Question

Problem 9 (for all students) Monte Carlo simulation is often used to analyze probabilities of failure of geotechnical engineering structures. a) For a case, where the limit state function, G, can be described by three normally distributed random variables such that G-Xi-X2x X3, describe a step-by-step methodology (i.e. an algorithm; MatLab code is not needed) that calculates the probability of failure with Monte Carlo simulation. (3 p) I. Il Running the algorithm that you proposed in a), you can visualize the result as a histogram, as shown in the figure (next page). Explain how the probability of failure is visualized in the figure, estimate its value roughly from the figure and state the reasons for your answer. (2 p) In a Monte Carlo simulation, the number of simulations affect the computational time considerably. What is the likely computational result of the probability of failure, if the number of simulations is far too low? Why? (2 p) IlI. b) The probability of failure can also be visualized in a graph with the two random variables on the horizontal axes and probability density in the third direction (toward you). The second figure (next page) illustrates a case where the limit state function, G, is described by a resistance R and a load S. Both random variables are normally distributed. The joint probability density function is indicated through the contour lines (rings), such that the highest probability density is in the smallest ring. Explain in text how the figure illustrates the probability of failure. Indicate also the "failure region" on the figure (submit the figure with your solution) (2 p) I. II. Are R and S correlated? State the reasons for your answer. (1 p) II Redraw the joint probability density function (the rings) in the figure so that the new probability density function corresponds to a case with negativel correlated random variables. Explain in text the new shape of the joint probability density function. (2 p)

Answer 1

Here we identifying the importance of Monte Carlo Method (MCM) in today world.Monte Carlo Methods are tend to be simple, flexible, and scalable. When applied to physical systems, Monte Carlo techniques can reduce complex models to a set of basic events and interactions, opening the possibility to encode model behaviour through a set of rules which can be efficiently implemented on a computer. This in turn allows much more general models to be implemented and studied on a computer than is possible using analytic methods.For example, the complexity of a simulation program for a machine repair facility would typically not depend on the number of machines or repairers involved. Finally, Monte Carlo algorithms are eminently parallelizable, when various parts can be run independently. This allows the parts to be run on different computers and/or processors, therefore significantly reducing the computation time.the MCM has evolved from a “last resort” solution to a leading methodology that permeates much of contemporary science, finance, and engineering. Uses of the MCM Monte Carlo simulation is the generation of random objects or processes by means of a computer.In many cases, however, the random objects in Monte Carlo techniques are introduced “artificially” in order to solve purely deterministic problems.

Here are some typical uses of the MCM: 1 Sampling. Here the objective is to gather information about a random object by observing many realizations of it. An example is simulation modelling, where a random process mimics the behaviour of some real-life system, such as a production line or telecommunications network.The MCM is a powerful tool for the optimization of complicated objective functions. In many applications these functions are deterministic and randomness is introduced artificially in order to more efficiently search the domain of the objective function. Randomness of MCM is its strength.For example, when employed for randomized optimization, the randomness permits stochastic algorithms to naturally escape local optima — enabling better exploration of the search space — a quality which is not usually shared by their deterministic counterparts.In addition, modern statistics increasingly relies on computational tools such as resampling and MCMC to analyze very large and/or high dimensional data sets.We list some important areas of application. • Industrial Engineering and Operations Research. Typical applications involve the simulation of inventory processes, job scheduling, vehicle routing, queueing networks, and reliability systems. The MCM is also used increasingly in the design and control of autonomous machines and robots.Monte Carlo techniques now play an important role in materials science, where they are used in the development and analysis of new materials and structures, such as organic LEDs,organic solar cells and Lithium-Ion batteries. In Economics & Finance as well MCM had proved its potential.. As financial products continue to grow in complexity, Monte Carlo techniques have become increasingly important tools for analysing them. The MCM is not only used to price financial instruments, but also plays a critical role in risk analysis. The use of Monte Carlo techniques in financial option pricing was popularized.MCM has dramatically changed the way in which Statistics is used in today’s analysis of data. The increasing complexity of data (“big data”) requires radically different statistical models and analysis techniques from those that were used 20–100 years ago.By using Monte Carlo techniques, the statistician is no longer restricted to use basic (and often inappropriate) models to describe data. Now any probabilistic model that can be simulated on a computer can serve as the basis for a statistical analysis.MCM samplers construct a Markov process which converges in distribution to a desired high-dimensional density.

Most Monte Carlo techniques have evolved directly from methods developed in the early years of computing. These methods were designed for machines with a single (and at that time, powerful) processor. Modern high performance computing, however, is increasingly shifting towards the use of many processors running in parallel.

It feels that much work remains to be done in Adaptive Monte Carlo Algorithms. Many Monte Carlo algorithms are reflexive in the sense that they use their own random output to change their behavior. Examples of these algorithms include most genetic algorithms and the cross-entropy method. These algorithms perform very well in solving many complicated optimization and estimation problems. However, the theoretical properties of these estimators are often hard or impossible (using current mathematical tools) to study but this doesn't stop MCM to prove its importance in the fields of engineering, finance, statistics and many more....