  Kernel based methods provide a way to reconstruct potentially
high-dimensional functions from meshfree samples, i.e., sampling points and
corresponding target values. A crucial ingredient for this to be successful is
the distribution of the sampling points. Since the computation of an optimal
selection of sampling points may be an infeasible task, one promising option is
to use greedy methods.
  Although these methods may be very effective, depending on the specific
greedy criterion the chosen points might quickly lead to instabilities in the
computation. To circumvent this problem, we introduce and investigate a new
class of \textit{stabilized} greedy kernel algorithms, which can be used to
create a scale of new selection strategies.
  We analyze these algorithms, and in particular we prove convergence results
and quantify in a precise way the distribution of the selected points. These
results allow to prove, in the case of certain Sobolev kernels, that the
algorithms have optimal stability and optimal convergence rates, including for
functions outside the native space of the kernel. The results also apply to the
case of the usual $P$-greedy algorithm, significantly improving
state-of-the-art results available in the literature. Illustrative experiments
are presented that support the theoretical findings.
