Spectral Models of Random Fields in Monte Carlo Methods

01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. Spectral models were developed in the 1970s and have appeared to be very promising for various applications. Nowadays, spectral models are extensively used for stochastic simulation in atmosphere and ocean optics, turbulence theory, analysis of pollution transport for porous media, astrophysics, and other fields of science. The spectral models presented in this monograph represent a new class of numerical methods aimed at simulation of random processes and fields. The book is divided into four chapters, which deal with scalar spectral models and some of their applications, vector-valued spectral models, convergence of spectral models, and problems of optimisation and convergence for functional Monte Carlo methods. Furthermore, the monograph includes four appendices, in which auxiliary information is presented and additional problems are discussed. The book will be of value and interest to experts in Monte Carlo methods, as well as to those interested in the theory and applications of stochastic simulation.

Table of Contents:
Part 1 Approximate modelling of homogeneous Gaussian fields on the basis of spectral decomposition: spectral models of random processes and fields; basic principles of constructing spectral models - generalized scheme, about numerical analysis of the error, examples of spectral models of stationary processes, examples of spectral models for isotropic fields on a plane, spectral models for isotropic fields in three-dimensional space; technique of successive refinement of spectral models on the same probability space; description of the algorithm - auxillary statements and examples, conditional spectral models; statement of the problem - method of solving the problem, on realization of numerical algorithm; specialized models for isotropic fields on a k-dimensional space and on a sphere; models of isotropic fields on a k-dimensional space - spectral models of isotropic fields on a sphere; certain applications of scalar spectral models; simulation of clouds - spectral model of the sea surface undulation; further remarks; nonhomogeneous spectral models - approximate modelling of Gaussian vectors of stationary type by discrete Fourier transform. Part 2 Spectral models for vector-valued fields: spectral representations; spectral representations for complex-valued vector random fields, spectral representations or real-valued vector random fields; isotropy; simulation of random harmonics; complex-valued harmonics - real-valued harmonics, about simulation of complex-valued Gaussian vectors; spectral models of homogeneous Gaussian vector-valued fields; examples of simulation; gradient of isotropic scalar field, solenoidal and potential isotropic random fields, vector-valued isotropic fields on plane and in space. Part 3 Convergence of spectral models of random fields in Monte Carlo methods: conditions of weak convergence in spaces C and Cp; weak convergence in the space of continuous functions, weak convergence of probability measures in space of continuously differentiable functions; convergence of spectral models of Gaussian homogeneous fields; spectral models - weak convergence of spectral models in spaces of continuously differentiable functions. (Part ocntents).