Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Abstract

Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of over-damped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker-Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker-Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.

Continuous limits for constrained ensemble Kalman filter

“…Proposition 2.8 guarantees that the constrained optimization problem (10) is solved by the unconstrained ensemble Kalman filter when the constraint A is linear, independently on the linearity of the model G. As consequence, analysis and continuous limits of the ensemble Kalman filter with linear equality constraints coincide with the recent results [5,17,21,30,31] for unconstrained problems, and in the following we will focus on nonlinear constraints only.…”

“…Using Proposition 3.3, the following results hold true for (17), in which we recall that the dependence of the algebraic equation on the multipliers is given implicitly by the differential variables, namely A(u j (t)) is in fact A(u j (t; λ j )). Proof.…”

“…Proof. Continuous differentiability of the vector valued function A with respect to the dynamical variable u j gives a sufficient condition to the right hand side of the dynamical equation in (17) to be Lipschitz continuous with respect to u j and λ j , for each fixed j = 1, . .…”

“…We recall that these conditions are also sufficient if the feasible set is convex, see Corollary 2.6. Now we study the large time behavior of the solution of the DAE system (17).…”

“…In particular, in the following we will investigate the Ensemble Kalman Filter (EnKF). While this method has already been introduced more than ten years ago [14], recent theoretical progress [4,5,9,10,11,17,21,30,31] have been done. In order to set up the mathematical formulation of the EnKF we consider to have the finite dimensional case X = R d and Y = R K , with d, K ∈ N.…”