Monte Carlo method in statistical physics

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The quantum Monte Carlo method : application to problems in statistical physics

The quantum Monte Carlo method : application to problems in statistical physics

To examine the boundary condition problems discussed above we have performed importance sampled calculation with a system of 108 particles. A time step of A t = 0.05 x 10“ *'5S W as used and the initial ensemble contained 100 systems distributed according to the variational wave function.. In Figure 4.4 we show the effects of extrapolating the radial distribution function obtained from this run by using the variational distribution. Again the results are compared with GFMC values. Good agreement between the diffusion Monte Carlo and GFMC results is generally observed. The two simulations were performed at slightly different densities, p p^C = 0 . 4 and p GFMC = 0*^01» so the small variations in the results are probably associated with this density difference. Agreement between the predicated eigenvalues (including only long range corrections) is also good, eDMC = “6.78 ± 0.06 and E q ^^ q = -6.743 ± 0.033 K/molecule. Whitlock et a l . (1979) have obtained a perturbation estimate of the three body correction, and at this density they give <V^ b > = 0.206 ± 0.002 K/molecule or about 3 l of the two body values given above. When the three body correction is made both the quantum Monte Carlo calculations give ground state energies which are approximately 0.5 K/molecule higher than the experimental value EeXp = -7.00 K/molecule (Roach, Ketterson and Woo (1970)). This discrepency is a result of the inadequacy of the Lennard-Jones potential (Whitlock et al. (1981) ).
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Research on cold chain in food industry in China

Research on cold chain in food industry in China

Monte Carlo analysis has played an important role for many years in the investigation of statistical estimators whose properties cannot be adequately determined through mathematical techniques alone. Monte Carlo methods have been used for centuries, but only in the past several decades has the technique gained the status of a full-fledged numerical method capable of addressing the most complex applications. The falling opportunity cost of computing, especially the greater availability of flexible spreadsheet software for microcomputers, makes Monte Carlo analysis feasible for an ever increasing number of practicing policy analysis (EPA, 1997). Monte Carlo is now used routinely in many diverse fields, from the simulation of complex physical phenomena such as radiation transport in the earth’s atmosphere and the simulation of the esoteric sub-nuclear processes in high energy physics experiments, to the life sciences such as DNA sequence assembly. In recent years, the Monte Carlo analysis is applied in the economic domain such as project management.
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Development of SUBSPACE-Based Hybrid Monte Carlo-Deterministic Algorithms for Reactor Physics Calculations.

Development of SUBSPACE-Based Hybrid Monte Carlo-Deterministic Algorithms for Reactor Physics Calculations.

In this section we demonstrate the sensitivity of the variance reduction results to the rank estimate. As discussed earlier, one could employ a rigorous method to estimate the exact rank of the matrix Ψ such as the range-finding algorithm described in the appendix. However in most applications employing Monte Carlo models, a small estimate of the rank should be sufficient. This is because the very first few singular values of the matrix Ψ display a significant decline with the rate of decline decreasing with increased rank. To illustrate this, the algorithm in the appendix is employed to estimate the first 30 singular values of the matrix Ψ . This could be achieved by executing the algorithm with different user-defined tolerance [36]. Notice that the singular values plotted in Fig. 4.11 fall down by three orders of magnitude by the time the tenth singular value is reached. After that, the singular values continue to fall down but at a much smaller rate. Given that the statistical uncertainties for the responses are expected to be in the 0.1% to 1% range, only the initial reduction in the singular values should be sufficient to estimate the rank.
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Ultra-fast Carrier Transport Simulation on the Grid. Quasi-Random Approach

Ultra-fast Carrier Transport Simulation on the Grid. Quasi-Random Approach

3. Quasirandom approach in SALUTE. Quasi-Monte Carlo methods and algorithms proved to be efficient in many areas ranging from physics to economy. We have applied quasirandom approach for studying quantum effects during ultra-fast carrier transport in semiconductors and quantum wires in order to reduce the error and to speedup the computations. Next, we have used scrambled sequences for two main reasons: (i) the problem is very complicated (the use of scrambling corrects the correlation problem found when we have used a purely quasi-Monte Carlo algorithm), and, (ii) the Grid implementation which needs parallel streams.
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How to Adjust the fGn Stochastic Model for Statistical Bias when Handling a Single Time Series; Application to Annual Flood Inundation

