Solving Fredholm Integral Equations of the Second Kind via Wasserstein Gradient Flows

TL;DR

The paper addresses solving Fredholm integral equations of the second kind on unbounded domains where the unknown is a density. It introduces a KL-regularized objective and derives a Wasserstein gradient flow and an associated McKean--Vlasov SDE to characterize the minimizer, followed by an interacting-particle scheme with Euler--Maruyama discretization to approximate the flow. The authors prove existence, uniqueness, and convergence results for the minimizers as regularization vanishes, establish propagation of chaos for the particle system, and derive practical error bounds to guide particle counts and time steps. Empirically, FE2kind-WGF demonstrates accuracy and stability across Gaussian benchmarks, KL expansions, and GP-SSM equilibria, and compares favorably with RJ-MCMC and Nyström methods, particularly on unbounded domains and in challenging regimes. The framework provides a scalable, adaptive alternative to fixed discretizations, with potential for high-dimensional extensions and applications in spatial statistics and Bayesian inference.

Abstract

Motivated by a recent method for approximate solution of Fredholm equations of the first kind, we develop a corresponding method for a class of Fredholm equations of the \emph{second kind}. In particular, we consider the class of equations for which the solution is a probability measure. The approach centres around specifying a functional whose gradient flow admits a minimizer corresponding to a regularized version of the solution of the underlying equation and using a mean-field particle system to approximately simulate that flow. Theoretical support for the method is presented, along with some illustrative numerical results.

Figures (7)

• Figure 1: Effect of reference measure on reconstruction accuracy as $\alpha$ increases. We compare $\mathop{\mathrm{ISE}}\nolimits$ and $\mathop{\mathrm{MSE}}\nolimits$ of both mean and variance.
• Figure 2: Accuracy of solutions as $\lambda$ increases on a toy Gaussian model. We compare FE2kind-WGF and RJ-MCMC with similar cost through the $\mathop{\mathrm{ISE}}\nolimits$ and the $\mathop{\mathrm{MSE}}\nolimits$ of the estimated variance.
• Figure 3: Comparison of Nyström method and FE2kind-WGF to reconstruct the first eigenfunction of common covariance kernels. Left panel: distribution of $\mathop{\mathrm{ISE}}\nolimits$ ratios ($\mathop{\mathrm{ISE}}\nolimits$ of Nyström method divide by $\mathop{\mathrm{ISE}}\nolimits$ of FE2kind-WGF). Middle and right panel: approximation of the eigenfunction of the largest eigenvalue for the exponential and squared exponential kernel.
• Figure 4: Predictive distribution for a 1-dimensional GP-SSM. Left: distribution recovered by FE2kind-WGF and Nyström. Right: Histogram of $x_{k+1}$ obtained by sampling $x_k$ from the predictive distribution provided by FE2kind-WGF.
• Figure 5: Decay of $\mathbf{W}_{1}$ along iterations for 4 initial distributions: a more diffuse Gaussian $\mathcal{N}(0, 2^2)$, a more concentrated one $\mathcal{N}(0, 0.1^2)$, the target $\mathcal{N}(0, 1^2)$ and a Uniform distribution centered at 0, $\textrm{Unif}[-1, 1]$.

Theorems & Definitions (20)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• Proposition 6