【作者】 朱平；

【导师】 吴庆标；

【作者基本信息】 浙江大学 ， 计算数学， 2010， 硕士

【摘要】 Quasi-Monte Carlo是基于Monte Carlo产生和发展起来的一种数值模拟算法,是一种基于“随机数”的计算方法,是计算数学,概率统计,运筹学,计算机科学的交叉性,边缘性的数值方法。它的实质是通过对建立的数学模型进行大量随机试验,利用概率论求得原始问题的近似解。拟蒙特卡罗方法求解得到的是确定性的误差,避免了蒙特卡罗方法得到概率误差的缺陷。在过去的十几年时间里,拟蒙特卡罗方法得到了快速发展,构造出了大量优质的低差异序列,如Halton序列,Sobol’s序列,Niderreiter的(t,m,s)网格和(t,s)序列等。它既能求解确定性的数学问题,也能求解随机性的问题。是系统分析和系统设计的强有力工具,已广泛应用于自然科学,工程技术和国民经济等各个领域。随着在工程和金融等社会科学领域的应用,出现了大量在高维情况下求解全局优化和积分等问题,对于这些问题常规的方法由于受维数的影响而不能得到很好的效果。本文主要讨论了拟蒙特卡罗方法在全局优化和无界加权积分问题中的应用。总共分为三部分。第一章主要概述了蒙特卡罗方法和拟蒙特卡罗方法的发展起源和基本思想。第二章阐述了基于多起始点自适应拟蒙特卡罗方法求解全局极值问题。拟蒙特卡罗方法被广泛应用于求解不可微优化问题。本章引进自适应拟随机搜索和多起始点算法的思想来求解全局极值问题；自适应拟随机搜索能够根据前一步的搜索结果自适应地选择下一步搜索方向和步长,将其应用到多起始点算法的主要迭代和完全迭代中,可以较好的平衡全局极值和局部极值问题。为了提高运算效率减少抽样,新的样本将根据主要迭代结果来取代较差个体。相比其它一些优化方法,本文方法具有更好的运算结果。第三章阐述了基于拒绝抽样的拟蒙特卡罗方法求解无界加权积分问题。本章讨论了用拟蒙特卡罗方法求解在积分域左边界存在奇异点的被积函数加权积分问题。对于分布函数可以分解为独立边际分布函数的加权积分,Hlawka Muck给出了收敛条件和H-偏差序列构造方法。我们取消了对分布函数的限制,证明了用拟蒙特卡罗拒绝抽样方法构造H-偏差序列的合理性。由于特征函数的不连续性会降低计算的精确度,我们将磨光技术引进到标准拒绝抽样方法中,在不改变积分值的情况下用B-样条拒绝抽样方法代替标准拒绝抽样,从而达到更高的模拟效果。更多还原

【Abstract】 This dissertation covers the applications of quasi-Monte Carlo methods in global optimization and unbounded weighted integration problems. The disser-tation consists of three chapter.Chapter 1 Introduce the development and main ideal of Monte Carlo and quasi Monte CarloChapter 2 Adaptive quasi-Monte Carlo methods for global optimization prob-lems based on the ideal of multistart.Quasi Monte Carlo method is widely applied in nondifferentiable optimization. We borrow the ideals of adaptive quasi random search and multistart algorithms to solve global optimization problems; Search direction and search step size are adjusted according to previous search result in the adaptive quasi random search method. Using the ideal of adaptive quasi random search for the major iterations and com-plete iteration of multistart algorithms, it’s well to balance the global and local problems. In order to speed up the method, the worse points will be replaced by the new samples according to results of iteration. Comparing with the other methods, our method has better result.Chapter 3 Quasi-Monte Carlo methods for unbounded integration problems based on the ideal of rejection sampling.In this article, we investigate quasi-Monte Carlo methods for weighted integration problems, in which the inte-grand is allowed to be unbounded at the lower boundary of the integration domain. Hlawka Muck proposed a method to generate the H-discrepancy sequences and proved convergence theorems for the distributions with in-dependent marginals, which we will extend to an arbitrary distribution. We proved that it’s sufficient for rejection sampling method in quasi-Monte Carlo to generate H-discrepancy sequences. Considering the accuracy may be lost due to the discontinuity of the characteristic functions, smoothing technique will be used to modify the standard rejection sampling methods. To effectively use rejection sampling method, B-spline rejection sampling methods is described. In the end, the rejection methods will be replaced by B-spline rejection methods without changing the value of integrals to simulate the unbounded weighted integration.更多还原