The ensemble Kalman inversion (EKI) method is widely used for solving inverse problems, leveraging ensemble-based techniques to iteratively refine parameter estimates. Despite its versatility, the accuracy of EKI is constrained by the subspace spanned by the initial ensemble, which may poorly represent the solution in cases of limited prior knowledge. This work addresses these limitations by optimising the subspace in which EKI operates, improving accuracy and computational efficiency. We derive a theoretical framework for constructing optimal subspaces in linear settings and extend these insights to nonlinear cases. A novel greedy strategy for selecting initial ensemble members is proposed, incorporating prior, data, and model information to enhance performance. Numerical experiments on both linear and nonlinear problems demonstrate the effectiveness of the approach, offering a significant advancement in the accuracy and scalability of EKI for high-dimensional and ill-posed problems.
