Determination of particle-size distributions from light-scattering measurement using constrained Gaussian process regression

In this work, we propose a methodology for robustly estimating particle-size distributions from optical scattering measurements using constrained Gaussian process regression. The estimation of particle-size distributions is commonly formulated as a Fredholm integral equation of the first kind, an ill-posed inverse problem characterized by instability due to measurement noise and limited data. To address this, we use a Gaussian process prior to regularize the solution and integrate a normalization constraint into the Gaussian process via two approaches: By constraining the Gaussian process using a pseudomeasurement and by using Lagrange multipliers in the equivalent optimization problem. To improve computational efficiency, we employ a spectral expansion of the covariance kernel using eigenfunctions of the Laplace operator, resulting in a computationally tractable low-rank representation without sacrificing accuracy. Additionally, we investigate two complementary strategies for hyperparameter estimation: a data-driven approach based on maximizing the unconstrained log-marginal likelihood, and an alternative approach where the physical constraints are taken into account. Numerical experiments demonstrate that the proposed constrained Gaussian process regression framework accurately reconstructs particle-size distributions, producing numerically stable, smooth, and physically interpretable results. This methodology provides a principled and efficient solution for addressing inverse scattering problems and related ill-posed integral equations.