Mathematical models Monte Carlo method

SABRE Method. Acronym for Simulated Approach to Bayesian Reliability Evaluation. An advanced approach to designing a reliability test program developed at PicArsn, the objective of which was to design a test program of minimum sample size for artillery fired atomic projectiles. Called the SABRE method, the program uses mathematical modeling, Monte Carlo simulation techniques, and Bayesian statistics. It is a sophisticated system devised to test items that cannot be tested because of their atomic nature. The aim is to determine the risk factor and to predict what will happen when the projectile is fired... 

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

The method for estimating parameters from Monte Carlo simulation, described in mathematical detail by Reilly and Duever (in preparation), uses a Bayesian approach to establish the posterior distribution for the parameters based on a Monte Carlo model. The numerical nature of the solution requires that the posterior distribution be handled in discretised form as an array in computer storage using the method of Reilly 2). The stochastic nature of Monte Carlo methods implies that output responses are predicted by the model with some amount of uncertainty for which the term "shimmer" as suggested by Andres (D.B. Chambers, SENES Consultants Limited, personal communication, 1985) has been adopted. The model for the uth of n experiments can be expressed by... 

The Monte Carlo method permits simulation, in a mathematical model, of stochastic variation in a real system. Many industrial problems involve variables which are not fixed in value, but which tend to fluctuate according to a definite pattern. For example, the demand for a given product may be fairly stable over a long time period, but vary considerably about its mean value on a day-to-day basis. Sometimes this variation is an essential element of the problem and cannot be ignored. 

When a mathematical model and the variation which affects it6 are both simple in form, it is sometimes possible to derive analytically any desired information relating to the behavior of the system. When this is the case the Monte Carlo method may offer little or no advantage. However, it many problems it is impractical to obtain the desired results entirely by analytical methods. It is in this situation that the Monte Carlo method becomes a most valuable tool. 

The Monte Carlo method subjects a mathematical model to the same 6 The two taken collectively may properly be considered a larger mathematical model. 

Fig. 2. The Monte Carlo method used in studying behavior of a mathematical model.

The Monte Carlo method is especially suited for use on a digital computer, particularly one of the stored-program type. The mathematical model and the distribution function, even if quite complicated, can be expressed on the computer and the necessary calculations are highly repetitive. Also, random numbers (or rather pseudorandom numbers) can be synthesized so that the computer procedure becomes fully automatic and self-contained (M9, S5). 

In summary, a promising new method, REPSWA, has been compared and contrasted to existing techniques. Due to its mathematical structure, REPSWA scales linear with system size and has been shown to perform well in model problems. It can easily be combined with parallel tempering and Hybrid Monte Carlo methods to form interesting and exciting novel sampling schemes. These will be described in future work. 

Science is in incessant evolution it grows with more precise theories and better instrumentation. The thermodynamic theories of polymers and polymeric systems move toward atomistic considerations for isomeric species modeled mathematically by molecular dynamics or Monte Carlo methods. At the same time good mean-field theories remain valid and useful—they must be remembered not only for the historical evolution of human knowledge, but also for the very practical reason of applicability, usefulness, and as tools for the understanding of material behavior. 

Monte Carlo methods (or Monte Carlo experiments) are used to simulate the probability of failure for a slope. Because of many coupled degrees of freedom such as the soil physical characteristics, statistical parameters and mathematical models, the Monte Carlo methods are especially useful to solve such a complicated system. The computational algorithms of the Monte Carlo method seek numerical solution by repeating random sampling. Its formulation is given by... 

The Monte Carlo method therefore simulates by means of a system model an individual sampling value. For evaluation of the results, the known procedures of mathematical statistics can be used. A very important instrument in Monte-Carlo-type simulation is the randomizer. It generates random values within the numerical interval (0,1). The allocation of the random number to a specific value of the random variables is effected via a given distribution function in accordance with Figure 3.20. 

Mathematical models describing formation of novolacs both in batch and continuous reactors have been developed by Frontini et al. [230] and Kumar et al. [231]. They consider the existence of at most one methylol group per molecule, and lump together all isomers with the same unit counts. A Monte Carlo method [232] can also be used in order to obtain a more detailed description at molecular level. 

The mathematical modeling of these polymers are based on either population balances [24-31] or Monte Carlo methods [32-35]. The output of these models includes a detailed description of the polymer architecture (sol MWD, gel fraction, cross-linking points, long chain branching). The models are rather complex and will not be discussed here. Nevertheless, for cases of practical interest, the average molecular weights can be calculated with a modest computational effort [36]. 

As opposed to a conventional numerical solution approach, which would start with the derivation of differential equations describing the mathematical model of the physical system, and then discretising the differential equations to solve a set of algebraic equations for the unknown state of the system, the Monte Carlo methods use random sampling techniques to arrive at a solution of the physical problem. Sometimes it is necessary to use other algebraic methods to manipulate the outcomes of Monte Carlo simulations. 

The biggest advantage of Markov chain Monte Carlo methods is that they allow the applied statistician to use more realistic models because he/she is not constrained by analytic or numerical tractability. Models that are based on the underlying situation can be used instead of models based on mathematical convenience. This allows the statistician to focus on the statistical aspects of the model without worrying about calculability. 

From a theoretical point of view, the Lion et al. model has the merit to approach the DSS effect by applying constitutive laws formulated on the basis of fractional calculus, in other terms by formulating the behavior of materials with respect to fractional time derivatives of stress and strain an approach that in principle requires only a small number of material constants to express the material properties in the time or the frequency domain. However, deriving model parameters from experimental data is not straightforward and, for instance Lion et al. had to use a stochastic Monte Carlo method to estimate the model parameters for a comparison with experimental data on 60 phr CB filled rubber compound. Moreover, mathematical handling of the above equations (see Appendix 5.5) shows that, like the Kraus model, this one exhibits also horizontal symmetry for the G curve and vertical symmetry for the G" curve, and is therefore not expected to perfectly meet experimental data, at least in its present state of development. 

In the next subsection, I describe how the basic elements of Bayesian analysis are formulated mathematically. I also describe the methods for deriving posterior distributions from the model, either in terms of conjugate prior likelihood forms or in terms of simulation using Markov chain Monte Carlo (MCMC) methods. The utility of Bayesian methods has expanded greatly in recent years because of the development of MCMC methods and fast computers. I also describe the basics of hierarchical and mixture models. 

Such Bayesian models could be couched in terms of parametric distributions, but the mathematics for real problems becomes intractable, so discrete distributions, estimated with the aid of computers, are used instead. The calculation of probability of outcomes from assumptions (inference) can be performed through exhaustive multiplication of conditional probabilities, or with large problems estimates can be obtained through stochastic methods (Monte Carlo techniques) that sample over possible futures. 

The second category of papers commonly found in the Journal of Molecular Evolution, accounting for about 5 percent of the total, concerns Mathematical models for evolution or new mathematical methods for comparing and interpreting sequence data. This includes papers with titles such as A Derivation of All Linear Invariants for a Nonbalanced Transversion Model 12 and Monte Carlo Simulation in Phylogenies ... 

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