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：本文针对系数函数不必为常数的含梯度依赖非线性项的一般半线性抛物型偏微分方程，提出了一种多层Picard逼近算法，并给出了算法的完整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，构造了一个特定的随机不动点方程（SFPE）。研究表明，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，该唯一粘性解的梯度存在且等于随机不动点方程唯一解的第二分量。进一步地，我们提供了至多300维的数值算例，验证所提多层Picard算法的实际适用性。