Spectral Models of Random Fields in Monte Carlo Methods

Abstract

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).