  The ensemble Kalman inversion (EKI), a recently introduced optimisation
method for solving inverse problems, is widely employed for the efficient and
derivative-free estimation of unknown parameters. Specifically in cases
involving ill-posed inverse problems and high-dimensional parameter spaces, the
scheme has shown promising success. However, in its general form, the EKI does
not take constraints into account, which are essential and often stem from
physical limitations or specific requirements. Based on a log-barrier approach,
we suggest adapting the continuous-time formulation of EKI to incorporate
convex inequality constraints. We underpin this adaptation with a theoretical
analysis that provides lower and upper bounds on the ensemble collapse, as well
as convergence to the constraint optimum for general nonlinear forward models.
Finally, we showcase our results through two examples involving partial
differential equations (PDEs).
