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.
\r\n\r\nThis 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.
", "doi": "10.7907/jkfa-cc97", "publication_date": "2026", "thesis_type": "senior_major", "thesis_year": "2026" }, { "id": "thesis:18721", "collection": "thesis", "collection_id": "18721", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05312026-165125878", "primary_object_url": { "basename": "Conger_Thesis_Final.pdf", "content": "final", "filesize": 5714083, "license": "other", "mime_type": "application/pdf", "url": "/18721/1/Conger_Thesis_Final.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Analysis of Large-Scale Systems: Coupled Multispecies Gradient Flows and Distributed Control", "author": [ { "family_name": "Conger", "given_name": "Lauren Elaine", "orcid": "0000-0002-2151-6044", "clpid": "Conger-Lauren-Elaine" } ], "thesis_advisor": [ { "family_name": "Hoffmann", "given_name": "Franca", "orcid": "0000-0002-1182-5521", "clpid": "Hoffmann-Franca" }, { "family_name": "Mazumdar", "given_name": "Eric V.", "orcid": "0000-0002-1815-269X", "clpid": "Mazumdar-Eric" }, { "family_name": "Doyle", "given_name": "John Comstock", "orcid": "0000-0002-1828-2486", "clpid": "Doyle-J-C" } ], "thesis_committee": [ { "family_name": "Wierman", "given_name": "Adam C.", "orcid": "0000-0002-5923-0199", "clpid": "Wierman-A-C" }, { "family_name": "Hoffmann", "given_name": "Franca", "orcid": "0000-0002-1182-5521", "clpid": "Hoffmann-Franca" }, { "family_name": "Mazumdar", "given_name": "Eric V.", "orcid": "0000-0002-1815-269X", "clpid": "Mazumdar-Eric" }, { "family_name": "Craig", "given_name": "Katy", "orcid": "0000-0001-6085-4022", "clpid": "Craig-Katy" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Large-scale multi-agent systems — from autonomous vehicle fleets and power grids to biological populations and machine learning algorithms — have grown too vast and heterogeneous for individual-level modeling or centralized control. This thesis develops new mathematical tools from two complementary perspectives: one that aggregates agents into evolving probability distributions and analyzes their collective behavior, and one that designs distributed controllers acting directly on individual components of large dynamical systems.
\r\n\r\nIn the first approach, we treat the joint distribution of n agent populations as evolving to minimize a collection of coupled cost functionals, yielding a coupled system of partial differential equations. The natural analytical framework is that of gradient flows in metric spaces — the infinite-dimensional analogue of gradient descent. While gradient flow theory is well-established for a single species, the multispecies setting presents new difficulties: the joint dynamics do not necessarily have a gradient flow structure, the natural candidate for a steady state is a Nash equilibrium rather than a minimizer, and strong displacement convexity of each individual energy is generally insufficient for convergence. We introduce λ-monotonicity for n-species systems in the Wasserstein-2 metric space, extending the game-theoretic notion of monotonicity from Euclidean space to the measure-valued setting. Under this condition, we prove exponential convergence to a unique steady state — which, when the dynamics arise from coupled gradient flows, is the unique Nash equilibrium of the associated n-player game. For existence of solutions in general metric spaces, we introduce the Variational Movement Scheme (VMS), a fully implicit discrete-time update whose solution is a Nash equilibrium at each step, and prove existence via a zeroth-order monotonicity condition combining barycentric κ-convexity and η-interaction dissipativity. Taking the discrete time step to zero yields a continuous evolution variational inequality and a contraction estimate at rate λ = κ − η. These results establish foundational multispecies gradient flow theory: existence of solutions and long-time behavior.
\r\n\r\nIn the second approach, we shift from a global to local perspective to design controllers for individual components of large-scale dynamical systems. Building on the System Level Synthesis (SLS) framework — which reparameterizes the controller via closed-loop maps (CLMs) from disturbances to states and inputs — we develop tools for distributed, globally optimal constrained control. First, we characterize controllability and observability under structural constraints (spatial locality, communication delays, actuation limits) through a constrained Gramian and controllability volume. We establish explicit rank conditions under which structural constraints incur no performance loss: locality and communication constraints can satisfy these conditions, while actuation delays monotonically reduce the reachable volume. Second, when system dynamics are unknown, we establish the first non-asymptotic convergence rate for set membership estimation (SME) under general convex disturbance supports, proving diameter convergence with rate Õ(1/T), which improves on the Õ(1/√T) rate of least squares estimation in sample complexity. Third, we extend SLS to infinite-dimensional evolution equations over Hilbert spaces, establishing weak-form CLMs, an operator-theoretic system level parameterization, and an equivalence between CLMs and linear feedback controllers — enabling an optimize-then-discretize paradigm for PDE control that avoids the state-space explosion of the standard discretize-then-optimize approach. We include a biology-inspired example that shows how our method can provide better control than spatial discretization. These results build a toolset for implementing distributed, local controllers with rigorous stability guarantees.
", "doi": "10.7907/vy1e-w250", "publication_date": "2026", "thesis_type": "phd", "thesis_year": "2026" } ]