  It is one of the most challenging problems in applied mathematics to
approximatively solve high-dimensional partial differential equations (PDEs).
In particular, most of the numerical approximation schemes studied in the
scientific literature suffer under the curse of dimensionality in the sense
that the number of computational operations needed to compute an approximation
with an error of size at most $ \varepsilon > 0 $ grows at least exponentially
in the PDE dimension $ d \in \mathbb{N} $ or in the reciprocal of $ \varepsilon
$. Recently, so-called full-history recursive multilevel Picard (MLP)
approximation methods have been introduced to tackle the problem of
approximately solving high-dimensional PDEs. MLP approximation methods
currently are, to the best of our knowledge, the only methods for parabolic
semi-linear PDEs with general time horizons and general initial conditions for
which there is a rigorous proof that they are indeed able to beat the curse of
dimensionality. The main purpose of this work is to investigate MLP
approximation methods in more depth, to reveal more clearly how these methods
can overcome the curse of dimensionality, and to propose a generalised class of
MLP approximation schemes, which covers previously analysed MLP approximation
schemes as special cases. In particular, we develop an abstract framework in
which this class of generalised MLP approximations can be formulated and
analysed and, thereafter, apply this abstract framework to derive a
computational complexity result for suitable MLP approximations for semi-linear
heat equations. These resulting MLP approximations for semi-linear heat
equations essentially are generalisations of previously introduced MLP
approximations for semi-linear heat equations.
