---
references:
- id: AlexanderianFast2016
  accessed:
    - year: 2022
      month: 1
      day: 6
  author:
    - family: Alexanderian
      given: Alen
    - family: Petra
      given: Noemi
    - family: Stadler
      given: Georg
    - family: Ghattas
      given: Omar
  citation-key: AlexanderianFast2016
  container-title: SIAM Journal on Scientific Computing
  container-title-short: SIAM J. Sci. Comput.
  DOI: 10.1137/140992564
  ISSN: 1064-8275
  issue: '1'
  issued:
    - year: 2016
      month: 1
      day: 1
  page: A243-A272
  publisher: Society for Industrial and Applied Mathematics
  title: >-
    A Fast and Scalable Method for A-Optimal Design of Experiments for
    Infinite-dimensional Bayesian Nonlinear Inverse Problems
  type: article-journal
  URL: http://arxiv.org/abs/1410.5899
  volume: '38'

- id: AlexanderianOptimal2021
  accessed:
    - year: 2022
      month: 1
      day: 6
  author:
    - family: Alexanderian
      given: Alen
  citation-key: AlexanderianOptimal2021
  container-title: arXiv:2005.12998 [math]
  issued:
    - year: 2021
      month: 1
      day: 28
  title: >-
    Optimal Experimental Design for Infinite-dimensional Bayesian Inverse
    Problems Governed by PDEs: A Review
  type: article-journal
  URL: http://arxiv.org/abs/2005.12998

- id: BorgonovoModel2012
  accessed:
    - year: 2020
      month: 11
      day: 9
  author:
    - family: Borgonovo
      given: E.
    - family: Castaings
      given: W.
    - family: Tarantola
      given: S.
  citation-key: BorgonovoModel2012
  collection-title: >-
    Emulation techniques for the reduction and sensitivity analysis of complex
    environmental models
  container-title: Environmental Modelling & Software
  container-title-short: Environmental Modelling & Software
  DOI: 10.1016/j.envsoft.2011.06.006
  ISSN: 1364-8152
  issued:
    - year: 2012
      month: 6
      day: 1
  language: en
  page: 105-115
  title: "Model emulation and moment-independent sensitivity analysis: An application to\_environmental modelling"
  type: article-journal
  volume: '34'

- id: BreslowApproximate1993
  accessed:
    - year: 2016
      month: 9
      day: 2
  author:
    - family: Breslow
      given: N. E.
    - family: Clayton
      given: D. G.
  citation-key: BreslowApproximate1993
  container-title: Journal of the American Statistical Association
  container-title-short: Journal of the American Statistical Association
  DOI: 10.2307/2290687
  ISSN: 0162-1459
  issue: '421'
  issued:
    - year: 1993
  page: 9-25
  title: Approximate Inference in Generalized Linear Mixed Models
  type: article-journal
  URL: >-
    https://www.math.ntnu.no/inla/r-inla.org/case-studies/Breslow-Clayton/breslow-clayton-1993.pdf
  volume: '88'

- id: Bui-ThanhGentle2012
  author:
    - family: Bui-Thanh
      given: Tan
  citation-key: Bui-ThanhGentle2012
  issued:
    - year: 2012
  title: A Gentle Tutorial on Statistical Inversion using the Bayesian Paradigm
  type: speech
  URL: http://users.ices.utexas.edu/~tanbui/PublishedPapers/BayesianTutorial.pdf

- id: ChungEfficient2014
  accessed:
    - year: 2023
      month: 11
      day: 10
  author:
    - family: Chung
      given: Julianne
    - family: Chung
      given: Matthias
  citation-key: ChungEfficient2014
  container-title: Inverse Problems
  container-title-short: Inverse Problems
  DOI: 10.1088/0266-5611/30/11/114009
  ISSN: 0266-5611, 1361-6420
  issue: '11'
  issued:
    - year: 2014
      month: 11
      day: 1
  page: '114009'
  title: >-
    An Efficient Approach for Computing Optimal Low-Rank Regularized Inverse
    Matrices
  type: article-journal
  URL: http://arxiv.org/abs/1404.1610
  volume: '30'

- id: CorderoGeneral2021
  accessed:
    - year: 2023
      month: 11
      day: 10
  author:
    - family: Cordero
      given: Alicia
    - family: Soto-Quiros
      given: Pablo
    - family: Torregrosa
      given: Juan R.
  citation-key: CorderoGeneral2021
  container-title: Applied Mathematics and Computation
  container-title-short: Applied Mathematics and Computation
  DOI: 10.1016/j.amc.2021.126381
  ISSN: 0096-3003
  issued:
    - year: 2021
      month: 11
      day: 15
  page: '126381'
  title: >-
    A general class of arbitrary order iterative methods for computing
    generalized inverses
  type: article-journal
  volume: '409'

