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 Pd 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.
Todorov et al. (2026) studied this question.