In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation.

翻译：本文提出了一种适用于一般半线性抛物型偏微分方程的多层Picard近似算法，该类方程具有梯度依赖非线性项，且其系数函数不要求为常数。我们还对该算法进行了完整的收敛性与复杂性分析。为获得主要结果，我们基于Feynman-Kac表示与Bismut-Elworthy-Li公式，引入了一个特定的随机不动点方程。我们证明了所考虑的偏微分方程存在唯一的粘性解，该解与随机不动点方程唯一解的第一分量一致。此外，该偏微分方程唯一粘性解的梯度存在，且与随机不动点方程唯一解的第二分量一致。