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Neufeld and Wu (arXiv:2310.12545) developed a multilevel Picard (MLP)algorithm which can approximately solve general semilinear parabolic PDEs withgradient-dependent nonlinearities, allowing also for coefficient functions ofthe corresponding PDE to be non-constant. By introducing a particularstochastic fixed-point equation (SFPE) motivated by the Feynman-Kacrepresentation and the Bismut-Elworthy-Li formula and identifying the first andsecond component of the unique fixed-point of the SFPE with the uniqueviscosity solution of the PDE and its gradient, they proved convergence oftheir algorithm. However, it remained an open question whether the proposed MLPschema in arXiv:2310.12545 does not suffer from the curse of dimensionality. Inthis paper, we prove that the MLP algorithm in arXiv:2310.12545 indeed canovercome the curse of dimensionality, i.e. that its computational complexityonly grows polynomially in the dimension $d\in \mathbb{N}$ and the reciprocalof the accuracy $\varepsilon$, under some suitable assumptions on the nonlinearpart of the corresponding PDE.

Multilevel Picard approximations overcome the curse of dimensionality in  the numerical approximation of general semilinear PDEs with  gradient-dependent nonlinearities