Academic Report of School of Mathematical Sciences [2025] No. 158

(Series Report for High-Level University Construction No. 1259)

Title：A derivative-free localized stochastic method for very high-dimensional semi-linear parabolic PDEs

Speaker： Bihao Su , Associate Researcher (Hainan University)

Time：10:30-11:30, Dec.26, 2025

Location：Huiwen Building 2433

Abstract：We develop a mesh-free, derivative-free, matrix-free, and highly parallel localized stochastic method for high-dimensional semi-linear 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 ∇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 ∇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 Δt. 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 practical solvability frontier of ultra-high dimensional PDEs but also maintains rigorous error control and ease of implementation.

Speaker Profile：Bihao Su graduated from Shanghai University of Finance and Economics in June 2024 and is currently an Associate Researcher at the School of Mathematics and Statistics, Hainan University. His research focuses on numerical solutions for PDEs and numerical methods for financial models, with a primary emphasis on stochastic simulation algorithms. His work has been published in journals including Mathematics of Computation, SIAM Journal on Numerical Analysis, and SIAM Journal on Scientific Computing.

Faculty and students are welcome to attend!

Invited by: Jingchao Li  

School of Mathematical Sciences

December 25, 2025