Uncertainty in the solution of linear operator equations (Q797962)

The author introduces and explores a new measure of accuracy for the approximate solution of a linear operator equation \(Lx=y\) in normed spaces. Instead of the error bound \(\| x-\hat x\|\) he defines the uncertainty \(U_{\epsilon}(\hat x)=\sup\{\| z-\hat x\|:z\in S_{\epsilon}\}\) where \(S_{\epsilon}=\{x\in X:\| Lx-y\|\leq \epsilon\}\). This notion seems to be very useful especially in the case of ill-posed problems. Numerical examples demonstrate the applicability of the theory to some practical cases.