# Top PDF Simulation and the Monte Carlo Method

### Simulation and the Monte Carlo Method

As a rule Monte Carlo methods are not competitive with classical numerical methods for solving systems of linear equations (some special cases where Monte Carlo meth- ods can b[r]

### Reliability Simulation of System with Periodic Test Based on Monte Carlo Method

[1-4] introduce the basic Monte Carlo theory and the Monte Carlo method from two perspectives and gives the corresponding state transition equations. The direct simulation method is based on components and completes the state transition of the system by sampling the transition time (life or repair time) of each component; the indirect simulation method is based on system and uses the sum of all component transition rates as the transition rate of the whole system. Meanwhile, discrete sampling is used to determine the transition component and the transition states. The two methods are applicable in normal circumstances. However, it is more convenient to use direct simulation method when there is correlation between components or the failure rate changes with the states.

### A MONTE CARLO METHOD SIMULATION OF THE EUROPEAN FUNDS THAT CAN BE ACCESSED BY ROMANIA IN 2014-2020

The absence of a classical econometric model of forecasting that can be fully validated, due to the lack of a comprehensive database over an acknowledged minimum of terms needed (e.g. the Durbin–Watson test, which requires a series of data of at least 15 terms, being relevant in this respect) required the authors to build and make use of another solution, i.e. the alternative of simulation using the Monte Carlo method. The practical need may require an estimate, forecast or decision in signifi cant situations of uncertainty, which, according to several opinions and EViews of the scientifi c literature of the last two decades (Jackel, 2002; Glasserman, 2004; Robert & Casella, 2004; Del Moral, Doucet, & Jasra, 2006; Mun, 2006; Creal, 2012) conduces to the implementation of

### Simulation for Callable Convertible Discount Bonds with Monte Carlo Method

of simulation using monte Carlo method. The path simulation is carried out according to the constraint conditions of callable convertible discount bond. The theoretical value is obtained by a large number of experiments. With the increase of convertible bond constraints, the path will become much more complex. The main purpose of this article is to provide some ideas for the pricing of China's convertible bonds.

### Molecular Simulation of Asphaltene Aggregation in Crude Oil by Monte Carlo Method

regions [16]. This model has a large overall area that gives it very distinct aggregation properties. It is reasonable that real asphaltene fractions might have varying degrees of both types of molecules or a significant amount of intermediate structures. In this article, the aggregation behavior of colloidal asphaltene-resin-solvent systems are described and size distribution of asphaltene aggregates are predicted using Monte Carlo simulation. Asphaltene molecules with resin compounds are considered in crude oil. The basis of this method is minimization of total molecular energies in the crude oil. Two different potential functions are used in studying the intermolecular interactions. The effect of oil media on these interactions is considered by introducing some parameters in the applied potential functions.

### Simulation of neutron flux in silicon, cadmium and plumbum using Monte Carlo method

Monte Carlo is a computer programming for solution to radiation transports equation. It is based on random sampling method to solve the mathematical diffusion theory and give the answers in mean of statistical distribution. The randomness of statistical probability of the result is where the Monte Carlo get its name, just like the games of chances in the world’s famous casino. The Monte Carlo method was developed during the World War II at Los Alamos in which the nuclear weapon becomes crucial. It was developed by mathematician, John von Neumann together with Nicholas metropolis, Stan Frankel, Enrico Fermi, Stan Ulam and many others. This method can be used successfully only after the development of electronics computer at time around 1940s and 1950s.

### Torsional path-integral Monte Carlo method for the quantum simulation of molecular systems

In this research, we successfully apply the uncoupled winding number formulation of path integral Monte Carlo theory to the torsional degrees of freedom in the molecules ethane, n-butane, n-octane, and enkephalin. This torsional PIMC technique offers a significant reduction in computational cost for systems in which vibrational degrees of freedom may be safely neglected. Employment of the PIMC m ethod is simplified by the observation th a t contributions to calculated properties will be negligible for winding numbers greater than zero. For a simple ethane model potential, the PIMC result recovers the exact internal energy value obtained with a variational technique. For n-butane, n-octane, and enkephalin, the PIMC converged to the quantum me­ chanical limit with only two or three Trotter beads. All studied molecules exhibited significant quantum mechanical contributions to their internal energy expectation values according to the PIMC technique.

