Adaptive Monte Carlo Methods And Fixed Width Condence Interval

In many fields, such as math, physics and engineer, we need to calculate integrals in multiple dimension spaces, for which conventional numerical methods are not efficient. One of the main reasons is that terrible curse of dimensionality will appear in multiple dimensional problems. According to theory of statistics, these integrals can be presented with expectation of some random variables which can be estimated via Monte Carlo Method or Quasi-Monte Carlo Method. Then result is got with reliable probability. An important idea of the two methods lays on the fact mean of sample will converge to expectation of random variables when sample size is getting bigger. However, there are some defects with the two methods. First, random sampling is based on asymptotic theories which can’t provide an accurate sample size, so there are no evidence to confirm the expectation of random variables dropping in the confidence interval with given probability. On the other hand, based on complexity theory, estimation of multiple dimension integral with deterministic sequences depends on prior information and this information is extremely hard to compute, in spite of accurate error analysis being performed.In this thesis, considering the multi-variable integral, a new adaptive Monte Carlo Method is proposed which have no disadvantages mentioned above. With the assumption of finite kurtosis of integrand, this method constructs accurate con-fidence interval (not asymptotic) of random variables’expectation with the given probability and determines specific Monte Carlo sample size. More specifically, there are two stages in the method. In the first, we get a sample with Cantelli inequal-ity on which upper bound estimation of integrand variance is adaptive to depend. Second, with the result of upper bound got in last stage, confidence interval of ran-dom variables’ expectation is constructed by Berry-Esseen theory. There are two advantages in this method. One is the exact confidence interval of random variables’ expectation is guaranteed under our theory for the given probability, which means more precision achieved. Another merit is expanding this theory to Quasi-Monte Carlo Method is a significantly valuable work. With the assumption of limited bound of integrand kurtosis, convergence efficiently speeds up as well as prior infor-mation needs are reduced by standard error analysis techniques with deterministic sequences. As a result, computing efficiency is improved greatly and computing complexity is reduced. Third, the method is little limited due to the assumption which is satisfied for all random variables whose four moments exist. Furthermore, function space of this assumption is closed with scalar multiplication.The thesis includes four chapters. In the first, the introduction of research background and subject is important part. Next, the new method and its theory are elaborated. Following is examples and simulation results. In the last chapter, we°Summarize and give conclusion as well as outline future work.