Comparative Analysis of New Unbiased and Biased Monte Carlo Algorithms for the Fredholm Integral Equation of the Second Kind

This paper presents a comprehensive stochastic framework for solving Fredholm integral equations of the second kind, combining biased and unbiased Monte Carlo approaches with modern variance-reduction techniques. Starting from the Liouville-Neumann series representation, several biased stochastic algorithms are formulated and analyzed, including the classical Markov Chain Monte Carlo (MCM), Crude Monte Carlo (CMCM), and quasi-Monte Carlo methods based on modified Sobol sequences (MCA-MSS-1, MCA-MSS-2, and MCA-MSS-2-S). Although these algorithms achieve substantial accuracy improvements through stratification and symmetrisation, they remain limited by the systematic truncation bias inherent in finite-series approximations. To overcome this limitation, two unbiased stochastic algorithms are developed and investigated: the classical Unbiased Stochastic Algorithm (USA) and a new enhanced version, the Novel Unbiased Stochastic Algorithm (NUSA). The NUSA method introduces adaptive absorption control and kernel-weight normalization, yielding significant variance reduction while preserving exact unbiasedness. Theoretical analysis and numerical experiments confirm that NUSA provides superior stability and accuracy, achieving uniform sub-10−3 relative errors in both one- and multi-dimensional problems, even for strongly coupled kernels with ∥K∥L2≈1. Comparative results show that the proposed NUSA algorithm combines the robustness of unbiased estimation with the efficiency of quasi-Monte Carlo variance reduction. It offers a scalable, general-purpose stochastic solver for high-dimensional Fredholm integral equations, making it well-suited for advanced applications in computational physics, engineering analysis, and stochastic modeling.