Applications Of Wavelet Theory In Solving Ill-Posed Problems

As a new mathematics embranchment, wavelet started at S.Mallat and Y.Meyer's work in the middle and latter half of the 1980s, which was to construct the wavelet base in a general way (multiresolution analysis).Later the wavelet got a drastic development and raised an upsurge in applications, such as signal processing, image analysis, singular check, marginal analysis, numerical solution of differential equations and so on. With the rapid development of wavelet in the field of numerical analysis, an increasing number of mathematical researchers pay attention to the applications of wavelet in solving ill-posed problems.In this thesis, we describes the basic theory of wavelet in detail, and draws the theory of wavelet and GMRES algorithm into solving ill-posed problems to carry out a series of studies in numerical algorithms, which have the abilities of fast convergence and high precision of the procedure. Then, we combine GMRES algorithm with Tikhonov regularization method to solve ill-posed problems, analyze their mutual relationship, and then propose the algorithm of solving the large scale ill-posed problems. Numerical simulations show that the algorithm is effective in programming. Comparing the relative error and absolute error of GMRES method with the wavelet transform method to solve ill-posed problems, numerical simulations indicate the former has higher precision while longer running time than the latter.Based on the two-grid iterative method, the theoretical analysis of Jacobi precondition, symmetric Gauss-Seidel precondition and Schur complement conjugate gradient are studied thoroughly, then we propose the algorithms of symmetric Gauss-Seidel precondition and Schur complement conjugate gradient.So we improve Schur complement conjugate gradient algorithm with the basic theory of wavelet, and then propose the algorithm of wavelet transform applied to solve ill-posed problems. Also, the algorithm is applied to solve the inverse gravity problem and the inverse blackbody radiation problem and compared with some other methods. At last, numerical simulations from different aspects illustrate that, the wavelet transform method adopted in this thesis is effective and feasible in solving ill-posed problems.