  The exploration of complex physical or technological processes usually
requires exploiting available information from different sources: (i) physical
laws often represented as a family of parameter dependent partial differential
equations and (ii) data provided by measurement devices or sensors. The amount
of sensors is typically limited and data acquisition may be expensive and in
some cases even harmful. This article reviews some recent developments for this
"small-data" scenario where inversion is strongly aggravated by the typically
large parametric dimensionality. The proposed concepts may be viewed as
exploring alternatives to Bayesian inversion in favor of more deterministic
accuracy quantification related to the required computational complexity. We
discuss optimality criteria which delineate intrinsic information limits, and
highlight the role of reduced models for developing efficient computational
strategies. In particular, the need to adapt the reduced models -- not to a
specific (possibly noisy) data set but rather to the sensor system -- is a
central theme. This, in turn, is facilitated by exploiting geometric
perspectives based on proper stable variational formulations of the continuous
model.
