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- ItemStein variational gradient descent: Many-particle and long-time asymptotics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Nüsken, Nikolas; Renger, D. R. MichielStein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: emphvariational inference and emphMarkov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the emphStein-Fisher information (or emphkernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.
- ItemDynamical phase transitions for flows on finite graphs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Gabrielli, Davide; Renger, D. R. MichielWe study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.
- ItemLarge deviations for Markov jump processes with uniformly diminishing rates(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Agazzi, Andrea; Andreis, Luisa; Patterson, Robert I. A.; Renger, D. R. MichielWe prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics.
- ItemGradient flows for bounded linear evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Renger, D. R. Michiel; Schindler, StefanieWe study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambda-convexity.
- ItemAnisothermal chemical reactions: Onsager--Machlup and macroscopic fluctuation theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Renger, D. R. MichielWe study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the large-deviation rate of the microscopic invariant measure. The second result is an application of modern Onsager-Machlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance.
- ItemUntangling dissipative and Hamiltonian effects in bulk and boundary driven systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Renger, D. R. Michiel; Sharma, UpanshuUsing the theory of large deviations, macroscopic fluctuation theory provides a framework to understand the behaviour of non-equilibrium dynamics and steady states in emphdiffusive systems. We extend this framework to a minimal model of non-equilibrium emphnon-diffusive system, specifically an open linear network on a finite graph. We explicitly calculate the dissipative bulk and boundary forces that drive the system towards the steady state, and non-dissipative bulk and boundary forces that drives the system in orbits around the steady state. Using the fact that these forces are orthogonal in a certain sense, we provide a decomposition of the large-deviation cost into dissipative and non-dissipative terms. We establish that the purely non-dissipative force turns the dynamics into a Hamiltonian system. These theoretical findings are illustrated by numerical examples.
- ItemFast reaction limits via $Gamma$-convergence of the flux rate functional(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Peletier, Mark A.; Renger, D. R. MichielWe study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as 1/∈, and we prove the convergence in the fast-reaction limit ∈ → 0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.
- ItemVariational structures beyond gradient flows: A macroscopic fluctuation-theory perspective(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Patterson, Robert I. A.; Renger, D. R. Michiel; Sharma, UpanshuMacroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.
- ItemFast Reaction Limits via Γ-Convergence of the Flux Rate Functional(New York, NY [u.a.] : Springer Science + Business Media B.V., 2023) Peletier, Mark A.; Renger, D. R. MichielWe study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as 1/ϵ, and we prove the convergence in the fast-reaction limit ϵ→0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.