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Evensen., called the prior, was evolved in time by running the model and now is to be updated to account for new data.It is natural to assume that the error distribution of the data is known; data have to come with an error estimate, otherwise they are meaningless.However, the ensemble members are not in general independent except in the initial ensemble, since every En KF step ties them together.They are deemed to be approximately independent, and all calculations proceed as if they actually were independent.So, take a stroll down memory lane to remember all of our past Word of the Year selections.

It is an opportunity for us to reflect on the language and ideas that represented each year.The En KF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting.En KF is related to the particle filter (in this context, a particle is the same thing as ensemble member) but the En KF makes the assumption that all probability distributions involved are Gaussian; when it is applicable, it is much more efficient than the particle filter.Since in reality the values of physical fields at distant locations are not that much correlated, the covariance matrix is tapered off artificially based on the distance, which gives rise to localized En KF algorithms.These methods modify the covariance matrix used in the computations and, consequently, the posterior ensemble is no longer made only of linear combinations of the prior ensemble.Replicate the data Instead of computing the inverse of a matrix and multiplying by it, it is much better (several times cheaper and also more accurate) to compute the Cholesky decomposition of the matrix and treat the multiplication by the inverse as solution of a linear system with many simultaneous right-hand sides.Since the ensemble covariance is rank deficient (there are many more state variables, typically millions, than the ensemble members, typically less than a hundred), it has large terms for pairs of points that are spatially distant.Without loss of generality, the solution is proposed to solve the electric field integral equation (EFIE) in this work.Numerical tests demonstrate the effectiveness of the proposed solution for the electromagnetic analysis, especially for multiscale structures. The convergence rate is determined for Runge-Kutta discretizations of nonlinear control problems.The analysis utilizes a connection between the Kuhn Tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle.