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, if the PDE admits a strong solution, then the gradient of the unique solution of the PDE 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梯度依赖非线性半线性抛物型偏微分方程的多层Picard近似算法，其系数函数无需为常数。我们还对所提算法进行了完整的收敛性与复杂度分析。为了得到主要结果，我们考虑了一个由Feynman-Kac表示和Bismut-Elworthy-Li公式导出的特定随机不动点方程（SFPE）。我们证明所研究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，若该PDE存在强解，则其唯一解的梯度与随机不动点方程唯一解的第二分量一致。