# A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion

arXiv: https://arxiv.org/abs/1308.1313 · HTML: https://mlsanity.com/paper/1308.1313/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2013.

## Built on by (most cited first, 30 of 32)

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- [Further analysis of multilevel Stein variational gradient descent with an application to the Bayesian inference of glacier ice models](https://mlsanity.com/paper/2212.03366/) (2022)
- [Posterior sampling with CNN-based, Plug-and-Play regularization with applications to Post-Stack Seismic Inversion](https://mlsanity.com/paper/2212.14595/) (2022)
- [Object based Bayesian full-waveform inversion for shear elastography](https://mlsanity.com/paper/2305.06646/) (2023)
- [Survey of Deep Learning and Physics-Based Approaches in Computational Wave Imaging](https://mlsanity.com/paper/2410.08329/) (2024)
- [Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problems](https://mlsanity.com/paper/2502.13105/) (2025)
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## Abstract (arXiv)

We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their uncertainty, the governing forward problem and its uncertainty, and a prior probability distribution describing uncertainty in the parameter field, find the posterior probability distribution over the parameter field. The prior must be chosen appropriately in order to guarantee well-posedness of the infinite-dimensional inverse problem and facilitate computation of the posterior. Furthermore, straightforward discretizations may not lead to convergent approximations of the infinite-dimensional problem. And finally, solution of the discretized inverse problem via explicit construction of the covariance matrix is prohibitive due to the need to solve the forward problem as many times as there are parameters. Our computational framework builds on the infinite-dimensional formulation proposed by Stuart (A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), pp. 451-559), and incorporates a number of components aimed at ensuring a convergent discretization of the underlying infinite-dimensional inverse problem. The framework additionally incorporates algorithms for manipulating the prior, constructing a low rank approximation of the data-informed component of the posterior covariance operator, and exploring the posterior that together ensure scalability of the entire framework to very high parameter dimensions. We demonstrate this computational framework on the Bayesian solution of an inverse problem in 3D global seismic wave propagation with hundreds of thousands of parameters.