- id: DashtiBayesian2015
  accessed:
    - year: 2022
      month: 1
      day: 17
  author:
    - family: Dashti
      given: Masoumeh
    - family: Stuart
      given: Andrew M.
  citation-key: DashtiBayesian2015
  container-title: arXiv:1302.6989 [math]
  issued:
    - year: 2015
      month: 7
      day: 2
  title: The Bayesian Approach To Inverse Problems
  type: article-journal
  URL: http://arxiv.org/abs/1302.6989

- id: FoongInBetween2019
  accessed:
    - year: 2021
      month: 8
      day: 23
  author:
    - family: Foong
      given: Andrew Y. K.
    - family: Li
      given: Yingzhen
    - family: Hernández-Lobato
      given: José Miguel
    - family: Turner
      given: Richard E.
  citation-key: FoongInBetween2019
  container-title: arXiv:1906.11537 [cs, stat]
  issued:
    - year: 2019
      month: 6
      day: 27
  title: '''In-Between'' Uncertainty in Bayesian Neural Networks'
  type: article-journal
  URL: http://arxiv.org/abs/1906.11537

- id: GiordanoConsistency2020
  accessed:
    - year: 2022
      month: 1
      day: 13
  author:
    - family: Giordano
      given: Matteo
    - family: Nickl
      given: Richard
  citation-key: GiordanoConsistency2020
  container-title: Inverse Problems
  container-title-short: Inverse Problems
  DOI: 10.1088/1361-6420/ab7d2a
  ISSN: 0266-5611
  issue: '8'
  issued:
    - year: 2020
      month: 8
  language: en
  page: '085001'
  publisher: IOP Publishing
  title: >-
    Consistency of Bayesian inference with Gaussian process priors in an
    elliptic inverse problem
  type: article-journal
  volume: '36'

- id: GrumittFlow2023
  accessed:
    - year: 2024
      month: 1
      day: 5
  author:
    - family: Grumitt
      given: Richard D. P.
    - family: Karamanis
      given: Minas
    - family: Seljak
      given: Uroš
  citation-key: GrumittFlow2023
  issued:
    - year: 2023
      month: 9
      day: 20
  language: en
  publisher: arXiv
  title: >-
    Flow Annealed Kalman Inversion for Gradient-Free Inference in Bayesian
    Inverse Problems
  type: article
  URL: https://arxiv.org/abs/2309.11490v1
  version: '1'

- id: ImmerImproving2021
  accessed:
    - year: 2021
      month: 8
      day: 23
  author:
    - family: Immer
      given: Alexander
    - family: Korzepa
      given: Maciej
    - family: Bauer
      given: Matthias
  citation-key: ImmerImproving2021
  container-title: International Conference on Artificial Intelligence and Statistics
  event-title: International Conference on Artificial Intelligence and Statistics
  ISSN: 2640-3498
  issued:
    - year: 2021
      month: 3
      day: 18
  language: en
  page: 703-711
  publisher: PMLR
  title: Improving predictions of Bayesian neural nets via local linearization
  type: paper-conference
  URL: https://proceedings.mlr.press/v130/immer21a.html

- id: KaipioStatistical2005
  accessed:
    - year: 2021
      month: 8
      day: 23
  author:
    - family: Kaipio
      given: Jari
    - family: Somersalo
      given: E.
  citation-key: KaipioStatistical2005
  collection-title: Applied Mathematical Sciences
  DOI: 10.1007/b138659
  event-place: New York
  ISBN: 978-0-387-22073-4
  issued:
    - year: 2005
  language: en
  publisher: Springer-Verlag
  publisher-place: New York
  title: Statistical and Computational Inverse Problems
  type: book
  URL: https://www.springer.com/gp/book/9780387220734

- id: KaipioStatistical2007
  accessed:
    - year: 2021
      month: 8
      day: 23
  author:
    - family: Kaipio
      given: Jari
    - family: Somersalo
      given: Erkki
  citation-key: KaipioStatistical2007
  container-title: Journal of Computational and Applied Mathematics
  container-title-short: Journal of Computational and Applied Mathematics
  DOI: 10.1016/j.cam.2005.09.027
  ISSN: '03770427'
  issue: '2'
  issued:
    - year: 2007
      month: 1
  language: en
  page: 493-504
  title: >-
    Statistical inverse problems: Discretization, model reduction and inverse
    crimes
  type: article-journal
  volume: '198'

