Monte Carlo And Quasi-monte Carlo Method

1 have studied a couple of topics on Monte Carlo and quasi-Monte Carlo methods. This dissertation covers its applications in integration, optimization and simulation.Chapter 1 and 2 are the basic knowledge to use Monte Carlo and quasi-Monte Carlo methods. Chapter 1 presents the error bounds of Monte Carlo and quasi-Monte Carlo integration methods. By comparing these two methods, we show the advantages of quasi-Monte Carlo method. We also introduce the standard Monte Carlo random search for optimization. The last but not least application is Metropolis algorithms which is the origin of Monte Carlo method. Because the random numbers generators are the key of Monte Carlo methods and quasi-Monte Carlo methods. Chapter 2 describes the pseudo-random number generators and quasi-random number generators. How to generate non-uniform random number from its distributed function is also introduced.Chapter 3 introduces B-spline smoothed rejection sampling method. The standard rejection sampling method which is introduced in chapter 2 is closely related to the problem of quasi-Monte Carlo integration of characteristic functions, whose accuracy may be lost due to the discontinuity of the characteristic functions. We use B-spline smoothing technique to smooth the characteristic function without changing the integral quantity and get a differentiable weight function. The method considerably improves the quality of sampling points. We apply the B-spline smoothed rejection sampling method to importance sampling. Numerical experiments show that the error size O(N-1) is regained by using the B-spline smoothed rejection method for quasi-Monte Carlo estimate. The error bound of Monte Carlo method using B-spline smoothed importance sampling is also better than that of thestandard Monte Carlo method. So the B-spline smoothed rejection sampling method is indirectly proved to be superior to the standard rejection sampling method.Chapter 4 is about the Monte Carlo integration. We get a theoretical error of the fine antithetic variables Monte Carlo(FAMC) method for multidimensional integration. The error size of FAMC is O(N-(1/2+2/s)) for functions having second continuous derivative, where s is the dimension of the integrand. We also give the theoretical error result of antithetic variable Monte Carlo(AMC) method for multi-variable functions whose degree is no more than two. The constant before O(N-1/2) is less than that of the MC method. We realize the parallel algorithm in C for the FAMC and AMC methods. The results of the numerical experiments coincide with the theoretical results very well.Chapter 5 introduces adaptive monte carlo method(AQMC) for global optimization. AQMC algorithm progresses in the nondifferentiablc optimization. First, we develop the local search such that the search direction, search radius and numbei of search points are adjusted according to the previous search result. Second, we introduce the ideas of population and generate new individuals according to population evolution degree. Because the search procedure will be adjusted according to the previous result, the method not only speeds up the random search but also balances the global and local demands (adaptive equalization).Chapter 6 combines the genetic programming with AQMC optimization method to solve the prediction problems. There are many complex systems in real life. In order to analyze, design and predict the system, we often want to model the dynamic systems of ordinary differential equations according to the observed data. We use genetic programming to optimize the the right hand functions of the ordinary differential equations. Adaptive quasi-Monte Carlo optimization methods are used to optimize the coefficients of the functions. The program for the prediction of electrical power consumption of Hangzhou city shows that the hybrid method is powerful.In Chapter 7, we combine the Monte Carlo simulation and optimiza-tion. We first introduce the Monte Carlo simulation of light transport in tissue, explain how to generate...