Stochastic gradient descent based variational inference for infinite-dimensional inverse problems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915833526091776 |
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| author | Sui, Jiaming Jia, Junxiong Li, Jinglai |
| author_facet | Sui, Jiaming Jia, Junxiong Li, Jinglai |
| contents | This paper introduces two variational inference approaches for infinite-dimensional inverse problems, developed through gradient descent with a constant learning rate. The proposed methods enable efficient approximate sampling from the target posterior distribution using a constant-rate stochastic gradient descent (cSGD) iteration. Specifically, we introduce a randomization strategy that incorporates stochastic gradient noise, allowing the cSGD iteration to be viewed as a discrete-time process. This transformation establishes key relationships between the covariance operators of the approximate and true posterior distributions, thereby validating cSGD as a variational inference method. We also investigate the regularization properties of the cSGD iteration and provide a theoretical analysis of the discretization error between the approximated posterior mean and the true background function. Building on this framework, we develop a preconditioned version of cSGD to further improve sampling efficiency. Finally, we apply the proposed methods to two practical inverse problems: one governed by a simple smooth equation and the other by the steady-state Darcy flow equation. Numerical results confirm our theoretical findings and compare the sampling performance of the two approaches for solving linear and non-linear inverse problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08380 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic gradient descent based variational inference for infinite-dimensional inverse problems Sui, Jiaming Jia, Junxiong Li, Jinglai Numerical Analysis This paper introduces two variational inference approaches for infinite-dimensional inverse problems, developed through gradient descent with a constant learning rate. The proposed methods enable efficient approximate sampling from the target posterior distribution using a constant-rate stochastic gradient descent (cSGD) iteration. Specifically, we introduce a randomization strategy that incorporates stochastic gradient noise, allowing the cSGD iteration to be viewed as a discrete-time process. This transformation establishes key relationships between the covariance operators of the approximate and true posterior distributions, thereby validating cSGD as a variational inference method. We also investigate the regularization properties of the cSGD iteration and provide a theoretical analysis of the discretization error between the approximated posterior mean and the true background function. Building on this framework, we develop a preconditioned version of cSGD to further improve sampling efficiency. Finally, we apply the proposed methods to two practical inverse problems: one governed by a simple smooth equation and the other by the steady-state Darcy flow equation. Numerical results confirm our theoretical findings and compare the sampling performance of the two approaches for solving linear and non-linear inverse problems. |
| title | Stochastic gradient descent based variational inference for infinite-dimensional inverse problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.08380 |