From Biased to Unbiased: Theory and Benchmarks for a New Monte Carlo Solver of Fredholm Integral Equations

We investigate biased and unbiased Monte Carlo algorithms for solving Fredholm integral equations of the second kind and for estimating linear functionals of their solutions. Fredholm integral equations provide a common mathematical framework in uncertainty quantification, Bayesian inference, physics, finance, engineering modeling, telecommunication systems, signal processing, and other applied problems where system responses depend on distributed, uncertain, or noise-affected inputs. The comparison covers Crude Monte Carlo and Markov Chain Monte Carlo baselines, modified Sobol quasi–Monte Carlo schemes (MSS variants), the classical Unbiased Stochastic Algorithm (USA), and a new variance-controlled unbiased estimator, the Novel Unbiased Stochastic Algorithm (NUSA). NUSA preserves unbiasedness via a randomized-trajectory representation while improving stability through two mechanisms: adaptive absorption control, governed by a parameter $P_d$ that regulates the effective trajectory length, and kernel-weight normalization based on an auxiliary proposal density to curb heavy-tailed weight products. Extensive experiments in one- and multi-dimensional settings (including regular and discontinuous kernels and weak/strong coupling regimes) show that NUSA consistently reduces dispersion and achieves smaller errors than USA under identical sampling budgets. In representative tests, NUSA attains relative errors below ${10}^{-3}$ and improves average accuracy by approximately 30–50% compared with USA, while maintaining near-linear runtime scaling in N and competitive scaling with dimension. Although NUSA is moderately more expensive per run than USA, the variance reduction yields a superior accuracy–cost trade-off, especially near strong-coupling regimes and in higher dimensions where standard unbiased estimators become variance-limited.