Ensemble Kalman methods with constraints

Abstract

Ensemble Kalman methods constitute an increasingly important tool in both state and parameter estimation problems. Their popularity stems from the derivative-free nature of the methodology which may be readily applied when computer code is available for the underlying state-space dynamics (for state estimation) or for the parameter-to-observable map (for parameter estimation). There are many applications in which it is desirable to enforce prior information in the form of equality or inequality constraints on the state or parameter. This paper establishes a general framework for doing so, describing a widely applicable methodology, a theory which justifies the methodology, and a set of numerical experiments exemplifying it.

Continuous limits for constrained ensemble Kalman filter

“…A particular numerical method for solving (1) is the Ensemble Kalman Filter (EnKF), which has been originally introduced to estimate state variables, parameters, etc. of stochastic dynamical systems.…”

“…The estimations are based on system dynamics and measurement data that are possibly perturbed by known noise. Therefore, in order to apply the EnKF to the inverse problem (1), this is usually rewritten as a partially observed and artificial dynamical system based on state augmentation, e.g. cf.…”

“…This is a standard problem in the mathematical literature on the EnKF for inverse problems, e.g. see [1,22,30].…”

“…Inverse problems, in particular in view of a possible illposedness, have been discussed in vast amount of literature and we refer to [12] for an introduction and further references. 1 We are interested in particle methods for solving numerically (1). In particular, in the following we will investigate the Ensemble Kalman Filter (EnKF).…”

On the incorporation of box-constraints for ensemble Kalman inversion

“…This viewpoint allows to straightforwardly include constraints on the parameters and states. In [14], the authors suggest a new approach to handle linear equality and inequality constraints for the EnKF and EKI by reparameterizing the solution of the optimization problem in the range of the covariance, i.e. by seeking the solution of the optimization problem in a subspace defined by the initial ensemble.…”