### 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.

### DIRECT SIMULATION OR ANALOG MONTE CARLO

A simple example clarifies the nature of the Monte Carlo method in its simplest unsophisticated form as a simple analog experiment, which can be carried out in close relationship to the physics involved. Suppose one is asked to determine the value of π. Several approaches can be adopted for this purpose, simpler than the Buffon Needle’s approach used by Laplace.

### Bandwidth simulation as a Monte Carlo queuing system

Abstract. One of the issues raised in cloud computing is the dat- acenter locating problem and one of the effective factors in design- ing data center locations is amount of data volume and referrals to it. This depends on the number of customers or users who are going to use that data center, which is a probable issue. In this paper, the bandwidth simulation is described by considering the bandwidth system as a queuing system and simulating it by Monte Carlo method. We explain how to simulate the bandwidth con- sumption in different static and dynamic simulation states for a real computer system and we show that the bandwidth required at an Endpoint in cloud computing can be calculated with different Gamma distribution parameters.

### Monte Carlo Simulation in Radionuclide Therapy Dosimetry

Monte Carlo [MC] method is a modeling tool, capable of achieving a close adherence to reality, concerning the analysis of complex systems. Generally is a method for estimation of the solution of mathematical problems by means of random numbers. MC techniques have been proposed for solving Boltzmann equation by simulating the interaction of par- ticles with matter, using stochastic methods. This method obtains solutions by simulating individual particles and recording some aspects (named tallies) of their average behavior rather than solving an explicit equation. The information required from tallies depends on user’s request. It is generally attributed to scientists working on the nuclear weap- ons in Los Alamos during the 1940s. However, its roots go back much further. The idea of simulation could be attributed to Compte Georges Louis Leclerc de Buffon in 1772. Monte Carlo method was first introduced by Snyder at Oak Ridge National Laboratory in order to assess the fraction of photon and electron energy emitted from radionuclides in source tissues, deposited in various target tissues. This was the concept of absorbed frac- tion defined within the later conceived Medical Internal Radiation Dose (MIRD) method of Internal Dosimetry.

### Exact Monte Carlo simulation of killed diffusions

Since we don’t know the law at time T of the killed diﬀusion it is clear that the explicit computation of ν is not possible and we resort to Monte Carlo methods to estimate ν . In this simulation study, we investigate the performance of the estimator of ν produced by the Exact Monte Carlo method (hereafter E 1). The plots in Figure 1 propose a comparison between E 1 and the estimators based on the continuous Euler scheme ( E 2) and on the discrete Euler scheme ( E 3). In particular, given a Monte Carlo sample suﬃciently large (10 6 ), for dif- ferent choices of the starting point y 0 and the barriers’ values a and b , we have computed the estimates of E 1 (dotted line) and the estimates produced by E 2 and E 3 for diﬀerent discretization intervals. Then we have plotted the values of E 2 (cross) and E 3 (circle ) versus the number of discretization intervals. As we expected, the values of E 2 and E 3 converge to E 1 as the number of discretization interval increases. Indeed it was shown by Gobet (2000) that, for killed diﬀusions, the weak approximation error of Euler schemes decreases to 0 as the number of discretization intervals increases. When the Monte Carlo sam- ple size is large enough, Monte Carlo error is negligible and the estimated values are aﬀected mainly by the discretization error. In this context the distance be- tween the values of E 2 and E 3 and the dotted line is a good representation of the (weak) discretization error aﬀecting the Euler schemes and their conver- gence to the dotted line reﬂects the theoretical convergence of the corresponding expected values. Furthermore, according to the conclusions of Gobet, we notice that the estimates based on the continuous Euler scheme show better conver- gence than the estimates based on the discrete Euler scheme.

