On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Computational Science

Background:

• Linear inverse problems are fundamental in various scientific fields.
• Traditional regularization methods like Tikhonov regularization have limitations.
• Ensemble Kalman inversion (EKI) is an emerging technique for inverse problems.

Purpose of the Study:

• To present a deterministic Ensemble Kalman Inversion (EKI) as a regularization technique for linear inverse problems.
• To introduce a novel sampling scheme for EKI based on the Nyström method.
• To develop an adaptive EKI that couples sample size with the regularization parameter.

Main Methods:

• Interpreting EKI as a low-rank approximation of Tikhonov regularization.
• Implementing a Nyström-based sampling scheme for enhanced practical performance.
• Formulating an adaptive EKI with a coupled sample size and regularization parameter.
• Utilizing the discrepancy principle as a stopping criterion.

Main Results:

• The proposed Nyström-based sampling scheme improves the practical performance of EKI.
• The adaptive EKI formulation, coupled with the discrepancy principle, achieves order-optimal regularization.
• Numerical comparisons demonstrate the effectiveness of the proposed methods for Radon transform inverse problems.

Conclusions:

• Deterministic EKI provides an effective regularization strategy for linear inverse problems.
• The Nyström method enhances EKI's practical applicability.
• The adaptive EKI scheme offers theoretical guarantees for order-optimal solutions.