A Modified LSQR Algorithm And Its Hybrid Algorithm For Solving Large-Scale Linear Discrete Ill-Posed Inverse Problems

Ill-posed inverse problems are widely used in a variety of scientific applications in modern times.Classical regularization methods are effective to compute a stable solu-tion for this kind of problems,but many of these methods are inadequate or insufficient for solving large-scaled problems.The feature, which iterative methods characterize in numerical computing, reflects the advanced advantage for calculating these problems: fast convergence in computing; The matrix is never altered but only "touched" via the matrix-vector products with A and AT; atomic operations,which are simple to par-allelize. All these advantages are very adequate to compute the solution for large-scaled problems. Still, the "semi-convergence" phenomenon obstructs these methods to be utilized in application, since computed solutions are not convergence to the true so-lution in general good way. This work addresses these limitations by developing and implementing the iterative regularization in order to get better computed solutions.We introduce some basic knowledge to these ill-posed problems,especially give more theoretical analysis to LSQR iterative method.In fact, it is an excellent method in fields of classical ill-posed problems and image reconstruction. Through Lanczos bidiagonalization, it shows a fast convergence speed to the true solution, but the be-havior of "semi-convergence" is obvious.The analysis of noise in application and the character of the Lanczos type method in computation,makes us to give an advanced it-erative method based on Lanczos bidiagonalization method, called LSD iteration.It is a good method, that develops and weakens the "semi-convergence" behavior of LSQR method.So we can make the iterative solutions close to the true in more iterative steps. Furthermore, more advanced schemes for LSD method are introduced and discussed, such as:reorthogonalization, Hybrid, etc.At last, the classical numerical examples in ill-posed and image reconstruction problems are illustrated to compare with the LSQR iterative method. By the numerical examples, the performance of the new developed method shows what we wished:effectively, weaken the semi-convergence behavior of LSQR.This gives interesting insight into stabling the semi-convergence of iterative regulariza-tion.This new method can replaced the LSQR method in computing the solutions to discrete ill-posed inverse problems in practice.