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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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