Randomized exponential transformation algorithm for solving the stochastic problems of gamma-ray transport theory

2020 ◽  
Vol 35 (3) ◽  
pp. 153-162
Author(s):  
Ilia N. Medvedev ◽  
Gennadii A. Mikhailov
Keyword(s):  
Markov Chain ◽  
Radiation Transfer ◽  
Gamma Ray ◽  
Transport Theory ◽  
Numerical Study ◽  
Thick Layer ◽  
Exponential Transformation ◽  
Stochastic Problems ◽  
Transformation Algorithm ◽  
Simulation Algorithms

AbstractThe paper presents a new algorithm of exponential transformation and its randomized modification with branching of a Markov chain trajectory for solving the problem of gamma-ray transport. Based on the example of radiation transfer in water, numerical study of presented algorithms is performed in comparison with standard simulation algorithms. The study of the influence of medium stochasticity on the probability of gamma-quanta passing through a thick layer of the substance is also carried out.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Transport ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transport Theory ◽  
Sufficient Conditions ◽  
Continuous Time Random Walks ◽  
Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

scholarly journals Dividend maximization in a hidden Markov switching model

2015 ◽  
Vol 32 (3-4) ◽  
pp. 143-158
Author(s):  
Michaela Szölgyenyi
Keyword(s):  
Markov Chain ◽  
Partial Information ◽  
Numerical Study ◽  
Insurance Company ◽  
Markov Switching Model ◽  
Switching Model ◽  
Surplus Process ◽  
Dividend Payments ◽  
Current State ◽  
Optimal Value

Abstract In this paper we study the valuation problem of an insurance company by maximizing the expected discounted future dividend payments in a model with partial information that allows for a changing economic environment. The surplus process is modeled as a Brownian motion with drift. This drift depends on an underlying Markov chain the current state of which is assumed to be unobservable. The different states of the Markov chain thereby represent different phases of the economy. We apply results from filtering theory to overcome uncertainty and then we give an analytic characterization of the optimal value function. Finally, we present a numerical study covering various scenarios to get a clear picture of how dividends should be paid out.

Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating

1997 ◽  
Vol 29 (01) ◽  
pp. 92-113 ◽  
Author(s):  
Frank Ball ◽  
Sue Davies
Keyword(s):  
Markov Chain ◽  
Ion Channel ◽  
State Space ◽  
Continuous Time ◽  
Markov Models ◽  
Numerical Study ◽  
Channel Gating ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean[N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

Universal modification of vector weighted method of correlated sampling with finite computational cost

2019 ◽  
Vol 34 (1) ◽  
pp. 43-55
Author(s):  
Ilya N. Medvedev
Keyword(s):  
Markov Chain ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Radiation Transfer ◽  
Computational Cost ◽  
Linear Functionals ◽  
Weighted Estimator ◽  
Matrix Weights ◽  
Correlated Sampling

Abstract The weighted method of dependent trials or weighted method of correlated sampling (MCS) allows one to construct estimators for functionals based on the same Markov chain simultaneously for a given range of the problem parameters. Choosing an appropriate Markov chain, it is necessary to take into account additional conditions providing the finiteness of the computational cost of weighted MCS. In this paper we study the issue of finite computational cost of the method of correlated sampling (MCS) in application to evaluation of linear functionals of solutions to a set of systems of 2nd kind integral equations. A universal modification of the vector weighted MCS is constructed providing the branching of chain trajectory according to elements of matrix weights. It is proved that the computational cost of the constructed algorithm is bounded in the case the base functionals are also bounded. The results of numerical experiments using the modified weighted estimator are presented for some problems of the theory of radiation transfer subject to polarization.

The method of similar trajectories with branching according to parametric maximum of the auxiliary weight

2016 ◽  
Vol 31 (6) ◽  
Author(s):  
Ilia N. Medvedev
Keyword(s):  
Markov Chain ◽  
Numerical Study ◽  
Computational Cost ◽  
Transfer Theory ◽  
Standard Problem

AbstractThe weighted method of similar trajectories (MST) allows one to construct estimators of functionals on a single Markov chain simultaneously for a given range of parameters of the problem. Choosing an appropriate Markov chain, we take into account additional conditions providing the finiteness of MST variance. A modification of the weighted MST with branching of chain trajectory is constructed in the paper according to the parametric maximum of the auxiliary weight. It is proved that the computational cost of this algorithm is bounded if the basis functionals are also bounded. Numerical study of the efficiency of the modified MST in comparison with analog modelling was carried out on the example of the standard problem of transfer theory on estimation of the probability of albedo and transmission of a particle.

scholarly journals A Confront between Amati and Combo Correlations at Intermediate and Early Redshifts

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1118
Author(s):  
Marco Muccino
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Predictive Power ◽  
Energy Equation ◽  
Gamma Ray ◽  
Zero Curvature ◽  
Level 3 ◽  
The Difference ◽  
Best Fit ◽  
The Universe

I consider two gamma-ray burst (GRB) correlations: Amati and Combo. After calibrating them in a cosmology-independent way by employing Beziér polynomials to approximate the Observational Hubble Dataset (OHD), I perform Markov Chain Monte Carlo (MCMC) simulations within the Λ CDM and the wCDM models. The results from the Amati GRB dataset do not agree with the standard Λ CDM model at a confidence level ≥ 3 – σ . For the Combo correlation, all MCMC simulations give best-fit parameters which are consistent within 1– σ with the Λ CDM model. Pending the clarification of whether the diversity of these results is statistical, due to the difference in the dataset sizes, or astrophysical, implying the search for the most suited correlation for cosmological analyses, future investigations require larger datasets to increase the predictive power of both correlations and enable more refined analyses on the possible non-zero curvature of the Universe and the dark energy equation of state and evolution.

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