Kinetic Methods for Inverse Problems
For researchers in inverse problems and filtering, this work provides a theoretical foundation and computational efficiency improvement for ensemble methods, though the results are incremental and lack concrete numerical comparisons.
The authors study the mean-field limit of the Ensemble Kalman Filter for inverse problems, deriving a kinetic equation that enables stability analysis and proposing a stable modification leading to a Fokker-Planck-type equation. The kinetic methods reduce computational complexity in the large-ensemble limit, as illustrated with literature examples.
The Ensemble Kalman Filter method can be used as an iterative numerical scheme for parameter identification or nonlinear filtering problems. We study the limit of infinitely large ensemble size and derive the corresponding mean-field limit of the ensemble method. The solution of the inverse problem is provided by the expected value of the distribution of the ensembles and the kinetic equation allows, in simple cases, to analyze stability of these solutions. Further, we present a slight but stable modification of the method which leads to a Fokker-Planck-type kinetic equation. The kinetic methods proposed here are able to solve the problem with a reduced computational complexity in the limit of a large ensemble size. We illustrate the properties and the ability of the kinetic model to provide solution to inverse problems by using examples from the literature.