  Elucidating electrostatic surface potentials contributes to a deeper
understanding of the nature of matter and its physicochemical properties, which
is the basis for a wide field of applications. Scanning quantum dot microscopy,
a recently developed technique allows to measure such potentials with atomic
resolution. For an efficient deployment in scientific practice, however, it is
essential to speed up the scanning process. To this end we employ a
two-degree-of-freedom control paradigm, in which a Gaussian process is used as
the feedforward part. We present a tailored online learning scheme of the
Gaussian process, adapted to scanning quantum dot microscopy, that includes
hyperparameter optimization during operation to enable fast and precise
scanning of arbitrary surface structures. For the potential application in
practice, the accompanying computational cost is reduced evaluating different
sparse approximation approaches. The fully independent training conditional
approximation, used on a reduced set of active training data, is found to be
the most promising approach.
