Inverse Random Source Problem For Parabolic Equations

With the development of science and technology,a growing number of people have realized that mathematical physical systems inevitably contain random perturbations,thus random terms need to be added into deterministic mathematical models.Therefore,stochastic partial differential equations(SPDEs)become a burgeoning field of computa-tional and applied mathematics.Recently,research on the inverse problem of determin-istic partial differential equations(PDEs)is in full swing,and the corresponding inverse problem of SPDEs is one of the hot spots,among which inverse source problem is an important issue.The theoretical analysis and numerical computations of inverse random source problem for parabolic equations are difficult because of the randomness of models of SPDEs.This dissertation focuses on a typical category of inverse random source prob-lem for parabolic equations,which plays an important role in the detection and control of soil,underwater and atmosphere contamination.This thesis concerns stochastic parabolic equation as follows ut(x,t)+Au(x,t)=f(x)+?Wx,where Wx=?Dr(x,y)Wydy.Given a deterministic function f(x)in the source term and the kernel function r(x,y)of the colored noise Wx,determining u(x,t)is the direct problem.The inverse problem is to simultaneously reconstruct f(x)and r(x,y)from the measurements at the final time.This dissertation considers direct and inverse problems individually.For the direct problem,series expansion of the solution is obtained through eigenfunction method.And it is shown,based on stochastic analysis,that this solution is a unique weak solution,which has some regularity properties and is Holder continuous with regard to temporal and spatial variables respectively through given suitable source term conditions.For the inverse problem,the mean of the measured data is taken to gain the series expansion of f(x),thus achieving uniqueness and conditional stability for exact solution.A numerical solution of f(x)is acquired via truncation method,and error estimations for numerical and exact solutions is computed.Meanwhile,to obtain the series expression of r(x,y),the variance of the measured data is taken.Then uniqueness and conditional stability for exact solution of r(x,y)are achieved.A regularized solution of r(x,y)via truncated regularization method is obtained.Thus error between exact and regularized solutions of r(x,y)is estimated via a-prior and a-posterior regularization pa-rameter selection strategies respectively.One-and two-dimensional numerical examples,are presented to demonstrate the validity of the proposed method.