A vector regularization method to solve linear inverse problems

作者：Liu Chein Shan^*

来源：Inverse Problems in Science and Engineering, 2014, 22(5): 765-786.

DOI：10.1080/17415977.2013.823415

• linear inverse problems
• vector regularization method
• invariant manifold
• vector regularization iterative algorithm
• direct inversion method

The linear inverse problem is discretized to be an n-dimensional ill-posed linear equations system Bx = b. In the present paper, an invariant manifold defined in terms of the square norm of a residual vector r := Bx - b is used to derive an iterative algorithm with a fast descent direction Ar, which is close to, but not exactly equal to, the best descent direction B(-1)r. The matrix A is obtained by using a vector regularization method together with a matrix conjugate gradient method to find the right inversion of B: BA = I-n. The vector regularization iterative algorithm is proven to be Lyapunov stable, and the direct inversion method with solution expressed by x = Ab converges fast. The accuracy and efficiency of them are verified through the numerical tests of linear inverse problems under a large random noise.

• 出版日期2014