How to Adjust the fGn Stochastic Model for Statistical Bias when Handling a Single Time Series; Application to Annual Flood Inundation

A simple scaling model known as the fractional Gaussian noise is often chosen for the description of several annual (and of larger) scale hydroclimatic processes exhibiting the Hurst phenomenon. An important characteristic of such model is the induced large statistical bias, i.e. the deviation of a statistical characteristic (e.g. variance) from its theoretical discretized value. Most studies in literature perform stochastic modelling by equating the sampling second order dependence structure with the expected value of the estimator of a stochastic model. However, this is justified only when many realizations (i.e. many time series) of a single process are available. In case where we have a single realization we should model the mode estimator of the dependence structure of the desired stochastic model instead, otherwise we may overestimate the extremeness of a realization, e.g. flood event. In this study, we show an innovative way of handling the statistical bias for an fGn process when analyzing one time series. Particularly, we conduct a thorough Monte-Carlo analysis based on the climacogram (i.e., marginal distribution of a scaled process, with focus on the second central moment of variance that is shown to be the least uncertain from the rest central moments) of an fGn process and we propose to equate the 25% quartile (and not the expected) value of the modeled climacogram with the sampling one to correctly adjust the model for bias.
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Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff

Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff

allow us to quantify an improvement in computational complexity when standard Monte Carlo is replaced by the multi-level version of [4]; they also explain the numerical results presented there. Promising topics for further work in this area include the analysis of (a) the weak error rate α in (1.7) for path-dependent options, (b) methods with higher strong order, and (c) quasi-Monte Carlo.

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Reliability Assessment of Bearings Based on Performance Degradation Values under Small Samples

Reliability Assessment of Bearings Based on Performance Degradation Values under Small Samples

Nowadays, the problem of small datasets is attracting increasing attention. Bootstrapping [16] is a good method to enlarge the sample sizes. Many engineers and scholars use it to raise the precision of the parameters’ estimation. Structural reliability was assessed by applying the bootstrapping method, according to [17]. Li et al. [18] pointed out that the method was useful for statistics with an unknown distribution and datasets with small sample size. The Bootstrapping method and Monte Carlo simulation were applied to evaluating the uncertainty of failure rate estimation in engineering problems [19]. The Monte Carlo method is another widely used method in engineering and statistics. That method and fault tree analysis were applied to analysis of the reliability for a wastewater treatment plant [20]. The Monte Carlo simulation was also applied to the solution of the population balance equations, and the accuracy and the optimal sampling in Monte Carlo solutions of the equations have been discussed [21]. A multilevel Monte Carlo method was proposed to estimate the uncertainty in pore-scale and digital rock physics problems [22].
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Monte Carlo Simulations

Monte Carlo Simulations

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Monte Carlo methods

Monte Carlo methods

If the choice of q is not obvious, we recommend the use of an adaptive strategy, such as population Monte Carlo. A description of population MC and an application to model selection in cosmology can be found in [24]. Basically, first make a wild guess q (0) for q, say a Gaussian with a large variance. Apply importance sampling a first time to obtain an estimate of π and fit a Gaussian q (1) to this estimate of π. Now re-apply importance sampling with q (1) as a proposal, and re-fit a new Gaussian q (2) to π, etc. After T iterations, q (T ) should be a good proposal distribution for importance sampling. Of course, you can apply this procedure with other candidate proposals than Gaussians, you should indeed choose a family of distributions among which you think you may find a good approximation of π . If you have reasons to believe that π is bimodal, for example, you should probably fit a mixture of two distributions as in [24] rather than a Gaussian, which is unimodal. Usually, with the right choice of family of distributions, a few iterations are enough to get a reasonable q, and you can stop when q (t) does not change a lot with t.
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Evaluation of Investment Risks in CBA with Monte Carlo Method