### Reliability of chatter stability in CNC turning process by Monte Carlo simulation method

In general, mechanical reliability refers to the ability of mechanical products to complete the specified function in the exact required service period, and it can be a measurable value. The two-common method for measuring the reliability are the moments method and Monte Carlo simulation. Comparing Monte Carlo simulation with moment method, the former one has the advantage of high universality and high precision [11]. However, Monte Carlo method is based on a large number of simulation and calculations, which is not suitable for predicting the reliability of complex numerical models as finite element method. As the case in this study does not involve complicated numerical model, Monte Carlo simulation is an appropriate tool to be used here, since numerical control cutting width is an explicit expression. Therefore, the chatter stability of the CNC turning operation was predicted and compared in the term of cutting width. Thus, the reliability of the turning system stability can be described as a multidimensional integral, given by:

### A Monte-Carlo Method for Optimal Portfolios

Cet article établit des résultats nouveaux sur (i) la structure des portefeuilles optimaux, (ii) le comportement des termes de couverture et (iii) les méthodes numériques de simulation en la matière. Le fondement de notre approche repose sur l'obtention de formules explicites pour les dérivées de Malliavin de processus de diffusion, formules qui simplifient leur simulation numérique et facilitent le calcul des composantes de couverture des portefeuilles optimaux. Une de nos procédures utilise une transformation des processus sous- jacents qui élimine les intégrales stochastiques de la représentation des dérivées de Malliavin et assure l'existence d'une approximation faible exacte. Cette transformation améliore alors la performance des méthodes de Monte-Carlo lors de l'implémentation numérique des politiques de portefeuille dérivées par des méthodes probabilistes. Notre approche est flexible et peut être utilisée même lorsque la dimension de l'espace des variables d'états sous-jacentes est large. Cette méthode est appliquée dans le cadre de modèles bivariés et trivariés dans lesquels l'incertitude est décrite par des mouvements de diffusion pour le prix de marché du risque, le taux d'intérêt et les autres facteurs d'importance. Après avoir calibré le modèle aux données nous examinons le comportement du portefeuille optimal et des composantes de couverture par rapport aux paramètres tels que l'aversion au risque, l'horizon d'investissement, le taux d'intérêt et le prix de risque du marché. Nous démontrons l'importance des demandes de couverture. L'aversion au risque et l'horizon d'investissement émergent comme des facteurs déterminants qui ont un impact substantiel sur la taille du portefeuille optimal et sur ses propriétés économiques.

### Distributed Monte Carlo Simulation

5.2 Future work We now suggest possible improvements to the current system. In the current system, simulation performed for each input job is independent i.e., if two users submit jobs with same input parameters the system cannot identify that the two jobs are alike and performs the simulation twice, once for each of the jobs. To minimize such unnecessary use of the computing resources and to make the system smart, data mining and pattern recognition techniques can be used to identify and categorize jobs. Using these techniques, we can add capabilities to the system such that it checks if the input parameters of a new job exactly match with the input parameters of an existing job (which is already executed or is in the process of execution), and then provide the results without actually performing the simulation for the new job. The other case is when the input parameters of new job partially match the parameters of the existing jobs. In this case, the system can take the available SNR points data and simulate for the SNR points for which data is not readily available in the system. Only when the above two cases fail, simulation for the job can be performed, thus saving the computing resources.

