Nesterov acceleration for ensemble Kalman inversion and variants

Ensemble Kalman inversion (EKI) is a derivative-free, particle-based optimization method for solving inverse problems. It can be shown that EKI approximates a gradient flow, which allows the application of methods for accelerating gradient descent. Here, we show that Nesterov acceleration is effective in speeding up the reduction of the EKI cost function on a variety of inverse problems. We also implement Nesterov acceleration for two EKI variants, unscented Kalman inversion and ensemble transform Kalman inversion. Our specific implementation takes the form of a particle-level nudge that is demonstrably simple to couple in a black-box fashion with any existing EKI variant algorithms, comes with no additional computational expense, and with no additional tuning hyperparameters. This work shows a pathway for future research to translate advances in gradient-based optimization into advances in gradient-free Kalman optimization.

SV was supported through a Summer Undergraduate Research Fellowship (SURF) at Caltech. EB was supported by the Foster and Coco Stanback Postdoctoral Fellowship at Caltech. EB and ORAD were supported by the Schmidt Foundation and by the Office of Naval Research (Grant No. N00014-23-1-2654). We thank Andrew Stuart for helpful suggestions.