# Uncertainty Quantification and Experimental Design for Large-Scale Linear Inverse Problems under Gaussian Process Priors

arXiv: https://arxiv.org/abs/2109.03457 · HTML: https://mlsanity.com/paper/2109.03457/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 67 references that are in the map.
- First posted on arXiv 8 Sep 2021.

## Builds on (most cited first, 15 of 15)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Exact Gaussian Processes on a Million Data Points](https://mlsanity.com/paper/1903.08114/) (2019)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [A supermartingale approach to Gaussian process based sequential design of experiments](https://mlsanity.com/paper/1608.01118/) (2016)
- [Goal-Oriented Optimal Design of Experiments for Large-Scale Bayesian Linear Inverse Problems](https://mlsanity.com/paper/1802.06517/) (2018)
- [Adaptive Design of Experiments for Conservative Estimation of Excursion Sets](https://mlsanity.com/paper/1611.07256/) (2016)
- [The linear conditional expectation in Hilbert space](https://mlsanity.com/paper/2008.12070/) (2020)
- [Bayesian model inversion using stochastic spectral embedding](https://mlsanity.com/paper/2005.07380/) (2020)
- [Quantifying uncertainties on excursion sets under a Gaussian random field prior](https://mlsanity.com/paper/1501.03659/) (2015)
- [Evaluating the squared-exponential covariance function in Gaussian processes with integral observations](https://mlsanity.com/paper/1812.07319/) (2018)
- [Learning excursion sets of vector-valued Gaussian random fields for autonomous ocean sampling](https://mlsanity.com/paper/2007.03722/) (2020)
- [Deep kernel learning for integral measurements](https://mlsanity.com/paper/1909.01844/) (2019)

## Built on by (most cited first, 2 of 2)

- [Non-Sequential Ensemble Kalman Filtering using Distributed Arrays](https://mlsanity.com/paper/2311.12909/) (2023)
- [On the Usage of Gaussian Process for Efficient Data Valuation](https://mlsanity.com/paper/2506.04026/) (2025)

## Abstract (arXiv)

We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covariance matrices can become too large to be stored. By leveraging results about sequential disintegrations of Gaussian measures, we are able to introduce an implicit representation of posterior covariance matrices that reduces the memory footprint by only storing low rank intermediate matrices, while allowing individual elements to be accessed on-the-fly without needing to build full posterior covariance matrices. Moreover, it allows for fast sequential inclusion of new observations. These features are crucial when considering sequential experimental design tasks. We demonstrate our approach by computing sequential data collection plans for excursion set recovery for a gravimetric inverse problem, where the goal is to provide fine resolution estimates of high density regions inside the Stromboli volcano, Italy. Sequential data collection plans are computed by extending the weighted integrated variance reduction (wIVR) criterion to inverse problems. Our results show that this criterion is able to significantly reduce the uncertainty on the excursion volume, reaching close to minimal levels of residual uncertainty. Overall, our techniques allow the advantages of probabilistic models to be brought to bear on large-scale inverse problems arising in the natural sciences.