# A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems

arXiv: https://arxiv.org/abs/1308.6221 · HTML: https://mlsanity.com/paper/1308.6221/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2013.

## Built on by (most cited first, 19 of 19)

- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Differentiable Programming for Earth System Modeling](https://mlsanity.com/paper/2208.13825/) (2022)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modeling](https://mlsanity.com/paper/2301.11402/) (2023)
- [Adaptive operator learning for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2310.17844/) (2023)
- [Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction](https://mlsanity.com/paper/2106.04170/) (2021)
- [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023)
- [Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Data](https://mlsanity.com/paper/2303.11379/) (2023)
- [A variational neural network approach for glacier modelling with nonlinear rheology](https://mlsanity.com/paper/2209.02088/) (2022)
- [Stratospheric aerosol source inversion: Noise, variability, and uncertainty quantification](https://mlsanity.com/paper/2409.06846/) (2024)
- [Variational Inference at Glacier Scale](https://mlsanity.com/paper/2108.07263/) (2021)
- [Gaussian Invariant Markov Chain Monte Carlo](https://mlsanity.com/paper/2506.21511/) (2025)
- [Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss](https://mlsanity.com/paper/2510.15664/) (2025)
- [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)
- [Object based Bayesian full-waveform inversion for shear elastography](https://mlsanity.com/paper/2305.06646/) (2023)
- [From Displacements to Distributions: A Machine-Learning Enabled Framework for Quantifying Uncertainties in Parameters of Computational Models](https://mlsanity.com/paper/2403.03233/) (2024)
- [Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs](https://mlsanity.com/paper/2603.03922/) (2026)
- [SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation](https://mlsanity.com/paper/2603.22468/) (2026)

## Abstract (arXiv)

We address the numerical solution of infinite-dimensional inverse problems in the framework of Bayesian inference. In the Part I companion to this paper (arXiv.org:1308.1313), we considered the linearized infinite-dimensional inverse problem. Here in Part II, we relax the linearization assumption and consider the fully nonlinear infinite-dimensional inverse problem using a Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of sampling high-dimensional pdfs arising from Bayesian inverse problems governed by PDEs, we build on the stochastic Newton MCMC method. This method exploits problem structure by taking as a proposal density a local Gaussian approximation of the posterior pdf, whose construction is made tractable by invoking a low-rank approximation of its data misfit component of the Hessian. Here we introduce an approximation of the stochastic Newton proposal in which we compute the low-rank-based Hessian at just the MAP point, and then reuse this Hessian at each MCMC step. We compare the performance of the proposed method to the original stochastic Newton MCMC method and to an independence sampler. The comparison of the three methods is conducted on a synthetic ice sheet inverse problem. For this problem, the stochastic Newton MCMC method with a MAP-based Hessian converges at least as rapidly as the original stochastic Newton MCMC method, but is far cheaper since it avoids recomputing the Hessian at each step. On the other hand, it is more expensive per sample than the independence sampler; however, its convergence is significantly more rapid, and thus overall it is much cheaper. Finally, we present extensive analysis and interpretation of the posterior distribution, and classify directions in parameter space based on the extent to which they are informed by the prior or the observations.