Abstract. The ensemble Kalman filter (EnKF) is a widely used methodology for state estimation in partial, noisily observed dynamical systems, and for parameter estimation in inverse problems. Despite its widespread use in the geophysical sciences, and its gradual adoption in many other areas of application, analysis of the method is in its infancy. Furthermore, much of the existing analysis deals with the large ensemble limit, far from the regime in which the method is typically used. The goal of this paper is to analyze the method when applied to inverse problems with fixed ensemble size. A continuous-time limit is derived and the long-time behavior of the resulting dynamical system is studied. Most of the rigorous analysis is confined to the linear forward problem, where we demonstrate that the continuous time limit of the EnKF corresponds to a set of gradient flows for the data misfit in each ensemble member, coupled through a common pre-conditioner which is the empirical covariance matrix of the ensemble. Numerical results demonstrate that the conclusions of the analysis extend beyond the linear inverse problem setting. Numerical experiments are also given which demonstrate the benefits of various extensions of the basic methodology.

Based on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite-dimensional, separable Banach spaces proposed in Schwab and Stuart (2012 Inverse Problems 28 045003), we develop a practical computational algorithm whose convergence rates are provably higher than those of Monte Carlo (MC) and Markov chain Monte Carlo methods, in terms of the number of solutions of the forward problem. In the formulation of Schwab and Stuart, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown diffusion coefficients from noisy observations comprising linear functionals of the system's response. The sparsity of the generalized polynomial chaos representation of the posterior density being implied by sparsity assumptions on the class of the prior (Schwab and Stuart 2012), we design, analyze and implement a class of adaptive, deterministic sparse tensor Smolyak quadrature schemes for the efficient approximate numerical evaluation of expectations under the posterior, given data. The proposed, deterministic quadrature algorithm is based on a greedy, iterative identification of finite sets of most significant, 'active' chaos polynomials in the posterior density analogous to recently proposed algorithms for adaptive interpolation (Chkifa et al 2012 Report 2012-NN, 2013. Convergence rates for the quadrature approximation are shown, both theoretically and computationally, to depend only on the sparsity class of the unknown, but are bounded independently of the number of random variables activated by the adaptive algorithm. Numerical results for a model problem of coefficient identification with point measurements in a diffusion problem confirm the theoretical results.

We present an analysis of ensemble Kalman inversion, based on the continuous time limit of the algorithm. The analysis of the dynamical behaviour of the ensemble allows us to establish well-posedness and convergence results for a fixed ensemble size. We will build on the results presented in [26] and generalise them to the case of noisy observational data, in particular the influence of the noise on the convergence will be investigated, both theoretically and numerically. We focus on linear inverse problems where a very complete theoretical analysis is possible. Applicable AnalysisIf the coefficients of the noise are of the size of α k , the right hand side becomes zero and the claim follows.

The ensemble Kalman inversion is widely used in practice to estimate unknown parameters from noisy measurement data. Its low computational costs, straightforward implementation, and non-intrusive nature makes the method appealing in various areas of application. We present a complete analysis of the ensemble Kalman inversion with perturbed observations for a fixed ensemble size when applied to linear inverse problems. The well-posedness and convergence results are based on the continuous time scaling limits of the method. The resulting coupled system of stochastic differential equations allows to derive estimates on the long-time behaviour and provides insights into the convergence properties of the ensemble Kalman inversion. We view the method as a derivative free optimization method for the least-squares misfit functional, which opens up the perspective to use the method in various areas of applications such as imaging, groundwater flow problems, biological problems as well as in the context of the training of neural networks.AMS classification scheme numbers: 65N21, 62F15, 65N75, 65C30, 90C56 for Hilbert spaces (H 1 , ·, · H 1 ), (H 2 , ·, · H 2 ) and z 1 ∈ H 1 , z 2 ∈ H 2 . The empirical means are given bythe minimum of n and the first exit time of e s at radius n. Then, for all n ∈ N, from (14) (after rebasing the integration interval from [0, t] to [s, s + t]) we obtainAs τ n → ∞, applying Fatou's lemma on the left hand side and applying the monotone convergence theorem on the right hand side givesProof. By Lemma Appendix A.4 we can directly take expectations in (14) to obtains 2 ds.Note that by dropping the non-negative mixed terms j = k and by using Jensen's and Young's inequalityProof. The idea of this proof is based on Theorem 4.6.2 in [33]. We define the stochastic Lyapunov functionThe generator applied to V fulfillsis monotonically decreasing.Proof. The assertions follow by arguments similar to the proof of Proposition 4.11.Thus, for all t, s ≥ 0, it follows similarly to the proof of Lemma 4.1 that,...,J converges to zero almost surely as t → ∞. Proof. We define the Lyapunov function V (r, t) = t β 1 J J j=1 |r (j) | 2 and obtain LV (r, t) ≤ βt β−1 J J j=1 |r (j) | 2 − t β 1 J J j=1 r (j) , C(r) + 1 t α + R B r (j) .Thus, LV (r, t) ≤ 1 J J j=1 |r (j) | 2 β − λ min t t α + R t β−1 .

We consider frequency-domain acoustic scattering at a homogeneous star-shaped penetrable obstacle, whose shape is uncertain and modelled via a radial spectral parameterization with random coefficients. Using recent results on the stability of Helmholtz transmission problems with piecewise constant coefficients from [A. Moiola and E. A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Mathematical Models and Methods in Applied Sciences, 29 (2019), pp. 317-354] we obtain frequency-explicit statements on the holomorphic dependence of the scattered field and the far-field pattern on the stochastic shape parameters. This paves the way for applying general results on the efficient construction of high-dimensional surrogate models. We also take into account the effect of domain truncation by means of perfectly matched layers (PML). In addition, spatial regularity estimates which are explicit in terms of the wavenumber k permit us to quantify the impact of finite-element Galerkin discretization using high-order Lagrangian finite-element spaces.