- id: KennedyBayesian2001
  accessed:
    - year: 2020
      month: 8
      day: 23
  author:
    - family: Kennedy
      given: Marc C.
    - family: O’Hagan
      given: Anthony
  citation-key: KennedyBayesian2001
  container-title: 'Journal of the Royal Statistical Society: Series B (Statistical Methodology)'
  DOI: 10.1111/1467-9868.00294
  ISSN: 1467-9868
  issue: '3'
  issued:
    - year: 2001
  language: en
  page: 425-464
  title: Bayesian calibration of computer models
  type: article-journal
  volume: '63'

- id: KnapikBayesian2011
  accessed:
    - year: 2022
      month: 1
      day: 13
  author:
    - family: Knapik
      given: B. T.
    - family: Vaart
      given: A. W.
      non-dropping-particle: van der
    - family: Zanten
      given: J. H.
      non-dropping-particle: van
  citation-key: KnapikBayesian2011
  container-title: The Annals of Statistics
  container-title-short: Ann. Statist.
  DOI: 10.1214/11-AOS920
  ISSN: 0090-5364
  issue: '5'
  issued:
    - year: 2011
      month: 10
      day: 1
  language: en
  title: Bayesian inverse problems with Gaussian priors
  type: article-journal
  URL: http://arxiv.org/abs/1103.2692
  volume: '39'

- id: MosegaardMonte1995
  accessed:
    - year: 2022
      month: 9
      day: 7
  author:
    - family: Mosegaard
      given: Klaus
    - family: Tarantola
      given: Albert
  citation-key: MosegaardMonte1995
  container-title: 'Journal of Geophysical Research: Solid Earth'
  DOI: 10.1029/94JB03097
  ISSN: 2156-2202
  issue: B7
  issued:
    - year: 1995
  language: en
  page: 12431-12447
  title: Monte Carlo sampling of solutions to inverse problems
  type: article-journal
  URL: https://www.gfy.ku.dk/~klaus/papers/B4-MT1995-LaTeX.pdf
  volume: '100'

- id: MosegaardProbabilistic2002
  accessed:
    - year: 2022
      month: 1
      day: 13
  author:
    - family: Mosegaard
      given: Klaus
    - family: Tarantola
      given: Albert
  citation-key: MosegaardProbabilistic2002
  container-title: International Geophysics
  DOI: 10.1016/S0074-6142(02)80219-4
  ISBN: 978-0-12-440652-0
  issued:
    - year: 2002
  language: en
  page: 237-265
  publisher: Elsevier
  title: Probabilistic approach to inverse problems
  type: chapter
  URL: https://linkinghub.elsevier.com/retrieve/pii/S0074614202802194
  volume: '81'

- id: OHaganBayesian2006
  accessed:
    - year: 2020
      month: 8
      day: 23
  author:
    - family: O’Hagan
      given: A.
  citation-key: OHaganBayesian2006
  collection-title: >-
    The Fourth International Conference on Sensitivity Analysis of Model Output
    (SAMO 2004)
  container-title: Reliability Engineering & System Safety
  container-title-short: Reliability Engineering & System Safety
  DOI: 10.1016/j.ress.2005.11.025
  ISSN: 0951-8320
  issue: '10'
  issued:
    - year: 2006
      month: 10
      day: 1
  language: en
  page: 1290-1300
  title: 'Bayesian analysis of computer code outputs: A tutorial'
  type: article-journal
  volume: '91'

- id: PlumleeBayesian2017
  accessed:
    - year: 2020
      month: 9
      day: 9
  author:
    - family: Plumlee
      given: Matthew
  citation-key: PlumleeBayesian2017
  container-title: Journal of the American Statistical Association
  DOI: 10.1080/01621459.2016.1211016
  ISSN: 0162-1459
  issue: '519'
  issued:
    - year: 2017
      month: 7
      day: 3
  page: 1274-1285
  publisher: Taylor & Francis
  title: Bayesian Calibration of Inexact Computer Models
  type: article-journal
  URL: >-
    https://www.asc.ohio-state.edu/statistics/comp_exp/jour.club/Bayesian_calibration_of_inexact_computer_models_Plumlee_2017.pdf
  volume: '112'

