Dropout Ensemble Kalman inversion for high dimensional inverse problems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912318569316352 |
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| author | Liu, Shuigen Reich, Sebastian Tong, Xin T. |
| author_facet | Liu, Shuigen Reich, Sebastian Tong, Xin T. |
| contents | Ensemble Kalman inversion (EKI) is an ensemble-based method to solve inverse problems. Its gradient-free formulation makes it an attractive tool for problems with involved formulation. However, EKI suffers from the ''subspace property'', i.e., the EKI solutions are confined in the subspace spanned by the initial ensemble. It implies that the ensemble size should be larger than the problem dimension to ensure EKI's convergence to the correct solution. Such scaling of ensemble size is impractical and prevents the use of EKI in high dimensional problems. To address this issue, we propose a novel approach using dropout regularization to mitigate the subspace problem. We prove that dropout-EKI converges in the small ensemble settings, and the computational cost of the algorithm scales linearly with dimension. We also show that dropout-EKI reaches the optimal query complexity, up to a constant factor. Numerical examples demonstrate the effectiveness of our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16784 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dropout Ensemble Kalman inversion for high dimensional inverse problems Liu, Shuigen Reich, Sebastian Tong, Xin T. Numerical Analysis 65K10, 90C56, 65M32 Ensemble Kalman inversion (EKI) is an ensemble-based method to solve inverse problems. Its gradient-free formulation makes it an attractive tool for problems with involved formulation. However, EKI suffers from the ''subspace property'', i.e., the EKI solutions are confined in the subspace spanned by the initial ensemble. It implies that the ensemble size should be larger than the problem dimension to ensure EKI's convergence to the correct solution. Such scaling of ensemble size is impractical and prevents the use of EKI in high dimensional problems. To address this issue, we propose a novel approach using dropout regularization to mitigate the subspace problem. We prove that dropout-EKI converges in the small ensemble settings, and the computational cost of the algorithm scales linearly with dimension. We also show that dropout-EKI reaches the optimal query complexity, up to a constant factor. Numerical examples demonstrate the effectiveness of our approach. |
| title | Dropout Ensemble Kalman inversion for high dimensional inverse problems |
| topic | Numerical Analysis 65K10, 90C56, 65M32 |
| url | https://arxiv.org/abs/2308.16784 |