Interacting Particle Systems for Sampling from Non-Gaussian Targets

Sampling from a target distribution is a fundamental problem in applied mathematics, arising naturally in Bayesian inference and uncertainty quantification. Classical methods such as Markov Chain Monte Carlo can converge prohibitively slowly for complex, high-dimensional targets, and typically require access to gradient information that may be unavailable when the forward model is a black-box numerical solver. The Ensemble Kalman Sampler (EKS) addresses both difficulties, as it is a gradient-free interacting particle algorithm in which particles evolve under an empirical covariance-preconditioned force, making the dynamics affine invariant and well-adapted to the geometry of the target. As the number of particles tends to infinity, the empirical distribution of the EKS converges formally to a nonlinear PDE (the mean-field limit) which can be interpreted as a gradient flow in the Kalman-Wasserstein metric. However, rigorous justification of this mean-field limit has so far been confined to Gaussian or near-Gaussian targets, a restriction that precludes the majority of practical applications. This thesis develops a regularized variant of the EKS mean-field PDE, obtained by replacing the potential and entropy terms with mollified counterparts parameterized by a smoothing scale. This covariance-modulated blob flow inherits the gradient flow structure of EKS while admitting a rigorous analysis for general targets. We prove existence of weak solutions via a covariance-modulated JKO variational scheme, characterize the steady states and their bias, and prove convergence of the mollified solutions to the EKS mean-field PDE as we take the limit of the smoothing scale. Numerical experiments on benchmark targets confirm that the resulting particle algorithm is competitive with EKS in practice.