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Consistency of variational Bayesian inference for non-linear inverse problems of partial differential equations
- Publication Year :
- 2024
-
Abstract
- We consider non-linear Bayesian inverse problems of determining the parameter $f$. For the posterior distribution with a class of Gaussian process priors, we study the statistical performance of variational Bayesian inference to the posterior with variational sets consisting of Gaussian measures or a mean-field family. We propose certain conditions on the forward map $\mathcal{G}$, the variational set $\mathcal{Q}$ and the prior such that, as the number $N$ of measurements increases, the resulting variational posterior distributions contract to the ground truth $f_0$ generating the data, and derive a convergence rate with polynomial order or logarithmic order. As specific examples, we consider a collection of non-linear inverse problems, including the Darcy flow problem, the inverse potential problem for a subdiffusion equation, and the inverse medium scattering problem. Besides, we show that our convergence rates are minimax optimal for these inverse problems.<br />Comment: 75 pages
- Subjects :
- Mathematics - Statistics Theory
62G20
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2409.18415
- Document Type :
- Working Paper