Fast Algorithm For Solving The First Category Of Operator Equations

In this thesis, some fast rate convergence algorithms of applying Tikhonov reglarization method to solve the first kind operator equations in dealing with inverse problems, are proposed and discussed both in time domain and frequency domain.The algorithms in time domain are implemented by using a very fast interative scheme presented by the author to determine the regularization parameter, while in frequency domain, the algorithms integrate regularization method with Discrete Fourier Transform to realize the fast reconstruction and extrapolation of band-limited signals.Theoretical analysis and numerical tests on the 1-D and 2-D problems indicate that the new algorithms possess the advantages of high-efficiency, good numerical stability and the inhibitation function against the noise of high frequencies.Meanwhile, under the environment of MATLAB, an applied software, based on the above algorithms, is developed, tested and shown to be a good solver for solving large scale inverse problems.