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Abstract The Kuramoto model (KM) ofncoupled phase-oscillators is analysed in this work. The KM on a Cayley graph possesses a family of steady state solutions called twisted states. Topologically distinct twisted states are distinguished by the winding number $q\in \mathbb{Z}$. It is known that for the KM on the nearest-neighbour graph, aq-twisted state is stable if $|q|<n/4$. In the presence of small noise, the KM exhibits metastable transitions betweenq–twisted states. Specifically, a typical trajectory remains in the basin of attraction of a givenq-twisted state for an exponentially long time, but eventually transitions to the vicinity of another such state. In the course of this transition, it passes in close proximity of a saddle of Morse index 1, called a relevant saddle. In this work, we provide an exhaustive analysis of metastable transitions in the stochastic KM with nearest-neighbour coupling. We start by analysing the equilibria and their stability. First, we identify all equilibria in this model. Using the discrete Fourier transform and eigenvalue estimates for rank–1 perturbations of symmetric matrices, we classify the equilibria by their Morse indices. In particular, we identify all stable equilibria and all relevant saddles involved in the metastable transitions. Further, we use Freidlin–Wentzell theory and the potential-theoretic approach to metastability to establish the metastable hierarchy and sharp estimates of Eyring–Kramers type for the transition times. The former determines the precise order, in which the metastable transitions occur, while the latter characterises the times between successive transitions. The theoretical estimates are complemented by numerical simulations and a careful numerical verification of the transition times. Finally, we discuss the implications of this work for the KM with other coupling types including non-local coupling and the continuum limit asntends to infinity. 

This paper is concerned with the problem of counting solutions of stationary nonlinear Partial Differential Equations (PDEs) when the PDE is known to admit more than one solution. We suggest tackling the problem via a sampling-based approach. The method allows one to find solutions that are stable, in the sense that they are stable equilibria of the associated time-dependent PDE. We test our proposed methodology on the McKean–Vlasov PDE, more precisely on the problem of determining the number of stationary solutions of the McKean–Vlasov equation. This article is part of the theme issue ‘Partial differential equations in data science’. 

In the study of stochastic systems, the committor function describes the probability that a system starting from an initial configuration x will reach a set B before a set A. This paper introduces an efficient and interpretable algorithm for approximating the committor, called the “fast committor machine” (FCM). The FCM uses simulated trajectory data to build a kernel-based model of the committor. The kernel function is constructed to emphasize low-dimensional subspaces that optimally describe the A to B transitions. The coefficients in the kernel model are determined using randomized linear algebra, leading to a runtime that scales linearly with the number of data points. In numerical experiments involving a triple-well potential and alanine dipeptide, the FCM yields higher accuracy and trains more quickly than a neural network with the same number of parameters. The FCM is also more interpretable than the neural net. 

It has been proposed that classical filtering methods, like the Kalman filter and 3DVAR, can be used to solve linear statistical inverse problems. In the work of Iglesias, Lin, Lu, and Stuart (Commun. Math. Sci. 15(7):1867–1896, 2017), error estimates were obtained for this approach. By optimally tuning a regularization parameter in the filters, the authors were able to show that the mean squared error could be systematically reduced. Building on the aforementioned work of Iglesias, Lin, Lu, and Stuart, we prove that by (i) considering the problem in a weaker norm and (ii) applying simple iterate averaging of the filter output, 3DVAR will converge in mean square, unconditionally on the choice of parameter. Without iterate averaging, 3DVAR cannot converge by running additional iterations with a fixed choice of parameter. We also establish that the Kalman filter’s performance in this setting cannot be improved through iterate averaging. We illustrate our results with numerical experiments that suggest our convergence rates are sharp. 

Abstract We develop an efficient inversion method for thin high contrast scatterers when the contrast is on the order of the reciprocal of the thickness of the scatterer. We extend prior theory for the Helmholtz equation to arbitrary bounded domains and multiple scatterers in two and three dimensions to obtain a fast forward solver with complexity of one dimension lower. The lower-dimensional approximation is then paired with optimization to form the basis for parameter inversion. We show numerical results for the forward and inverse problems in two dimensions and describe extensions to Maxwell.