Neufeld and Wu (arXiv:2310.12545) developed a multilevel Picard (MLP) algorithm which can approximately solve general semilinear parabolic PDEs with gradient-dependent nonlinearities, allowing also for coefficient functions of the corresponding PDE to be non-constant. By introducing a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula and identifying the first and second component of the unique fixed-point of the SFPE with the unique viscosity solution of the PDE and its gradient, they proved convergence of their algorithm. However, it remained an open question whether the proposed MLP schema in arXiv:2310.12545 does not suffer from the curse of dimensionality. In this paper, we prove that the MLP algorithm in arXiv:2310.12545 indeed can overcome the curse of dimensionality, i.e. that its computational complexity only grows polynomially in the dimension $d\in \mathbb{N}$ and the reciprocal of the accuracy $\varepsilon$, under some suitable assumptions on the nonlinear part of the corresponding PDE.

翻译：Neufeld与Wu（arXiv:2310.12545）提出了一种多层次Picard（MLP）算法，可近似求解含梯度非线性的广义半线性抛物型偏微分方程，并允许对应偏微分方程的系数函数非常数。通过引入基于Feynman-Kac表示和Bismut-Elworthy-Li公式的特定随机不动点方程（SFPE），并将SFPE唯一不动点的第一、第二分量分别与偏微分方程的唯一粘性解及梯度对应，他们证明了该算法的收敛性。然而，arXiv:2310.12545中提出的MLP架构是否不受维度灾难影响仍是一个开放问题。本文证明，在对应偏微分方程非线性部分满足适当假设的条件下，arXiv:2310.12545中的MLP算法确实能克服维度灾难——其计算复杂度仅随维度$d\in \mathbb{N}$和精度倒数$\varepsilon$呈多项式增长。