Necessary conditions for variational regularization schemes (Q2861889)

The authors study in a general setting variational schemes for ill-posed problems, with a focus on necessary conditions for the Tikhonov method to be a regularization scheme. We present some details, and for this purpose consider a continuous mapping \( F:X \to Y \), where \( (X,\tau_X) \) and \( (Y,\tau_Y) \) are topological spaces. In addition, let \( \rho: Y \times Y\to [0,\infty] \) be a discrepancy functional which may be non-metric, in general, and \( R: X \to [0,\infty] \) denotes a stabilizing functional. The paper starts with a comparison of three variational methods for the regularization of an ill-posed problem \( F(x)= y^{\mathrm{exact}}\): the Tikhonov regularization \( T_{\alpha,y}(x) = \rho(F(x),y) + \alpha R(x) \to \min \), the Ivanov regularization \( \rho(F(x),y) \to \min \) subject to \( R(x) \leq \tau \), and the Morozov regularization \( R(x) \to \min \) subject to \( \rho(F(x),y) \leq \delta \). Here, minimization is always considered with respect to \( x \in X \). In addition \( \alpha > 0 \) and \( \tau > 0 \) holds, and \( \delta > 0 \) denotes the noise level for the right-hand side of \( F(x) = y^{\mathrm{exact}}\). The authors then present a definition for the Tikhonov functional \( T_{\alpha,y} \) to be a regularization scheme. This definition contains existence, stability and convergence conditions. The convergence part makes use of a sequential convergence structure \( \mathcal{S}\) on \( Y \) which maps, by definition, any element \(y \in Y \) to the set of sequences in \( Y \) converging to \( y \). In a theorem, conditions are given which guarantee that the Tikhonov functional \( T_{\alpha,y} \) is a regularization scheme in the sense indicated above. One of those conditions states that convergence \( y_n \to y \) with to respect to \( \mathcal{S} \) is equivalent to \( \rho(y_n,y) \to 0 \), and a continuity condition is also considered. Subsequently, the relations between the topological structure, the sequential convergence structure \( \mathcal{S}\), and the convergence and continuity conditions from the theorem considered above are investigated. Finally, the results are applied to discrepancy functionals \( \rho \) generated by Bregman distances and the Kullback-Leibler divergence, respectively.