  Max-infinitely divisible (max-id) processes play a central role in
extreme-value theory and include the subclass of all max-stable processes. They
allow for a constructive representation based on the componentwise maximum of
random functions drawn from a Poisson point process defined on a suitable
functions space. Simulating from a max-id process is often difficult due to its
complex stochastic structure, while calculating its joint density in high
dimensions is often numerically infeasible. Therefore, exact and efficient
simulation techniques for max-id processes are useful tools for studying the
characteristics of the process and for drawing statistical inferences. Inspired
by the simulation algorithms for max-stable processes, we here develop theory
and algorithms to generalize simulation approaches tailored for certain
flexible (existing or new) classes of max-id processes. Efficient simulation
for a large class of models can be achieved by implementing an adaptive
rejection sampling scheme to sidestep a numerical integration step in the
algorithm. We present the results of a simulation study highlighting that our
simulation algorithm works as expected and is highly accurate and efficient,
such that it clearly outperforms customary approximate sampling schemes. As a
byproduct, we also develop here new max-id models, which can be represented as
pointwise maxima of general location scale mixtures, and which possess flexible
tail dependence structures capturing a wide range of asymptotic dependence
scenarios.
