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.However, maintaining the covariance matrix is not feasible computationally for high-dimensional systems. En KFs represent the distribution of the system state using a collection of state vectors, called an ensemble, and replace the covariance matrix by the sample covariance computed from the ensemble.
This can be alleviated by regularization, such as penalization of states with large spatial gradients.
For problems with coherent features, such as hurricanes, thunderstorms, firelines, squall lines, and rain fronts, there is a need to adjust the numerical model state by deforming the state in space (its grid) as well as by correcting the state amplitudes additively.
In Data Assimilation by Field Alignment, Ravela et al.
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The ensemble Kalman filter (En KF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models.