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).

翻译：集成卡尔曼反演（EKI）是一种近期提出的用于求解反问题的优化方法，广泛应用于未知参数的高效且免导数估计。特别是在涉及不适定反问题和高维参数空间的情况下，该方案已展现出显著的成功。然而，在其一般形式中，EKI并未考虑约束条件，而这些约束通常源于物理限制或特定要求，至关重要。基于对数障碍方法，我们建议调整EKI的连续时间形式以纳入凸不等约束。我们通过理论分析支撑这一调整，给出了集成坍缩的下界与上界，以及针对一般非线性前向模型收敛到约束最优值的证明。最后，我们通过两个涉及偏微分方程（PDE）的示例展示了研究结果。