Summary: This article addresses the problem of compensating for discretization errors in inverse problems based on partial differential equation models. Multidimensional inverse problems are by nature computationally intensive, and a key challenge in practical applications is to reduce the computing time. In particular, a reduction by coarse discretization of the forward model is commonly used. Coarse discretization, however, introduces a numerical model discrepancy, which may become the predominant part of the noise, particularly when the data is collected with high accuracy. In the Bayesian framework, the discretization error has been addressed by treating it as a random variable and using the prior density of the unknown to estimate off-line its probability density, which is then used to modify the likelihood. In this article, the problem is revisited in the context of an iterative scheme similar to ensemble Kalman filtering, in which the modeling error statistics is updated sequentially based on the current ensemble estimate of the unknown quantity. Hence, the algorithm learns about the modeling error while concomitantly updating the information about the unknown, leading to a reduction of the posterior variance.