- id: SainsburyDaleFast2022
  accessed:
    - year: 2022
      month: 11
      day: 24
  author:
    - family: Sainsbury-Dale
      given: Matthew
    - family: Zammit-Mangion
      given: Andrew
    - family: Huser
      given: Raphaël
  citation-key: SainsburyDaleFast2022
  DOI: 10.48550/arXiv.2208.12942
  issued:
    - year: 2022
      month: 8
      day: 27
  number: arXiv:2208.12942
  publisher: arXiv
  title: >-
    Fast Optimal Estimation with Intractable Models using Permutation-Invariant
    Neural Networks
  type: article
  URL: http://arxiv.org/abs/2208.12942

- id: SarkkaSpatiotemporal2013
  author:
    - family: Särkkä
      given: Simo
    - family: Solin
      given: A.
    - family: Hartikainen
      given: J.
  citation-key: SarkkaSpatiotemporal2013
  container-title: IEEE Signal Processing Magazine
  DOI: 10.1109/MSP.2013.2246292
  ISSN: 1053-5888
  issue: '4'
  issued:
    - year: 2013
      month: 7
  number: '4'
  page: 51-61
  title: >-
    Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and
    Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering
  type: article-journal
  volume: '30'

- id: SchillingsAnalysis2017
  accessed:
    - year: 2022
      month: 8
      day: 23
  author:
    - family: Schillings
      given: Claudia
    - family: Stuart
      given: Andrew M.
  citation-key: SchillingsAnalysis2017
  container-title: SIAM Journal on Numerical Analysis
  container-title-short: SIAM J. Numer. Anal.
  DOI: 10.1137/16M105959X
  ISSN: 0036-1429
  issue: '3'
  issued:
    - year: 2017
      month: 1
      day: 1
  page: 1264–1290
  publisher: Society for Industrial and Applied Mathematics
  title: Analysis of the Ensemble Kalman Filter for Inverse Problems
  type: article-journal
  URL: http://arxiv.org/abs/1602.02020
  volume: '55'

- id: SchwabSparse2012
  author:
    - family: Schwab
      given: C.
    - family: Stuart
      given: A. M.
  citation-key: SchwabSparse2012
  container-title: Inverse Problems
  DOI: 10.1088/0266-5611/28/4/045003
  ISSN: 0266-5611, 1361-6420
  issue: '4'
  issued:
    - year: 2012
      month: 4
      day: 1
  page: '045003'
  title: Sparse deterministic approximation of Bayesian inverse problems
  type: article-journal
  URL: http://homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart92.pdf
  volume: '28'

- id: SpantiniGoalOriented2017
  accessed:
    - year: 2023
      month: 11
      day: 10
  author:
    - family: Spantini
      given: Alessio
    - family: Cui
      given: Tiangang
    - family: Willcox
      given: Karen
    - family: Tenorio
      given: Luis
    - family: Marzouk
      given: Youssef
  citation-key: SpantiniGoalOriented2017
  container-title: SIAM Journal on Scientific Computing
  container-title-short: SIAM J. Sci. Comput.
  DOI: 10.1137/16M1082123
  ISSN: 1064-8275
  issue: '5'
  issued:
    - year: 2017
      month: 1
  page: S167-S196
  publisher: Society for Industrial and Applied Mathematics
  title: Goal-Oriented Optimal Approximations of Bayesian Linear Inverse Problems
  type: article-journal
  URL: http://arxiv.org/abs/1607.01881
  volume: '39'

- id: SpantiniOptimal2015
  accessed:
    - year: 2022
      month: 1
      day: 28
  author:
    - family: Spantini
      given: Alessio
    - family: Solonen
      given: Antti
    - family: Cui
      given: Tiangang
    - family: Martin
      given: James
    - family: Tenorio
      given: Luis
    - family: Marzouk
      given: Youssef
  citation-key: SpantiniOptimal2015
  container-title: SIAM Journal on Scientific Computing
  container-title-short: SIAM J. Sci. Comput.
  DOI: 10.1137/140977308
  ISSN: 1064-8275
  issue: '6'
  issued:
    - year: 2015
      month: 1
      day: 1
  page: A2451-A2487
  publisher: Society for Industrial and Applied Mathematics
  title: Optimal Low-rank Approximations of Bayesian Linear Inverse Problems
  type: article-journal
  URL: http://arxiv.org/abs/1407.3463
  volume: '37'

- id: StuartInverse2010
  author:
    - family: Stuart
      given: Andrew M
  citation-key: StuartInverse2010
  container-title: Acta Numerica
  DOI: 10.1017/S0962492910000061
  ISSN: 1474-0508, 0962-4929
  issued:
    - year: 2010
  language: en
  page: 451–559
  title: 'Inverse problems: A Bayesian perspective'
  type: article-journal
  URL: http://resolver.caltech.edu/CaltechAUTHORS:20161111-112136150
  volume: '19'

