# Monte Carlo Method (MCM)

## Top PDF Monte Carlo Method (MCM): ### Uncertainty analysis for a wave energy converter: the Monte Carlo method

A key reason why WEC experimentation literature lacks uncertainty analysis is due to the challenge of modelling a complex WEC system in a complex environment [6, 8]. Oftentimes it is not straightforward to, first, provide a mathematical description of the model and, second, evaluate how variables and their associated uncertainties influence the model’s behaviour. In the latter case, such an evaluation requires propagating the uncertainties of each variable through the data reduction equations (DRE’s) that describe the model. This task can be difficult or inconvenient when the model contains many variables with multiple DRE’s – common for most WECs. While the wave energy community has recently provided general guidance on the principles and use of uncertainty analysis as it relates to WEC experiments [6, 9], treatment of other methods for propagating uncertainty is lacking. In particular, the Joint Committee for Guides in Metrology (JCGM) provides a supplement to the “Guide to the expression of uncertainty in measurement” (GUM)  concerned with the propagation of uncertainty using the Monte Carlo method (MCM) . The MCM is a practical alternative to the GUM uncertainty framework based on the law of propagation of uncertainty. It can be applied to overcome various problems in uncertainty evaluation, for example, when the probability density function (PDF) for the output quantity is not a Gaussian distribution; when a model is arbitrarily complex; or when it is difficult or inconvenient to provide the partial derivatives of the model, as needed by the law of propagation of uncertainty. ### On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method

In this section, we employ the proposed relaxed Monte Carlo method with importance sampling (say RMCIS) to compute the numerical solution of some examples and compare it with their exact solutions. The numerical re- sults are presented in Table 1 and Table 2, where AE means absolute error for ϕ p ( ) ( x p = 1, 2) . We plot the Table 1. Numerical results of Example 1 with k = 5, h = 0.2 and N = 100 . ### 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. ### 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. ### 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. ### The quantum Monte Carlo method : application to problems in statistical physics

Section 3.) of Chapter 2 and the curves presented in Figure 3.5 are the results of superimposing many segments of the random walk trajectory. As discussed in Chapter 2, estimates of the expectation values of the potential and kinetic energy can be obtained from the asymptotic behaviour of the curves presented in Figures 3.5. The values of these quantities are <V> = 2370 ± 50 K/molecule and <T> = 3400 ± 100 K/molecule giving the total ground state energy of the water dimer as <E0 > = 5770 ± 100 K/molecule. To within the statistical fluctuations inherent in these calculations this agrees with the average value of Vref <vref> = 5730 ± 20 K/molecule obtained during the run. As discussed earlier, this agreement provides a self-consistent check of the quantum Monte Carlo method and indicates that the Virial Theorem is satisfied. The eigenvalue estimate calculated from the average energy reference has less statistical uncertainty than the value obtained from the sum of the potential and kinetic energies since the evaluation does not rely on the ( j ) 02 generation procedure. ### A Simple Monte Carlo Method for the Calculation of Efficiency Limit for Current

Monte Carlo method is a tool used to simulate systems with many coupled degrees of freedom. In this technique, a large number of points are scattered uniformly over a defined interval as inputs before computation is done on each point. The method is useful to obtain numerical solutions for problems which is analytically complicated. The most common application of the Monte Carlo method technique is conducted for integration. Apart from this, it is also used to minimize (or maximize) functions of some vector with a significant number of dimensions. To obtain accurate solutions, a large number of stochastic events is needed to be reproduced. Due to this, the decision lead to an increase of computing time . Therefore, several algorithms are suggested to reduce the variance and execution time. Hence, it is suggested that a proper sampling technique can help to improve enormously the performance of a Monte Carlo program. In importance sampling scheme certain values of the random input variables in a simulation have more impact on the parameter that is being estimated than others. Therefore, if these "important" values are emphasized frequentlythrough proper sampling technique more, it is expected that the estimator variance can be reduced. ### Mean exit times and the multilevel Monte Carlo method

4.2. Multilevel Monte Carlo. Giles in  considered the problem of computing some function of the ﬁnal-time SDE solution, E [ f ( X ( T ))], where f is globally Lipschitz, a problem that arises naturally in ﬁnancial option pricing. He showed the remarkable fact that it is possible to reduce the complexity of the standard Euler–Maruyama/Monte Carlo method from O −3 to O −2 (log ) 2 ; see also  for related ideas. Further work has extended ### RISK ASSESSMENT WITH THE USE OF THE MONTE-CARLO METHOD

The developed numerical model was used to calculate the zone of chemical contamination in case of accidental ammonia emission at the pump- ing station located near the Bashmachka settlement (Fig. 1). Based on expert data analysis, it has been determined that ammonia emission can range from 200 to 500 kg. Based on the Monte-Carlo method, it is estimated that, within this range, the highest probability of accident ammonia emission (19%) corresponds to a 300 kg emission. Therefore, this emission was taken for the computational experi- ment. Since there is some inertia at the pump stop, it is clear that the emission will occur within ### The Monte Carlo method to find eigenvalues and eigenvectors

 Bratley, P., Fox, L.B. and Schrage, L.E.: A Guide to Simulation, Springer Verlag, 1987.  Ciuiu, D.: Solving Linear Systems of Equations and Differential Equations with Partial Derivatives by the Monte Carlo Method using Service Systems, Analele UniversităŃii Bucure8ti , (2004), 93 104. ### Lattice switching Monte Carlo method for crystals of flexible molecules

where the primes denote quantities evaluated after the trial move. Evaluating the acceptance probability hence necessi- tates computing the number of overlaps in both lattices after every trial move which does not introduce overlaps in the active lattice. This is easily accomplished with shared memory parallelism over the two synchronized Markov chains. The trial volume or lattice vector moves are biased in a similar fashion. The function η is chosen to achieve uniform sampling in M. In our work we adopt the recursion scheme commonly employed in the Wang-Landau  method to compute this function. ### Applications of the Fixed-node Quantum Monte Carlo Method.

