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. Furthermore, we also provide a numerical example in up to $300$ dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.

翻译：本文针对系数函数无需为常数的、具有梯度依赖非线性的一般半线性抛物型偏微分方程，提出了一种多层Picard近似算法。我们还对该算法进行了完整的收敛性与复杂度分析。为获得主要结果，我们基于Feynman-Kac表示与Bismut-Elworthy-Li公式，考虑了一类特定的随机不动点方程。我们证明所研究的偏微分方程具有唯一的黏性解，且该解与随机不动点方程唯一解的第一个分量一致。此外，该偏微分方程唯一黏性解的梯度存在，且与随机不动点方程唯一解的第二个分量一致。最后，我们通过维度高达$300$的数值算例，验证了所提多层Picard算法的实际适用性。