Evaluation of Investment Risks in CBA with Monte Carlo Method

KORYTÁROVÁ JANA, POSPÍŠILOVÁ BARBORA. 2015. Evaluation of Investment Risks in CBA with Monte Carlo Method. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 63(1): 245–251. Investment decisions are at the core of any development strategy. Economic growth and welfare depend on productive capital, infrastructure, human capital, knowledge, total factor productivity and the quality of institutions. Decision-making process on the selection of suitable projects in the public sector is in some aspects more diffi cult than in the private sector. Evaluating projects on the basis of their fi nancial profi tability, where the basic parameter is the value of the potential profi t, can be misleading in these cases. One of the basic objectives of the allocation of public resources is respecting of the 3E principle (Economy, Eff ectiveness, Effi ciency) in their whole life cycle. The life cycle of the investment projects consists of four main phases. The fi rst pre-investment phase is very important for decision-making process whether to accept or reject a public project for its realization. A well-designed feasibility study as well as cost-benefi t analysis (CBA) in this phase are important assumptions for future success of the project. A future fi nancial and economical CF which represent the fundamental basis for calculation of economic eff ectiveness indicators are formed and modelled in these documents. This paper deals with the possibility to calculate the fi nancial and economic effi ciency of the public investment projects more accurately by simulation methods used.
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Applications of the Fixed-node Quantum Monte Carlo Method.

Applications of the Fixed-node Quantum Monte Carlo Method.

The Full Configuration Interaction (Full CI) method might build an exact wave function as a linear combination of the HF determinant and all excited determinants (i.e., the full spectrum of the particle-hole expansion) in a complete basis set (CBS) limit. However, there are a few drawbacks in the application of Full CI. First of all, unfortunately, the CBS limit is currently not possible; truncated basis sets are employed. Second, the number of excited determinants grows exponentially with system size and the number of basis functions. Therefore, the determinant space (or configuration space) also needs to be truncated, resulting in the truncated CI. Third, in either the Full CI or the truncated CI, the hamiltonian matrix is diagonalized to find the optimum linear combination of the determinants [79]. This full matrix diagonalization is a significantly time-consuming step of the method and can be another obstacle to considering a larger space of determinants. Finally, following a CI calculation, a space with a large number of determinants is generated. The construction of a trial wave function in QMC by using this s determinant expansion is done usually by selecting the determinants according to the absolute values of their (normalized) coefficients, not according to the energy drop that they cause.
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Mean exit times and the multilevel Monte Carlo method

Mean exit times and the multilevel Monte Carlo method

Abstract. Numerical methods for stochastic differential equations are relatively inefficient when used to ap- proximate mean exit times. In particular, although the basic Euler–Maruyama method has weak order equal to one for approximating the expected value of the solution, the order reduces to one half when it is used in a straightforward manner to approximate the mean value of a (stopped) exit time. Consequently, the widely used standard approach of combining an Euler–Maruyama discretization with a Monte Carlo simulation leads to a computationally expensive procedure. In this work, we show that the multilevel approach developed by Giles [Oper. Res., 56 (2008), pp. 607–617] can be adapted to the mean exit time context. In order to justify the algorithm, we analyze the strong error of the discretization method in terms of its ability to approximate the exit time. We then show that the resulting multilevel algorithm improves the expected computational complexity by an order of magnitude, in terms of the required accuracy. Numerical results are provided to illustrate the analysis.
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A Simple Monte Carlo Method for the Calculation of Efficiency Limit for Current

A Simple Monte Carlo Method for the Calculation of Efficiency Limit for Current

In this paper, I have calculated the limit efficiency and determined the optimal band‐gaps of a tandem solar cell system, consisting of n (n = 1, 2…10) homojunction cells using the detailed balance approach. The calculation employed Monte Carlo statistical technique to resolve equation system that is significantly potential to describe the current density versus voltage characteristics of each sub‐cell, to determine its maximum power point. The application of rejection technique is a potent tool to reduce the calculating time for current matched tandem solar cells. The highest estimated efficiency in this work is 80.35% reached for a stack of 10 junctions under fully concentrated AM1.5 D solar spectrum. This value is not far away from the upper theoretical efficiency limit of 86.8% predicted for an infinite number of junctions [4]. The obtained optimal energy gaps are in fair accord with most available data from literature [6‐11], though they are achieved using elaborate numerical techniques. Monte Carlo method could be improved by taking into account the sub‐cell thickness and its effect on matching the maximum current density.
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Simulation for Callable Convertible Discount Bonds with Monte Carlo Method

