| Abstract: |
| Inverse problems for PDEs are common in scientific disciplines and can be formulated as statistical inference problems via Bayes theorem. For large-scale problems, developing discretization-invariant algorithms is crucial, achievable by formulating methods in infinite-dimensional spaces. Restricting the variational family to the pushforward of a prior measure`s nonlinear transformation yields various variational inference methods. Overcoming singularity issues in infinite-dimensional function spaces, we develop two methods: infinite-dimensional Stein variational gradient descent (iSVGD) and functional normalizing flows (FNF). The transformations in both iSVGD and FNF involve a sequence of identity operator perturbations. In iSVGD, perturbation mappings are in a reproducing kernel Hilbert space, while in FNF, they are constructed with designed neural network architectures. We apply these algorithms to an inverse problem of the steady-state Darcy flow equation. Numerical results validate the theoretical analysis, show the algorithms` efficiency, and confirm their discretization-invariant properties. |
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