Numerical methods for ill-posed, linear problems

<p>A means of assessing the effectiveness of methods used in the numerical solution of various linear ill-posed problems is outlined. Two methods: Tikhonov' s method of regularization and the quasireversibility method of Lattès and Lions are appraised from this point of view.</p> <p>In the former method, Tikhonov provides a useful means for incorporating a constraint into numerical algorithms. The analysis suggests that the approach can be generalized to embody constraints other than those employed by Tikhonov. This is effected and the general "T-method" is the result.</p> <p>A T-method is used on an extended version of the backwards heat equation with spatially variable coefficients. Numerical computations based upon it are performed.</p> <p>The statistical method developed by Franklin is shown to have an interpretation as a T-method. This interpretation, although somewhat loose, does explain some empirical convergence properties which are difficult to pin down via a purely statistical argument.</p>