A Comparison of Stability and Convergence Properties of Techniques for Inverse Problems,
Abstract
This paper demonstrates severe problems in some instances with using an unconstrained algorithm to estimate the parameter q. When modified, either by regularizing the problem using Tikhonov regularization or by constraining the estimate set the algorithm does give good estimates. Unlike the unconstrained algorithm, both the Tikhonov and constrained algorithms are stable with respect to increasing M while holding N fixed. However as N is increased the estimates from the Tikhonov algorithm do not improve as much as do those of the constrained algorithm. The Tikhonov estimates are biased by the regularization of the cost functional, and never show all the detail of q when q has significant variation. Both the constrained and Tikhonov estimation algorithms are stable with respect to systematic errors in the input data, while, except when N is large, the unconstrained algorithm fails to give good results on even the exact data. For both the Tikhonov and constrained algorithms there are parameters which affect the algorithm's performance. For the constrained algorithm suitable constraints must be found while for the Tikhonov algorithm suitable values of a and b must be found. The constrained algorithm has the advantage that the constraints used here, i.e. limits on the slope of q, have an obvious meaning, and so may well be known in advance. In the Tikhonov algorithm b and a have no obvious meaning. They must be suggested by looking at the change in the estimate behavior as b and a change, and perhaps using some apriori knowledge about the shape of q to choose values of b and a that give an estimate that is neither too flat, nor too oscillatory.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1986
- Accession Number
- ADA174733
Entities
People
- D. W. Iles
- H. Thomas Banks
Organizations
- Brown University