An optimal tri-vector iterative algorithm for solving ill-posed linear inverse problems

作者：Liu Chein Shan^*

来源：Inverse Problems in Science and Engineering, 2013, 21(4): 650-681.

DOI：10.1080/17415977.2012.717077

• linear inverse problems
• ill-posed linear equations system
• conjugate gradient method
• generalized minimal residual method
• optimal tri-vector iterative algorithm
• invariant-manifold

A future cone in the Minkowski space, defined in terms of the square-norm of the residual vector for an ill-posed linear system to be solved, is used to derive an optimal tri-vector descent system of nonlinear ordinary differential equations (ODEs). Then, a simple Euler scheme is used to generate an iterative algorithm from these ODEs, of which the two parameters appeared are optimized from a properly defined merit function to accelerate the convergence speed in solving the ill-posed linear systems. The optimal tri-vector iterative algorithm (OTVIA) is fast convergent and accurate, which is proven by numerical tests of inverse problems, including the backward heat conduction problem, the Calderon inverse problem and the inverse Cauchy problems. By defining a suitable convergence rate, we assess the convergence speeds of OTVIA and the conjugate gradient method (CGM), which reveal that the performance of OTVIA is better than the CGM. Also by comparing the OTVIA with the generalized minimal residual method (GMRES), we observe that the OTVIA is better than the GMRES.

• 出版日期2013-6-1