The Stochastic Spectral Methods And Its Applications In Stochastic Boundary Problems

In the feld of scientifc computing, the stochastic computations is relativelynew area. In a wide range of practical applications such as fnancial engineering,Biomedicine, Electrical Engineering, it has undergone tremendous growth andrapid development. With the development of the theory of stochastic computa-tions and the hardware of computer, it highlights its importance in the scientifccomputational areas. As we all know, now we always use the defnitive modelswhen we try to do some research on the physical or fnancial problems. However,there exist more or less disturbs from the external cause. How to make a moreaccurate models forces us to make some innovations and correspondingly, thetheory of stochastic diferential equations can obtain dramatic progresses. Inorder to better comprehend the computation of stochastic diferential equation,we organize this thesis as a system of itself and some corresponding mathemat-ics models and numerical results are presented in each chapter. At the end ofthe thesis, we use the theory that is described above to deal with a stochasticboundary problem and obtain a satisfying numerical simulate result.For an easier understanding of the stochastic numerical computation, a briefintroduction of the theory of stochastic equation is presented and we also numer-ically simulate the standard brownian motivation, Karhunen-Loeve expansion,generalized polynomial chaos. In the second chapter, we interpret the forefron-t study results such as stochastic Galerkin method and stochastic collocationmethod respectively and present the numerical results. Under the guidance of the preparation; in the last chapter, we make a complete independent innova-tion. It is well known that the stochastic boundary problem is an importantkind of problem that needs to be dealt with. In this chapter, we employ thetechnique of sparse grid stochastic collocation method to simulate a stochasticboundary problem under the model of acoustic obstacle scattering and obtain avery well simulate result. From the numerical tables, we can fnd that the ad-vantage of sparse grid method over the Monte Carlo method, in which we onlyneed relatively low computational cost to obtain a rather excellent numericalresult.