Polarized Ensemble Kalman Inversion: A Derivative-Free Method for Multimodal Inverse Problems with Non-asymptotic Analysis

Abstract

Inverse problems often admit multiple solutions consistent with the same noisy, indirect observations, especially with sparse data and/or strongly nonlinear forward models. Ensemble Kalman inversion (EKI) provides a robust, derivative-free framework to solve inverse problems, but its global empirical covariances couple all ensemble members across distinct solution modes, limiting its effectiveness for multimodal problems. We introduce polarized ensemble Kalman inversion (PEKI), which constructs distance-weighted local covariances for each ensemble member. This localization weakens contributions from distant, data-consistent regions and enables different clusters of ensemble members to evolve toward distinct solution modes. For the practical algorithm with finite ensemble size and polarization radius, we also establish non-asymptotic bounds and perform numerical studies, showing that cross-cluster effects decay exponentially with cluster separation relative to the polarization radius. For linear forward maps, we further establish particle-wise residual contraction and derive an exact cluster-mean decomposition that separates cross-cluster leakage, within-cluster gain variation, and observation perturbations. These results further motivate a one-step procedure for choosing the polarization radius by reusing stored forward-model evaluations. We studied the performance of PEKI on several numerical experiments, including low-dimensional nonconvex benchmarks, a problem equipped with many-mode landscape, and a high-dimensional inverse problem governed by Navier--Stokes equations. The results confirm the theoretical exponential decay of cross-cluster effects and show that PEKI can serve as a robust and derivative-free method for inverse problems with multiple solution modes.