A New Gradient Method for Ill-Posed Problems

ABSTRACT

In this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method.

KEYWORDS:

• Discrepancy principle
• gradient methods
• Landweber iteration
• Linear and nonlinear ill-posed problems
• minimal error method
• steepest descent method
• stopping rule

AMS SUBJECT CLASSIFICATION:

• 65J20
• 65R30
• 47J06
• 47A52