  This paper examines issues of data completion and location uncertainty,
popular in many practical PDE-based inverse problems, in the context of option
calibration via recovery of local volatility surfaces. While real data is
usually more accessible for this application than for many others, the data is
often given only at a restricted set of locations. We show that attempts to
"complete missing data" by approximation or interpolation, proposed in the
literature, may produce results that are inferior to treating the data as
scarce. Furthermore, model uncertainties may arise which translate to
uncertainty in data locations, and we show how a model-based adjustment of the
asset price may prove advantageous in such situations. We further compare a
carefully calibrated Tikhonov-type regularization approach against a similarly
adapted EnKF method, in an attempt to fine-tune the data assimilation process.
The EnKF method offers reassurance as a different method for assessing the
solution in a problem where information about the true solution is difficult to
come by. However, additional advantage in the latter approach turns out to be
limited in our context.
