms from the uniform, with the weight proportional to the posterior density/mass function. This produces independent random samples with high approximation accuracy, as well as nice theoretical guarantees. Our method shows compelling advantages in performance and accuracy, compared to the state-of-the-art Markov chain Monte Carlo and approximations such as variational Bayes and normalizing flow. We illustrate this approach via several challenging applications, such as sampling from multi-modal distribution, estimating sparse signals in high dimension, and soft-thresholding of a graph with a prior on the degrees. Supplementary materials for this article, including the source code and additional comparison with popular alternative algorithms are available on the journal website.

Critical transition to a non-chaotic regime in isotropic turbulence

We study the properties of homogeneous and isotropic turbulence in higher spatial dimensions through the lens of chaos and predictability using numerical simulations. We employ both direct numerical simulations and numerical calculations of the eddy damped quasi-normal Markovian closure approximation. Our closure results show a remarkable transition to a non-chaotic regime above the critical dimension, $d_c$ , which is found to be approximately 5.88. We relate these results to the properties of the energy cascade as a function of spatial dimension in the context of the idea of a critical dimension for turbulence where Kolmogorov's 1941 theory becomes exact.

Degradation and reliability of multi-function systems using the hazard rate matrix and Markovian approximation

The study presents a new method to model the degradation and reliability of multi-function complex systems using the hazard rate matrix and Markovian approximation. The system is hierarchically decomposed into a set of states according to the states of its functions, and the hazard rate matrix is proposed to describe the failure rates of the functions. By using Markovian approximation, the elements of the hazard rate matrix can be expressed as a set of dynamical equations involving the failure information about the functions. Consequently, the dynamical behavior of the degradation for the whole system and its functions can be determined using the observed failure data of those functions instead of the lifetime data of the whole system, eliminating the need for lifetime testing data of the whole system in statistical-based reliability assessment. The overall method is illustrated using an example problem, and is further applied to a power system degradation problem to demonstrate its usage in realistic engineering applications.

Effective Field Theory for jet substructure in heavy ion collisions

I develop an Effective Field Theory (EFT) framework to compute jet substructure observables for heavy ion collision experiments. As an example, I consider dijet events that accompany the formation of a weakly coupled long lived Quark Gluon Plasma (QGP) medium in a heavy ion collision and look at an observable insensitive to jet selection bias: the simultaneous measurement of jet mass along with the transverse momentum imbalance between the jets that are groomed to remove soft radiation. Treating the jet as an open quantum system, I write down a factorization formula within the SCET (Soft Collinear Effective Theory) framework in the forward scattering regime. The physics of the medium is encoded in a universal soft field correlator while the jet-medium interaction is captured by a medium induced jet function. The factorization formula leads to a Lindblad type equation for the evolution of the reduced density matrix of the jet in the Markovian approximation. The solution for this equation allows a resummation of large logarithms that arise due to the final state measurements imposed while simultaneously summing over multiple incoherent interactions of the jet with the medium.

A microscopic model of wave-function dephasing and decoherence in the double-slit experiment

The act of measurement on a quantum state is supposed to “dephase” (dephasing refers to the phenomenon that the states lose phase coherence; then the phases get randomized in interaction with a bath of other oscillators, which is referred to as “decoherence”), then “decohere” and “collapse” (or more precisely “register” and “reduce”) the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schrödinger’s equation. Progress has, however, been made in studying this problem with two main calculation tools—one uses a time-independent Hamiltonian, while a rather more general approach proving that decoherence occurs under some generic conditions. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the “consistent” or “decoherent”’ histories approach, studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the “environmental decoherence” approach studies the effect of the environment upon the quantum system, to explain wave-function decoherence which is produced by irreversible effects of various sorts. However, as we know, wave-function collapse is not related to thermal connection with the environment, rather, it is inherent to how measurements are performed by macroscopic apparata. In the “environmental decoherence” approach, one studies decoherence using a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix (post dephasing) and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system starting with “dephasing” followed by the effects of the environment with some differences from prior analyses. We start from the Schrödinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve (exactly) for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful, and it shows that the system wave-function will explicitly “un-collapse” if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, this “dephasing” quickly leads to “decoherence”—collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. This has been attempted previously and our calculation is particularly simple and calculable. We make some connections to the work by Linden et al. while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner’s friend experiment, as well as the Mott experiment, which is explored in “Connection to some general theoretical results” and “Recurrence times” (especially the enumerated points in “Recurrence times”). The upshot is that as long as Wigner’s friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner’s friend does not even need to be conscious during the measurement that she conducts. It also allows one to reasonably interpret some of the more recent thought experiments proposed. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermi-golden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper—we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.

