Alternating Direction Method Of Multipliers For Linear Inverse Problems

In this thesis we propose an iterative method using alternating direction method of multipliers strategy to solve linear inverse problems in Hilbert spaces with a general convex penalty term,we prove that it is an iterative regularization method.We consider two cases according to the structures of the problems and use alternating direction method of multipliers,preconditioned alternating direction method of multipliers to solve the problems respectively.Numerical simulations are performed to test the validity and efficiency of two methodsIn Chapter 1 we introduce the background of inverse problems,the theory and algorithms of regularization methodIn Chapter 2 we review the prior knowledge needed in this paper:optimization methods and iterative regularization methods.Optimization methods include gradi-ent descent method,Uzawa method,augmented Lagrangian method and alternating direction method of multipliers.The iterative regularization methods include Landwe-ber method,semi-iterative method,iterative Tikhonov regularization method,Uzawa method and augmented Lagrangian methodIn Chapter 3 we propose the alternating direction method of multipliers to solve linear inverse problems.When the data is exact,we give a convergence analysis of alternating direction method of multipliers without assuming the existence of a Lagrange multiplier.When the data contains noise,we show that alternating direction method of multipliers is an iterative regularization method as long as it is terminated by a suitable stopping rule.In Chapter 4 we consider the large size problems that have no specific structure,in which case we can't solve the subproblem exactly in an efficient way.We propose the preconditioned alternating direction method of multipliers to overcome this problem Similar to Chapter 3,when the data is exact,we give a convergence analysis of pre-conditioned alternating direction method of multipliers without assuming the existence of a Lagrange multiplier.When the data contains noise,we consider two cases:noise level is either known or unknown.For each case,we propose a stop rule respectively.Moreover we show that the preconditioned alternating direction method of multipliers is also an iterative regularization method.