Diffusion quantum Monte Carlo (DMC) is a projector quantum Monte Carlo (QMC) method for solving quantum many-body prob- lems. Its name originates from the formal similarity between the Schrödinger’s equation in imaginary time and a classical diffusion equation. Based on stochastic sampling and evolution of particle conﬁgurations, DMC projects out the ground state of a given sym- metry from any trial wave function with nonzero overlap. In deal- ing with fermions, DMC has to adopt some strategy for overcoming the infamous fermion sign problem which would make it dramat- ically inefﬁcient for many-particle systems. It is common to cir- cumvent the sign problem by forcing the ground state nodes (zero-locus) to be the same as the nodes of the best available trial function. This nodal constraint ensures that the product of the ground state and the trial wave function becomes nonnegative throughout the conﬁguration space. This is known as the ﬁxed- node (FN) approximation [1–3]. ### RISK ASSESSMENT WITH THE USE OF THE MONTE-CARLO METHOD

Purpose. This work involves the development of a numerical model for the calculation of chemical contamina- tion zones in the event of ammonia accident at the pumping station, as well as a model for assessing the risk of dam- age and wound depth in the body in case of fragments scattering formed during the pipeline explosion at the pump- ing station. Methodology. To solve this problem, we used the mass transfer equation for the ammonia propagation in the air. A potential flow model is used to calculate the air flow velocity field in the presence of buildings at the ammonia pumping station. The numerical solution of the three-dimensional equation for the velocity potential is derived by the cumulative approximation method. When using this numerical model, the irregular field of wind flow velocity, the change in vertical atmospheric diffusion coefficient with altitude, the ammonia emission intensity, the ### An introduction to Monte Carlo method for the Boltzmann equation

[r] ### Increase the Throughput of the Shuffle Exchange Networks using Monte Carlo Method

The adaptable TDMA enhanced edge structure in perspective of Monte Carlo unpredictable testing procedure is used. It urges to anticipate a measure of lengths for each source focus in each diversion cycle. The method procedure relies upon randomized testing using assurance between time, occurs end up being more sensible after every amusement cycles. To deal with the blocking issues caused by nonattendance of advantage arranging, we use an improved time division multiple access edge to assign plan openings to source centre points in a notable sort of multiple interconnection networks, Shuffle-exchange orchestrate. Results demonstrate that by utilizing this approach, orchestrate faithful quality moreover, throughput of shuffle exchange networks with time division multiple access increase contemplated to general frameworks. Additional shuffle exchange networks with time division multiple access demonstrates better outcomes diverged from substitute frameworks. Thusly, improve more capable shuffle exchange networks with resource arranging framework. ### Reproducible Estimation of Osmotic Coeﬃcients Using the Inverse Monte Carlo Method

FIG. 1: Convergence of quantities derived from EPP-driven Monte Carlo simulation, as a function of the number of production MC steps attempted, using 200 ion pairs and attempting 8 million equilibration steps. See text for details on the EPPs used. Symbols denote where measurements were taken. Lines are included only as an aid to visualize trends. (a) Osmotic coefficient esti- mate. (b) Average energy per particle. (c) Short-range contribution to the virial. (d) Long-range contribution to the virial. ### Solving Method of Workspace for 6 DOF Robot Based on Vision

Abstract. In order to get the solution of the robot workspace, a method that based on the SimMechanics model in Matlab and the Monte-Carlo method was adopted to solve the robot workspace．Matlab/Simulink software was used to realize movement simulation of robot arms and got a compact workspace. Compared to traditional Monte-Carlo method, this method had the advantages of better speed in solving workspace，strong visibility and avoiding complicated mathematical operations, which provided an important basis for robot's obstacle avoidance planning. ### Monte Carlo Simulation Method To Predict The Charging Load Curve

3.1 The Generation of True Random Numbers Monte Carlo method is a method used to solve physical and mathematical problems by repeated statistical experiments. When addressing problems with Monte Carlo method, solutions are often constructed as mathematical expectations of a certain random variable. This random variable is derived from a hypothetical experiment on certain figures on a computer. The arithmetic mean value of its specific value is used as an approximate solution of the problem. It is worth noting that Monte Carlo Simulation has a high requirement for random numbers. Some pseudo-random numbers may bring about errors in the entire simulation and predicted results. Domestic and foreign scholars generally employ Matlab, Excel and other software to generate random numbers, which are pseudo- random numbers and will cause inaccuracy in some predicted results. Therefore, we use mixed congruential method to generate random numbers and carry out randomness test, to get true random numbers. ### Calibration of dynamic responses in a physiological model

Bell et al. (2008) have developed a model of magnesium dynamics in dairy cattle as part of a programme to develop a tool to assess the risk of hypomagnesaemic tetany in a dairy herd. Biological variation was modelled by implementing selected parameters of the model as distributions, and then using a Monte-Carlo method to generate the distribution of selected model response variables, such as the distribution of CSF and plasma Mg concentration in the simulated herd. The model is able to simulate the dynamic changes in plasma Mg concentration that occur over several days when animals are fed a low Mg diet, as in the experiment of McCoy et al. (2001) shown in Figure 1. 