  This chapter deals with kernel methods as a special class of techniques for
surrogate modeling. Kernel methods have proven to be efficient in machine
learning, pattern recognition and signal analysis due to their flexibility,
excellent experimental performance and elegant functional analytic background.
These data-based techniques provide so called kernel expansions, i.e., linear
combinations of kernel functions which are generated from given input-output
point samples that may be arbitrarily scattered. In particular, these
techniques are meshless, do not require or depend on a grid, hence are less
prone to the curse of dimensionality, even for high-dimensional problems.
  In contrast to projection-based model reduction, we do not necessarily assume
a high-dimensional model, but a general function that models input-output
behavior within some simulation context. This could be some micro-model in a
multiscale-simulation, some submodel in a coupled system, some initialization
function for solvers, coefficient function in PDEs, etc.
  First, kernel surrogates can be useful if the input-output function is
expensive to evaluate, e.g. is a result of a finite element simulation. Here,
acceleration can be obtained by sparse kernel expansions. Second, if a function
is available only via measurements or a few function evaluation samples, kernel
approximation techniques can provide function surrogates that allow global
evaluation.
  We present some important kernel approximation techniques, which are kernel
interpolation, greedy kernel approximation and support vector regression.
Pseudo-code is provided for ease of reproducibility. In order to illustrate the
main features, commonalities and differences, we compare these techniques on a
real-world application. The experiments clearly indicate the enormous
acceleration potential
