We develop a mesh-free, derivative-free, matrix-free, and highly parallel localized stochastic method for high-dimensional semilinear parabolic PDEs. The efficiency of the proposed method is built upon four essential components: (i) a martingale formulation of the forward backward stochastic differential equation (FBSDE); (ii) a small scale stochastic particle method for local linear regression (LLR); (iii) a decoupling strategy with a matrix-free solver for the weighted least-squares system used to compute $\nabla u$; (iv) a Newton iteration for solving the univariate nonlinear system in $u$. Unlike traditional deterministic methods that rely on global information, this localized computational scheme not only provides explicit pointwise evaluations of $u$ and $\nabla u$ but, more importantly, is naturally suited for parallelization across particles. In addition, the algorithm avoids the need for spatial meshes and global basis functions required by classical deterministic approaches, as well as the derivative-dependent and lengthy training procedures often encountered in machine learning. More importantly, we rigorously analyze the error bound of the proposed scheme, which is fully explicit in both the particle number $M$ and the time step size $\Delta t$. Numerical results conducted for problem dimensions ranging from $d=100$ to $d=10000$ consistently verify the efficiency and accuracy of the proposed method. Remarkably, all computations are carried out efficiently on a standard personal computer, without requiring any specialized hardware. These results confirm that the proposed method is built upon a principled design that not only extends the practically solvable range of ultra-high-dimensional PDEs but also maintains rigorous error control and ease of implementation.

翻译：本文提出了一种针对高维半线性抛物型偏微分方程的、无网格、无导数、无矩阵且高度并行的局部随机方法。该方法的效率建立在四个核心组成部分之上：(i) 前向-后向随机微分方程（FBSDE）的鞅表示；(ii) 用于局部线性回归（LLR）的小规模随机粒子方法；(iii) 采用无矩阵求解器的解耦策略，用于计算 $\nabla u$ 的加权最小二乘系统；(iv) 用于求解 $u$ 中单变量非线性系统的牛顿迭代法。与依赖全局信息的传统确定性方法不同，这种局部计算方案不仅能够显式地逐点评估 $u$ 和 $\nabla u$，更重要的是，它天然适用于跨粒子的并行化。此外，该算法避免了经典确定性方法所需的空间网格和全局基函数，也避免了机器学习中常见的依赖导数和冗长的训练过程。更重要的是，我们严格分析了所提方案的误差界，该误差界在粒子数 $M$ 和时间步长 $\Delta t$ 上都是完全显式的。针对问题维度从 $d=100$ 到 $d=10000$ 的数值实验结果一致验证了所提方法的效率和精度。值得注意的是，所有计算均在标准个人计算机上高效完成，无需任何专用硬件。这些结果证实，所提方法基于一种有原则的设计，不仅扩展了超高维偏微分方程的实际可解范围，同时保持了严格的误差控制和易于实现的特点。