Generalized Tikhonov regularization and modern convergence rate theory in Banach spaces (Q2845116)

The subject of this monograph is Tikhonov regularization for nonlinear ill-posed problems \( F(x) = y \) in a general setting, \( T_{\alpha}^z(x) = S(F(x),z) + \alpha \Omega(x) \to \min \) with respect to \( x \in X \). Here, \( F:X \to Y \) is the operator under investigation, where \( X \) and \( Y \) are Hausdorff spaces, and \( \Omega: X \to (-\infty,\infty] \) denotes a stabilizing functional. In addition, \( S: Y \times Z \to [0,\infty] \) denotes some fitting functional which measures the distance between the data \( z \in Z \) and the exact right-hand side \( y \in Y \) of the underlying equation \( F(x) = y \). Here, \( Z \) is another Hausdorff space (the data space).NEWLINENEWLINEPart I of this monograph is devoted to (a) existence of minimizers of \( T_{\alpha}^z \), (b) stability of minimizers of \( T_{\alpha}^z \) with respect to small perturbations of \( \alpha > 0 \) and the data \( z \), (c) convergence of minimizers of \( T_{\alpha}^z \) to an \( \Omega \)-minimizing \(S\)-generalized solution \( x^\dagger \in X \) of \( F(x) = y \) for appropriately chosen \( \alpha \), and (d) convergence rates for a~priori parameter choices and a posteriori parameter choices \( \alpha = \alpha(\delta) \). In the latter case, \( \delta > 0 \) denotes the noise level measured by some general functional, and an element \( x \in X \) is by definition an \(S\)-generalized solution of \( F(x) = y \) if and only if \( S(F(x),z) = 0 \) and \( S(y,z) = 0 \) holds for some \( z \in Z \). The approximation error is measured by some general functional \( E_{x^\dagger}: X \to [0,\infty] \), and smoothness conditions for \( x^\dagger \) are of variational form \( \beta E_{x^\dagger}(x) \leq \Omega(x) - \Omega(x^\dagger) + \varphi(S_Y(F(x),F(x^\dagger))) \) for each \( x \in M \), where \( \beta > 0 \) is some finite constant and \( M \subset X \) is an appropriately chosen set. In addition, \( \varphi: [0,\infty) \to [0,\infty) \) is some given function, and \( S_Y(y_1,y_2) = \inf_{z \in Z} (S(y_1,z) + S(y_2,z)) \) for \( y_1, y_2 \in Y \) is the distance function induced by the fitting functional \( S \). In the final section of Part I, the quantities \( x = \xi(\theta) \) and \( z = \zeta(\theta) \) are considered as results of random processes \( \xi: \Theta \to X \) and \( \zeta: \Theta \to Z \), respectively, where \( (\Theta,P) \) denotes some probability space. It is shown that, if \( x^\dagger \) satisfies a variational smoothness assumption, then the regularized solution \( x_\alpha^z \in \) argmin \( T_\alpha^z \) corresponding to observed data \( z \) is close to \( x^\dagger \) with a certain probability. More precisely, lower bounds for the conditional probability \( P_{\zeta|\xi=x^\dagger}(Z_\alpha^\varepsilon) = \int_{Z_\alpha^\varepsilon} p_{\zeta|\xi=x^\dagger}\, d \mu_Z \) are presented, where \( Z_\alpha^\varepsilon = \{ z \in Z: E_{x^\dagger}(x_\alpha^z) \leq \varepsilon \) for all \( x_\alpha^z \in \) argmin \( T_\alpha^z \} \). In addition, \( p_{\zeta|\xi=x^\dagger} \) is some conditional density function which contains a scaling parameter \( \alpha > 0 \), and \( \mu_Z \) is some measure on \( Z \). Here, a basic tool for the analysis is that maximization of the conditional density function \( p_{\zeta|\xi=x^\dagger} \) on \( X \) is equivalent to Tikhonov-type regularization as considered above, with some specific fitting functional \( S \).NEWLINENEWLINEPart II is devoted to the special case of Poisson distributed data. The resulting Tikhonov functional is considered thoroughly, both in a semi-discrete and a continuous setting, with an analysis of the resulting fitting functional and a derivation of an appropriate variational smoothness assumption, respectively. This part ends up with a comprehensive section containing numerical results.NEWLINENEWLINEIn Part III, the cross-connections between different smoothness concepts like source conditions, approximate source conditions, variational inequalities and approximate variational inequalities are presented. It turns out that each concept can be expressed by a variational smoothness assumption of the form considered in Part~I. The monograph concludes with an appendix containing the necessary mathematical tools from general topology, convex analysis, conditional probability densities and the Lambert \( W \) function.NEWLINENEWLINEThis monograph is an extended version of the author's PhD thesis. It provides a very readable introduction to the topic, including many examples, applications and numerical illustrations.