### OPTION PRICING USING MONTE CARLO SIMULATION

plan 2008; Jia 2009). Monte Carlo simulation gen- erates a sample by drawing from a hypothesised ana- lytical distribution. One of the biggest advantages is that successive replications generate a collection of samples with the same distributional properties as the original data (Everaert 2011; Gitman 2009). Though, there are some disadvantages too, as results depend on whether the distributional assumption is correct, there is a slow rate of convergence, it is very time-consuming and computationally intensive. Moreover, Monte Carlo simulation is attractive rela- tive to other numerical techniques because it is flex- ible, easy to implement and modify, and the error convergence rate is independent of the dimension of the problem (Charnes 2000). Since the convergence rate of Monte Carlo methods is generally independ- ent of the number of state variables, it is clear that they become viable as the underlying models (asset prices and volatilities, interest rates) and derivative contracts themselves (defined on path-dependent functions or multiple assets) become more compli- cated (Fu et al. 2001; Jia 2009). A key specification in Monte Carlo simulations is the probability distri- butions of the various sources of risk. The implica- tions of different investment policy decisions can be assessed through simulated time. In addition, Monte Carlo simulation is widely used to develop estimates of Value at Risk (DeFusco et al.. 2001). This method- ology simulates many times the profit and loss per- formance of the portfolio over a specified horizon. Boyle (1977) was the first one who proposed a Mon- te Carlo simulation approach for European option valuation. The method is based on the idea that sim- ulating price trajectories can approximate probabil- ity distributions of terminal asset values. Option cash flows are computed for each simulation run and then averaged. The discounted average cash flow using the risk free interest rate represents a point estimator of the option value.

### Atomistic Monte Carlo simulation of lipid membranes

bilayer arrangement can be efficiently simulated. Our starting structure is a crystalline-like DPPC bilayer with straight fatty acyl chains and identical conformations for the 64 lipid molecules that comprise our system (see Section 2.2 and Figure 5A). We wanted to determine whether our local-move MC technique is able to produce conformational moves large enough to create a structural transition from a crystalline to a fluid-like state in the bilayer. As shown in Figure 5B, already, after 10,000 MC steps, the high molecular order typical for the crystal-like structure is lost and acyl chains of individual PC molecules show large conformational differences. Individual molecules are tightly packed, which is due to the fact that the box size in the NPT-ensemble simulation is rapidly adjusted (see also the accompanying video sequence). After 20,000 MC cycles, fatty acyl chains became more disordered (Figure 5C), while after 40,000 MC steps, the acyl chains of the phospholipids show large structural variation compared to the starting configuration. These changes indicate that the bilayer system made a transition to a fluid lipid bilayer. The simulated membrane model system is driven in a state of slight undulations, as suggested by the wave-like appearance of the head group regions (Figure 5D). This mesoscopic organisation has previously been described for long MD simulations of fluid DPPC membranes [25]. However, due to the simple solvent representation used in our MC simulation, we cannot rule out that these phenomena are caused or at least influenced by the simple solvent description. Starting from the crystalline-like ordered structure shown in Figure 5A, we determined next whether our MC algorithm leads to equilibration of the DPPC bilayer in terms of system enthalpy. As shown in Figure 5E, the system enthalpy drops to the equilibrium value in less than 10,000 MC steps, which are simulated in about one day on a Pentium PC. Thus, no energy minimization of the bilayer is required as often performed prior to extensive MD simulations (see [114] as example). A plateau value around −250 kcal/mol is obtained, which pertains stable during the simulation run. The PDF of the system enthalpy, p k (H), is well approximated by a Gaussian function with mean −256.7 and SD = 54.1 kcal/mol (Figure 5F) [112]. Using again the relation between fluctuation in energy (or enthalpy, as the membrane simulation was done in the constant NPT-ensemble; see section 2.2, above) and the heat capacity, we can derive a value of c p = 821.63

### Multilevel Metric Invariance: A Monte Carlo Simulation

Chapter 3: Method Data Generation Two-level multiple-group multivariate normal data were generated in Mplus 7.1 (Muthén & Muthén, 1998-2011). A grouping variable was generated at level-two to represent treatment versus control classrooms. The individual-level and cluster-level factor structures were identical. That is, invariance was assumed between level-one and level-two, regardless of the grouping variable. The two groups had identical population parameters except for the factor loading parameter of noninvariance. The current study generated data for three separate models, referred to as Replication Study, Extension Study 1, and Extension Study 2 (see Figures 1 – 3).