Markovian approximations of stochastic Volterra equations with the fractional kernel

We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an N-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in N. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model.

Filter Functions for Quantum Processes under Correlated Noise.

Many qubit implementations are afflicted by correlated noise not captured by standard theoretical tools that are based on Markov approximations. While independent gate operations are a key concept for quantum computing, it is actually not possible to fully describe noisy gates locally in time if noise is correlated on times longer than their duration. To address this issue, we develop a method based on the filter function formalism to perturbatively compute quantum processes in the presence of correlated classical noise. We derive a composition rule for the filter function of a sequence of gates in terms of those of the individual gates. The joint filter function allows us to efficiently compute the quantum process of the whole sequence. Moreover, we show that correlation terms arise which capture the effects of the concatenation and, thus, yield insight into the effect of noise correlations on gate sequences. Our generalization of the filter function formalism enables both qualitative and quantitative studies of algorithms and state-of-the-art tools widely used for the experimental verification of gate fidelities like randomized benchmarking, even in the presence of noise correlations.

Non-Markovian open quantum system approach to the early Universe: Damping of gravitational waves by matter

By revising the application of the open quantum system approach to the early universe and extending it to the conditions beyond the Markovian approximation, we obtain a new non-Markovian quantum Boltzmann equation. Throughout the paper, we also develop an extension of the quantum Boltzmann equation to describe the processes that are irreversible at the macroscopic level. This new kinetic equation is, in principle, applicable to a wide variety of processes in the early universe. For instance, using this equation one can accurately study the microscopic influence of a cosmic environment on a system of cosmic background photons or stochastic gravitational waves. In this paper, we apply the non-Markovian quantum Boltzmann equation to study the damping of gravitational waves propagating in a medium consisting of decoupled ultra-relativistic neutrinos. For such a system, we study the time evolution of the intensity and the polarization of the gravitational waves. It is shown that, in contrast to intensity and linear polarization which are damped, the circular polarization (V-mode) of the gravitational wave (if present) is amplified by propagating through such a medium.

Nonlocal Markov approximation for mean field propagating in a medium with dielectric permittivity fluctuations in case of finite values of longitudinal correlation radius 2. Inhomogeneous background medium

The solution to the integro-differential (nonlocal) Markov equation for the mean field, developed in the companion paper, which takes account of finite values of the longitudinal correlation radius of fluctuations of the dielectric permittivity, is extended to the case of an arbitrary smoothly inhomogeneous background medium. The explicit solution to the problem is presented.

Nonlocal Markov approximation for mean field propagating in a medium with dielectric permittivity fluctuations in case of finite values of longitudinal correlation radius 1. Homogeneous background medium

Classic method of the diffusive Markov parabolic equation for the mean field in a stochastic medium is extended to the case of finite values of the longitudinal correlation radius of fluctuations. In the limiting case of δ-correlated fluctuations the solution constructed reproduces a well-known solution to the case of the diffusive Markov approximation for the mean field.

Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium.

We present a generalized Langevin equation (GLE) of motion that governs exactly the time evolution of phase-space observables in finite open systems described by classical Hamiltonians with explicitly time-dependent potentials. This formalism is based on the Mori-Zwanzig projection operator (PO) method with a time-independent Zwanzig PO within a Heisenberg (Lagrangian) picture and reduced description of Hamiltonian systems in terms of canonical relevant and irrelevant coordinates. We demonstrate that, similarly to closed systems, GLE dynamics in Hamiltonian systems in the presence of time-dependent potentials is determined by conservative, dissipative memory, and projected force fields, and that the memory functions relate to the projected force, which is a two-time process, in a way that is reminiscent of the equilibrium second fluctuation-dissipation relation. We further show that, in the most general case, the memory kernel depends on the relevant momentum gradients of the (Boltzmann) entropy of the irrelevant subsystem. Using two Zwanzig operators which are, respectively, functionals of the canonical and generalized canonical probability densities, we then derive what we call canonical and generalized canonical GLEs. Further, we can formulate the particle-based, coarse-grained (CG) GLE dynamics by transitioning to Jacobi coordinates which corresponds to a particle set partitioning of the Hamiltonian system. The obtained canonical CG GLE of motion for the relevant momenta is a generalization of the CG equation of motion known for closed systems. Also, using a Markovian approximation of the canonical CG GLE, we can extend the dissipative particle dynamics equation to open systems. A distinctive feature of our extension is a use of explicitly time-dependent frictions, which reflect the changes in the dissipation rate caused by time-dependent coupling to an external bath. Our GLE formalism and workflow constitute a general and viable framework that can be readily used as a starting point to rigorously formulate microscopically informed CG treatments for a variety of phenomena in externally forced systems far from equilibrium.

