Enhanced Feynman-Kac Formula for Nonlinear Uncertainty Quantification Problems

Uncertainty qualification (UQ) issues and partial differential equations (PDEs) are becoming hot topics in the fields of physic and engineering. There are a various number of strategies for handling unknown linear PDEs, including Monte Carlo and the polynomial regression approach. Nevertheless, nonlinear issues could not be effectively solved using existing approach. This paper focuses primarily on a novel method, an enhanced Feynman-Kac formula with a Least Squares Monte Carlo approach. Numerical experiments were conducted in our research to verify its efficacy and convergence. This paper discovers that the enhanced approach exhibits strong performance regarding nonlinear PDE and UQ scenarios with high accuracy and strong performance of convergence. In the future, this technique might be efficiently applied in the fields of engineering and finance in real-world scenarios.