Spectral Method In A Class Of Inverse Problems Of Mathematical Physics

In this dissertation, we mainly discuss on the application of spectral method to several inverse problems, such as inverse problem in fractional diffusion equation and seismic imaging.In Chapter 1, we introduce some preliminaries on spectral method, e.g., Legen-dre/Chebyshev polynomials and Gauss quadratures. Furthermore, we introduce several lemmas and inequalities in interpolation theory and Sobolev space.In Chapter 2, we first propose a novel spectral method for solving the Volterra type integral equation of the second kind. And we give a rigorous convergence analysis to the new scheme. Numerical results confirm the predicted spectral accuracy. Fur-thermore, we also mention some recent progress on this kind of method. Secondly we propose a new post processing skill based on this newly obtained scheme and apply it to several initial value problems such as ODE, Hamiltonian system and so on. At last of this chapter, we present a regular stability analysis of the post processing skills. With comparison to the existing spectral deferred correction method we found that its stable region is better than the explicit scheme but worse than implicit scheme.In Chapter 3, we introduce the spectral method into fractional diffusion equation. Since the fractional order of derivatives can be viewed as a kind of integral with weakly singular kernel, we firstly consider about the forward problem on fractional order dif-fusion equation. We obtained a generalized maximum principle and discrete maximum principle based on finite difference scheme. However the finite difference scheme can give only algebraic order of accuracy, hence we proposed a spectral method which yields total spectral accuracy both in time and space direction. Secondly we concern on the inverse problem in fractional diffusion equation. We establish a Carleman estimate in a special one dimensional case, i.e., the fractional order is half. With the Carleman estimate we can make a variety of applications, such as the conditional stability and inverse source problem and so on.In Chapter 4, a new regularization approaches based on the spectral method are introduced, which can invert the velocity value and the geometry of the interface simul-taneously. The unknown interfaces are parameterized by Legendre spectral expansion, and various regularization methods combined with traditional regularization parameters selections are utilized to solve the ill-conditioned algebraic equation system. Moreover, a new regularized algorithm with prior choice of regularizing parameters is employed to cut down the storage and computation time.