Optimal residual algorithms for linear operator equations

AbstractTraditionally, we measure the quality of an approximation to the solution of a linear operator equation by its error. However, the worst case error is sometimes an unsatisfactory measure of uncertainty, especially for ill-posed problems. In this paper, we propose that the residual be used instead of the error as our measure of uncertainty. We describe optimal information and ask to what extent linear algorithms can be optimal. These results are applied to the ill-posed problem of inverting a finite Laplace transform. In particular, we find that there are instances where there are no finite-residual linear algorithms for this problem, although the problem is convergent; i.e., there are nonlinear algorithms whose residual tends to zero