On my way to ENUMATH 11 in Leicester I stumbled upon the preprint Multi-parameter Tikhonov Regularisation in Topological Spaces by Markus Grasmair. The paper deals with fairly general Tikhonov functionals and its regularizing properties. Markus considers (nonlinear) operators {F:X\rightarrow Y} between two set {X} and {Y} and analyzes minimizers of the functional

\displaystyle T(x) = S(F(x),y) + \sum_k \alpha_k R_k(x).

The functionals {S} and {R_k} play the roles of a similarity measure and regularization terms, respectively. While he also treats the issue of noise in the operator and the multiple regularization terms, I was mostly interested in his approach to the general similarity measure. The category in which he works in that of topological spaces and he writes:

“Because anyway no trace of an original Hilbert space or Banach space structure is left in the formulation of the Tikhonov functional {T} […], we will completely discard all assumption of a linear structure and instead consider the situation, where both the domain {X} and the co-domain {Y} of the operator {F} are mere topological spaces, with the topology of {Y} defined by the distance measure {S}.”

The last part of the sentence is important since previous papers often worked the other way round: Assume some topology in {Y} and then state conditions on {S}. Nadja Worliczek observed in her talk “Sparse Regularization with Bregman Discrepancy” at GAMM 2011 that it seems more natural to deduce the topology from the similarity measure and Markus took the same approach. While Nadja used the notion of “initial topology” (that is, take the coarsest topology that makes the functionals {y\mapsto S(z,y)} continuous), Markus uses the following family of pseudo-metrics: For {z\in Y} define

\displaystyle d^{(z)}(y,\tilde y) = |S(z,y)-S(z,\tilde y)|.

Unfortunately, the preprint is a little bit too brief for me at this point and I did not totally get what he means with “the topology {\sigma} induced by the uniformity induced by the pseudo-metric”. Also, I am not totally sure if “pseudo-metric” is unambiguous.. However, the topology he has in mind seems to be well suited in the sense that {y^n\rightarrow y} if {S(z,y^n)\rightarrow S(z,y)} for all {z}. Moreover, the condition that {S(z,y)=0} iff {z=y} implies that {\sigma} is Hausdorff. It would be good to have a better understanding on how the properties of the similarity measure are related to the properties of the induced topology. Are there examples in which the induced topology is both different from usual norm and weak topologies and also interesting?

Moreover, I would be interested, in the relations of the two approaches: via “uniformities” and the initial topology…