  The recently introduced full-history recursive multilevel Picard (MLP)
approximation methods have turned out to be quite successful in the numerical
approximation of solutions of high-dimensional nonlinear PDEs. In particular,
there are mathematical convergence results in the literature which prove that
MLP approximation methods do overcome the curse of dimensionality in the
numerical approximation of nonlinear second-order PDEs in the sense that the
number of computational operations of the proposed MLP approximation method
grows at most polynomially in both the reciprocal $1/\epsilon$ of the
prescribed approximation accuracy $\epsilon>0$ and the PDE dimension $d\in
\mathbb{N}=\{1,2,3, \ldots\}$. However, in each of the convergence results for
MLP approximation methods in the literature it is assumed that the coefficient
functions in front of the second-order differential operator are affine linear.
In particular, until today there is no result in the scientific literature
which proves that any semilinear second-order PDE with a general time horizon
and a non affine linear coefficient function in front of the second-order
differential operator can be approximated without the curse of dimensionality.
It is the key contribution of this article to overcome this obstacle and to
propose and analyze a new type of MLP approximation method for semilinear
second-order PDEs with possibly nonlinear coefficient functions in front of the
second-order differential operators. In particular, the main result of this
article proves that this new MLP approximation method does indeed overcome the
curse of dimensionality in the numerical approximation of semilinear
second-order PDEs.
