Functional error estimators for the adaptive discretization of inverse problems (Q2832551)

The paper is concerned with abstract linear inverse problems consisting of the forward model \(Ay=Bu\) and the measurement equation \(Cy=g\). Here, \(u \in U\) is the unknown parameter, \(A\in L(Y,W^*)\), \(B\in L(U,W^*)\), \(C\in L(Y,G)\) are linear operators, \(G\), \(U\), \(W\), \(Y\) are Banach spaces, and instead of the true input data \(g\) an approximation \(g^{\delta}\) is given, \(\| g-g^{\delta}\|_G \leq \delta\). The authors investigate the Tikhonov regularization scheme \(\min_{u\in U, y\in Y, Ay=Bu} \{ \| Cy-g^{\delta}\|_G^2+R_{\alpha}(u) \}\) and its finite-dimensional counterpart, where \(\{ R_{\alpha} \}_{\alpha>0}\) is a family of proper, convex, lower semicontinuous functionals. A posteriori estimates for the closeness of solutions of the original Tikhonov problems to solutions of the discretized problems are derived. The theory is illustrated by an elliptic inverse source problem, numerical results are also presented.