The Ensemble Kalman inversion (EKI), proposed by Iglesias et al. for the solution of Bayesian inverse problems of type $y=A u^\dagger +\varepsilon$, with $u^\dagger$ being an unknown parameter and $y$ a given datum, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose initial empirical measure is sampled from the prior, evolving over an artificial time $t$ towards an approximate solution of the inverse problem. Using spectral techniques, we provide a complete description of the deterministic dynamics of EKI and their asymptotic behavior in parameter space. In particular, we analyze the dynamics of deterministic EKI and mean-field EKI. We demonstrate that the Bayesian posterior can only be recovered with the mean-field limit and not with finite sample sizes or deterministic EKI. Furthermore, we show that -- even in the deterministic case -- residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix.
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Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts.
We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. \"Osterreich. Akad. Wiss. Math.-Natur. Kl. 132 (1995), 3--10]. In particular, we see that the inverse $N(\varepsilon,d)$ of the minimal spherical dispersion is, for fixed $\varepsilon>0$, linear in the dimension $d$ of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse $\widetilde{N}(\varepsilon,d)$, our bounds are optimal with respect to the dependence on $\varepsilon$.
We show that a specific skew-symmetric form of hyperbolic problems leads to energy conservation and an energy bound. Next, the compressible Euler equations is transformed to this skew-symmetric form and it is explained how to obtain an energy estimate. Finally we show that the new formulation lead to energy stable and energy conserving discrete approximations if the scheme is formulated on summation-by-parts form.
The Wiener-Hopf equations are a Toeplitz system of linear equations that naturally arise in several applications in time series. These include the update and prediction step of the stationary Kalman filter equations and the prediction of bivariate time series. The celebrated Wiener-Hopf technique is usually used for solving these equations and is based on a comparison of coefficients in a Fourier series expansion. However, a statistical interpretation of both the method and solution is opaque. The purpose of this note is to revisit the (discrete) Wiener-Hopf equations and obtain an alternative solution that is more aligned with classical techniques in time series analysis. Specifically, we propose a solution to the Wiener-Hopf equations that combines linear prediction with deconvolution. The Wiener-Hopf solution requires the spectral factorization of the underlying spectral density function. For ease of evaluation it is often assumed that the spectral density is rational. This allows one to obtain a computationally tractable solution. However, this leads to an approximation error when the underlying spectral density is not a rational function. We use the proposed solution with Baxter's inequality to derive an error bound for the rational spectral density approximation.
We investigated how neural networks (NNs) understand physics using one-dimensional quantum mechanics. After training an NN to accurately predict energy eigenvalues from potentials, we used it to confirm the NN's understanding of physics from four different aspects. The trained NN could predict energy eigenvalues of a different potential than the one learned, focus on minima and maxima of a potential, predict the probability distribution of the existence of particles not used during training, and reproduce untrained physical phenomena. These results show that NNs can learn the laws of physics from only a limited set of data, predict the results of experiments under conditions different from those used for training, and predict physical quantities of types not provided during training. Since NNs understand physics through a different path than humans take, and by complementing the human way of understanding, they will be a powerful tool for advancing physics.
This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge--Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of explicit RK methods in [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal., 57 (2019), pp. 1158-1182]. The established energy identities provide a precise characterization on whether and how the energy dissipates in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we discover a unified energy identity for all the diagonal Pade approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices. The structure of the matrices is very complicated, rendering the discovery of the unified energy identity and the proof of the decomposition highly challenging. Our proofs involve the construction of technical combinatorial identities and novel techniques from the theory of hypergeometric series. Our framework is motivated by a discrete analogue of integration by parts technique and a series expansion of the continuous energy law. In some special cases, our analyses establish a close connection between the continuous and discrete energy laws, enhancing our understanding of their intrinsic mechanisms. Several specific examples of implicit methods are given to illustrate the discrete energy laws. A few numerical examples further confirm the theoretical properties.
In this paper, we study a non-local approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the non-local problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and non-local problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the non-local problem and that of the local one, both in continuous-time and Backward Euler time discretization. We then turn to studying continuum limits of non-local problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows, then almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.
To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.
This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Application of ensemble Kalman filters, for example use of the ensemble Kalman inversion (EKI) algorithm, has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI). The paper contains two main contributions. Firstly, we identify a novel stochastic dynamical system in which the parameter-to-data map is embedded. We present theory in the linear case to show exponential convergence of the mean of the filtering distribution to the solution of a regularized least squares problem. This is in contrast to previous work in which the EKI has been employed where the dynamical system used leads to algebraic convergence to an unregularized problem. Secondly, we show that the application of the UKI to this novel stochastic dynamical system yields improved inversion results, in comparison with the application of EKI to the same novel stochastic dynamical system.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
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