Iterative Solution of Regularization to Ill-conditioned Problems in Geodesy and Geophysics

Abstract

In geodesy and geophysics, many large-scale over-determined linear equations need to be solved which are often ill-conditioned. When the conjugate gradient method is used, their ill-conditioning effects to the solutions must be overcome, which is studied in this paper. Through the regularization ideas, the conjugate gradient method is improved, and the regularization iterative solution based on controlling condition number is put forward. Firstly by constructing the interference source vector, a new equation is derived with ill-condition diminished greatly, which has the same solution to the original normal equation. Then the new equation is solved by conjugate gradient method. Finally the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface. In the numerical experiments the new method is compared with LS, CG and Tikhonov methods, and its accuracy is the highest.

Keywords

• |ill-condition|regularization|condition number|interference source vector|iteration