Stochastic Numerics for Mathematical Physics
Stochastic Numerics for Mathematical Physics
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DOI：
10.1007/978-3-662-10063-9
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发表时间：
2004-07
期刊：
影响因子：
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通讯作者：
G. Milstein;G. Milstein;M. Tretyakov;G. Milstein
中科院分区：
文献类型：
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作者：
G. Milstein;G. Milstein;M. Tretyakov;G. Milstein
这本书是一个大幅度修订和扩大版，反映了自2004年第1版[314]出版以来随机数值的主要发展。新的课题包括，特别是，均方和弱近似的情况下，非全局Lipschitz系数的随机微分方程（SDES）;条件概率表示及其应用到实际的方差减少使用回归方法;多级蒙特卡罗方法;计算遍历极限和其他类别的几何积分分子动力学;正倒向随机微分方程的数值方法;抛物型随机偏微分方程的特征线逼近方法。在Chap. 1我们添加了第1.1节。6关于偏微分方程逼近的几乎处处收敛性。我们扩展了（1.4节）中关于偏微分方程非全局Lipschitz系数情形的均方逼近。具有非全局Lipschitz系数且具有唯一解的偏微分方程是一类非常重要的应用。第1.4节包括了一个版本的基本均方收敛定理的随机微分方程与非全局Lipschitz条件和例子显式和隐式格式收敛的非全局Lipschitz的情况下。第2章“弱逼近随机微分方程”的第1版现在分为两章：第一章。2“弱逼近随机微分方程：基础”和第二章。3“随机微分方程的弱逼近：特殊情况”。第二章.基础”包括如何构造弱近似的过程，主要的弱意义收敛定理，以及一般SDES的弱格式的例子。它还包括三个新的部分。新的第2.2节。5通过拒绝爆炸轨迹的概念解决了非全局Lipschitz情形下的偏微分方程的弱意义数值积分问题。由于拒绝“坏”轨迹的概念，我们可以选择一个合适的方法来解决非全局Lipschitz系数的SDES系统，考虑到本书中提出的所有已知的弱方法。新的第2.4节介绍了条件概率表示，并在此基础上，
This book is a substantially revised and expanded edition reflecting major developments in stochastic numerics since the 1st edition [314] was published in 2004. The new topics include, in particular, mean-square and weak approximations in the case of nonglobally Lipschitz coefficients of stochastic differential equations (SDEs); conditional probabilistic representations and their application to practical variance reduction using regression methods; the multi-level Monte Carlo method; computing ergodic limits and additional classes of geometric integrators used in molecular dynamics; numerical methods for forward-backward stochastic differential equations (FBSDEs); approximation of parabolic stochastic partial differential equations (SPDEs) based on the method of characteristics. In Chap. 1 we added Sect. 1.1. 6 on almost sure convergence of SDEs’ approximations. We extended the section (Sect. 1.4) on mean-square approximations in the case of nonglobally Lipschitz coefficients of SDEs. SDEs with nonglobally Lipschitz coefficients possessing unique solutions make up a very important class in applications. Section 1.4 includes a version of the fundamental mean-square convergence theorem for SDEs with nonglobally Lipschitz conditions and examples of explicit and implicit schemes convergent in the nonglobally Lipschitz case.Chapter 2 “Weak approximation of stochastic differential equations” of the 1st edition is now split into two chapters: Chap. 2 “Weak approximation for stochastic differential equations: foundations” and Chap. 3 “Weak approximation for stochastic differential equations: special cases”. Chapter 2 “... foundations” includes the procedure how to construct weak approximations, the main weak-sense convergence theorem, and examples of weak schemes for general SDEs. It also contains three new sections. The new Sect. 2.2. 5 addresses the problem of weak-sense numerical integration of SDEs in the nonglobally Lipschitz case via the concept of rejecting exploding trajectories. Due to the concept of rejecting “bad” trajectories, we can choose a suitable method for solving a system of SDEs with nonglobally Lipschitz coefficients, taking into account all the known weak methods presented in this book. The new Sect. 2.4 introduces conditional probabilistic representations and, based on them, practical