Simulation for Callable Convertible Discount Bonds with Monte Carlo Method

The Monte Carlo method is widely applied in many fields. It is applicable to multi-dimensional derivative securities pricing characteristics and easy to deal with the realistic characteristics of discrete coupon dividends, path dependence and other convertible bonds. As a result, it has gradually become one of the most effective methods in the pricing of convertible bonds. In a risk-neutral world, stock prices are largely subject to geometric Brownian motion. Therefore, we can conduct a large number of repeated random simulations on the change path of stock prices on T in the future, and then average the results of these simulations at a risk-free rate. In the end, we can get the value of the derivative.
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Monte Carlo Simulation Method To Predict  The Charging Load Curve

Monte Carlo Simulation Method To Predict The Charging Load Curve

For test statistic G , Grubbs derived its statistical distribution and gave a critical value G (1 − n ) ( ) n when the significant level a was 1% or 5%. G (1 − n ) ( ) n is known as Grubb’s exponent. It can be consulted from a spot check table. When test statistics G corresponding to the minimum x 1 or the maximum x n are greater than critical values, it is considered that corresponding x 1 or x n are suspected abnormal values and should be eliminated. The accuracy of Monte Carlo Simulation in processed data can be judged by the variance coefficient β [10] , namely
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Coded aperture imaging systems:past, present and future development   a review

Coded aperture imaging systems:past, present and future development a review

Scintillator based coded-aperture imaging has proven to be effective when applied for X- and gamma-ray detection. Adaptation of the same method for neutron imaging has resulted in a number of propitious sys- tems, which could be potentially employed for neutron detection in secu- rity and nuclear decommissioning applications. Recently developed scin- tillator based coded-aperture imagers reveal that localisation of neutron sources using this technique may be feasible, since pulse shape discrimina- tion algorithms implemented in the digital domain can reliably separate gamma-rays from fast neutron interactions occurring within an organic scintillator. Moreover, recent advancements in the development of solid organic scintillators make them a viable solution for nuclear decommis- sioning applications as they present less hazardous characteristics than currently dominating liquid scintillation detectors. In this paper exist- ing applications of coded-apertures for radiation detection are critically reviewed, highlighting potential improvements for coded-aperture based neutron source localisation. Further, the suitability of coded-apertures for neutron imaging in nuclear decommissioning is also assessed using Monte- Carlo modelling.
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Development of Monte Carlo Methods in Hypersonic Aerodynamics

Development of Monte Carlo Methods in Hypersonic Aerodynamics

The simulation of a collision is reduced to a statistical realization of the evolution of model during the time Δt rather than to the realization of the Boltzmann equation. The collision time in the Kac model is calculated in accordance with collision statistics in the ideal gas following the Bernoulli scheme. This scheme makes it possible to use a considerably smaller number of particles in a cell and a finer grid. The analysis shows that the computation results are almost independent of the number of particles in a cell down to two particles. The point is that the Boltzmann equation requires the molecular chaos assumption to be satisfied; however, for the number of particles in a cell that can be processed by modern computers, this assumption is satisfied only with a systematic error. By contrast, Kac does not rely on this assumption; therefore, the collision is calculated as a Markov process. On the other hand, as N → ∞ , the Kac model is completely equivalent to the spatially homogeneous Boltzmann equation. Thus, the approach developed by Belotserkovskii and Yanitskii provides a basis for constructing efficient numerical schemes for solving three-dimensional aerodynamic flow problems, and, on the other hand, it solves the important methodological problem of the equivalence of the numerical method and the solution of the kinetic equation.
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Uncertainty analysis for a wave energy converter: the Monte Carlo method

