  Ensemble Kalman inversion is a parallelizable methodology for solving inverse
or parameter estimation problems. Although it is based on ideas from Kalman
filtering, it may be viewed as a derivative-free optimization method. In its
most basic form it regularizes ill-posed inverse problems through the subspace
property: the solution found is in the linear span of the initial ensemble
employed. In this work we demonstrate how further regularization can be
imposed, incorporating prior information about the underlying unknown. In
particular we study how to impose Tikhonov-like Sobolev penalties. As well as
introducing this modified ensemble Kalman inversion methodology, we also study
its continuous-time limit, proving ensemble collapse; in the language of
multi-agent optimization this may be viewed as reaching consensus. We also
conduct a suite of numerical experiments to highlight the benefits of Tikhonov
regularization in the ensemble inversion context.