Age-based Markovian approximation of the G/M/1 queue

We extend the approach of Koole et al. (2012) [15] and Legros et al. (2018) [20] for the G/M/1 queue. The idea is to provide a Markovian approximation where a state represents the oldest customer's wait. This modeling is made possible by creating states with negative wait, representing an estimate of the time at which a new customer would arrive when the system is empty. We apply this method for performance evaluation and routing optimization. Finally, we further extend the model to the G/M/1+G queue.

An efficient feature selection framework based on information theory for high dimensional data

Feature selection plays a vital role in many fields, particularly in pattern recognition and bioinformatics, for selecting informative and relevant features from high dimensional datasets. The increase in dimensionality of data along with the existence of redundant and irrelevant features leads to challenging performance issues when processing and analysing the data. In this paper, an effective feature selection technique called mutual information and Monte Carlo based feature selection (MIMCFS) is proposed. It comprises of two stages. The first stage aims to select predominant features from the high dimensional data. The second stage involves elimination of redundant features that were selected in the first stage. For the purpose of implementing the first stage, a new feature selection strategy based on the approximate Markov blanket and the concept of mutual information is proposed to find out irrelevant and redundant features. In second stage, to avoid misjudgement of redundant features as relevant features, a new strategy based on Monte Carlo tree search technique is proposed in order to completely eradicate redundant features and to improve feature interaction. For experimental evaluation, eight benchmark microarray datasets including imbalanced ones pertaining to cancer analysis are used. Further, in order to compare and justify the performance of the proposed feature selection method, seven state-of-art feature selection techniques namely CFS, Relief, DISR, JMI, CMIM and CMI are employed. The outputs from these feature selection techniques are provided to three standard classifiers namely Naive Bayes, SVM and C4.5 in order to assess the significance of the selected features in building classification models. 10-fold cross validation is adopted to evaluate the classifiers. Accuracy, precision, recall, f-measure, standard deviation, statistical significance metrics are measured to quantify the classifier performance. Experimental results demonstrate the outstanding performance of the proposed algorithm when compared to that of the standard existing methods.

Exact non-Markovian permeability from rare event simulations

Permeation of compounds through membranes is important in biological and engineering processes, e.g., drug delivery through lipid bilayers, anesthetics, or chemical reactor design. Simulations at the atomic scale can provide insight in the diffusive pathways and they give estimates of the membrane permeability based on counting membrane transitions or on the inhomogeneous solubility-diffusivity model described by the Smoluchowski equation. For many permeants, permeation through a membrane is too slow to gather sufficient statistics with conventional molecular dynamics simulations, i.e., permeation is a rare event. Recent attempts to improve the description of the dynamics of such rare permeation events have been based on milestoning, which allows the study of processes at timescales beyond those achievable by straightforward molecular dynamics. The approach is not relying on an overdamped description, but, still, it uses a Markovian approximation which is only valid for small permeants that are not disruptive to the membrane structure. To overcome this fundamental limitation, we show here how replica exchange transition interface sampling (RETIS) can effectively be used on this problem by deriving an effective set of equations that relate the outcome of RETIS simulations and the permeability coefficient. In addition, we introduce two new path Monte Carlo (MC) moves specifically for permeation dynamics, that are used in combination with the ordinary path generating moves, which considerably increase the efficiency. The advantage of our method is that it gives exact results, identical to brute force molecular dynamics, but orders of magnitude faster.

Quantum Control of Coherent π-Electron Dynamics in Aromatic Ring Molecules

Herein we review a theoretical study of unidirectional π-electron rotation in aromatic ring molecules, which originates from two quasi-degenerate electronic excited states created coherently by a linearly polarized ultraviolet/visible laser with a properly designed photon polarization direction. Analytical expressions for coherent π-electron angular momentum, ring current and ring current-induced magnetic field are derived in the quantum chemical molecular orbital (MO) theory. The time evolution of the angular momentum and the ring current are expressed using the density matrix method under Markov approximation or by solving the time-dependent Schrödinger equation. In this review we present the results of the following quantum control scenarios after a fundamental theoretical description of coherent angular momentum, ring current and magnetic field: first, two-dimensional coherent π-electron dynamics in a non-planar (P)-2,2’-biphenol molecule; second, localization of the coherent π-electron ring current to a designated benzene ring in polycyclic aromatic hydrocarbons; third, unidirectional π-electron rotations in low-symmetry aromatic ring molecules based on the dynamic Stark shift of two relevant excited states that form a degenerate state using the non-resonant ultraviolet lasers. The magnetic fields induced by the coherent π-electron ring currents are also estimated, and the position dependence of the magnetic fluxes is demonstrated.