Uncertainty analysis for a wave energy converter: the Monte Carlo method

Abstract— Developing wave energy converter technology requires physical-scale model experiments. To use and compare such experimental data reliably, its quality must be quantified through an uncertainty analysis. To avoid uncertainty analysis problems for wave energy converter models, such as providing partial derivatives for time-varying quantities within numerous data reduction equations, we explored the use of a practical alternative: the Monte Carlo method (MCM). We first set out the principles of uncertainty analysis and the MCM. After, we present our application of the MCM for propagating uncertainties in a generic Oscillating Water Column wave energy converter experiment. Our results show the MCM is a straightforward and accurate method to propagate uncertainties in the experiment; thus, quantifying the quality of experimental data in terms of power performance. The key conclusion of this work is that, given the demonstrated relative ease in performing uncertainty analysis using the MCM, experimental results reported in the future literature of wave energy converter modelling should be accompanied by the uncertainty in those results. More broadly, this study aims to precipitate awareness among the wave energy community of the importance of quantifying the quality of modelling data through an uncertainty analysis. We therefore recommend future guidelines and specifications pertinent to uncertainty analysis for wave energy converters, such as those developed by the International Towing Tank Conference (ITTC) and International Electrotechnical Commission (IEC), to incorporate the MCM with a practical example.
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Criticality Analysis and Quality Appraisal of Innoson Injection Mould System

Criticality Analysis and Quality Appraisal of Innoson Injection Mould System

The current dynamic and turbulent manufacturing environment has forced companies that compete globally to change their traditional methods of conducting business. Recent developments in manufacturing and business operations have led to the adoption of preventive maintenance techniques that is based on systems and process that support global competitiveness. This paper employed Monte Carlo Normal distribution model which interacts with a developed Obudulu model to assess reliability and maintenance of Injection Moulding machine. The failure rate, reliability and standard deviations are reliability parameter used. Monte Carlo Normal distribution was used to analyse the reliability and failure rate of the entire system. The result shows that failure rate increases with running time accruing from wear due to poor lubrication systems; while system reliability decreases with increase time (years). Obudulu model was used to evaluate the variance ration of failure between system components under preventive maintenance and those outside preventive maintenance. The result shows that at reliability +0.3 and failure rate - 0.02, preventive maintenance should be done. Interaction between the Monte Carlo normal distribution and obudulu model shows that the total system reliability is 0.489 when maintained which is 49% and 0.412 (41%) when not maintained. Also quality of production increased during Preventive maintenance while system downtime reduced greatly. These models were programmed using Monte Carlo Excel tool package software, showing the graphs of reliability and failure rates for each system.
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Application of Monte Carlo method in Grid Computing and Allied Fields

Application of Monte Carlo method in Grid Computing and Allied Fields

1. Nuclear Reactor-Related Criticality Calculations: In a nuclear reactor, the structure containing the reactor is often subjected to an intense amount of radiation due to nuclear fission reactions. These radioactive particles impact the surrounding mechanical structure and some areas may receive most of the radioactive impact. These areas increase the vulnerability of the structure, and thus these areas have to be reinforced in order to ensure the integrity of the surrounding structure. The simulation of the radioactive particle trajectories allows for the discovery of the points of weakness that could exist due to the deformation over time or a poor design of the structure. The radiation particles generated during the nuclear process can be considered to be random. Furthermore, the trajectories of the radioactive particles after they are produced are also highly random due to the presence of air particles that interact with this radioactive particle. Due to this phenomenon, Monte Carlo methods can be used to simulate the overall reaction and in particular the impact on the surrounding structure. Based on the properties of the surrounding environment, the radioactive particles emitted would follow a nondeterministic trajectory. Monte Carlo can be used to simulate different trajectories caused of the radioactivity in the nuclear reactor core. The inherent random property of the trajectories is provided by the random number generator discussed. Similarly, the total number of trajectories to be simulated is divided among various nodes to reduce the overall time needed. However, this example differs from the previous as it has a three dimensional environment, as compared to a one dimensional problem, which has an exercise time T as the boundary. Furthermore, we have also to consider the structural properties of the structure exposed to radiation. These mechanical properties have to be inculcated into the three dimensional boundary when implementing the Monte Carlo method. This is unlike the previous example where the boundary is just a vertical line. The definition of the structural properties for modeling in the grid is also beyond the scope of this book. However, we have seen here that, by running the Monte Carlo simulations, we will be able to have a distribution of the areas where the impact of the radioactive particles is the most intense. This allows us to identify potential weakness in the structure and help to prevent the compromise of the structure surrounding the nuclear reactor core.
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