- id: TaitVariational2020
  accessed:
    - year: 2020
      month: 7
      day: 12
  author:
    - family: Tait
      given: Daniel J.
    - family: Damoulas
      given: Theodoros
  citation-key: TaitVariational2020
  container-title: arXiv:2006.15641 [cs, stat]
  issued:
    - year: 2020
      month: 6
      day: 28
  title: Variational Autoencoding of PDE Inverse Problems
  type: article-journal
  URL: http://arxiv.org/abs/2006.15641

- id: TarantolaInverse2005
  author:
    - family: Tarantola
      given: Albert
  citation-key: TarantolaInverse2005
  ISBN: 978-0-89871-572-9
  issued:
    - year: 2005
      month: 1
      day: 1
  language: en
  number-of-pages: '348'
  publisher: SIAM
  title: Inverse Problem Theory and Methods for Model Parameter Estimation
  type: book

- id: TarantolaMapping
  archive: Calibre
  author:
    - family: Tarantola
      given: Albert
  citation-key: TarantolaMapping
  title: Mapping Of Probabilities
  type: book

- id: TenorioIntroduction2017
  accessed:
    - year: 2023
      month: 11
      day: 10
  author:
    - family: Tenorio
      given: Luis
  citation-key: TenorioIntroduction2017
  collection-title: Mathematics in Industry
  DOI: 10.1137/1.9781611974928
  event-place: Philadelphia, PA
  ISBN: 978-1-61197-491-1 978-1-61197-492-8
  issued:
    - year: 2017
      month: 6
      day: 30
  language: en
  publisher: Society for Industrial and Applied Mathematics
  publisher-place: Philadelphia, PA
  title: >-
    An Introduction to Data Analysis and Uncertainty Quantification for Inverse
    Problems
  type: book

- id: TonoliniVariational2020
  accessed:
    - year: 2020
      month: 9
      day: 17
  author:
    - family: Tonolini
      given: Francesco
    - family: Radford
      given: Jack
    - family: Turpin
      given: Alex
    - family: Faccio
      given: Daniele
    - family: Murray-Smith
      given: Roderick
  citation-key: TonoliniVariational2020
  container-title: Journal of Machine Learning Research
  ISSN: 1533-7928
  issue: '179'
  issued:
    - year: 2020
  page: 1-46
  title: Variational Inference for Computational Imaging Inverse Problems
  type: article-journal
  URL: http://jmlr.org/papers/v21/20-151.html
  volume: '21'

- id: WackerLaplace2017
  accessed:
    - year: 2022
      month: 1
      day: 4
  author:
    - family: Wacker
      given: Philipp
  citation-key: WackerLaplace2017
  container-title: arXiv:1701.07989 [math]
  issued:
    - year: 2017
      month: 4
      day: 11
  language: en
  title: Laplace's method in Bayesian inverse problems
  type: article-journal
  URL: http://arxiv.org/abs/1701.07989

- id: WeiInnerloop2017
  accessed:
    - year: 2017
      month: 9
      day: 19
  author:
    - family: Wei
      given: Qi
    - family: Fan
      given: Kai
    - family: Carin
      given: Lawrence
    - family: Heller
      given: Katherine A.
  citation-key: WeiInnerloop2017
  container-title: arXiv:1709.01841 [cs]
  issued:
    - year: 2017
      month: 9
      day: 6
  title: An inner-loop free solution to inverse problems using deep neural networks
  type: article-journal
  URL: http://arxiv.org/abs/1709.01841

- id: YangBPINNs2021
  accessed:
    - year: 2022
      month: 7
      day: 26
  author:
    - family: Yang
      given: Liu
    - family: Meng
      given: Xuhui
    - family: Karniadakis
      given: George Em
  citation-key: YangBPINNs2021
  container-title: Journal of Computational Physics
  container-title-short: Journal of Computational Physics
  DOI: 10.1016/j.jcp.2020.109913
  ISSN: 0021-9991
  issued:
    - year: 2021
      month: 1
      day: 15
  language: en
  page: '109913'
  title: >-
    B-PINNs: Bayesian physics-informed neural networks for forward and inverse
    PDE problems with noisy data
  type: article-journal
  URL: http://arxiv.org/abs/2003.06097
  volume: '425'

- id: ZammitMangionWOMBAT2021
  accessed:
    - year: 2021
      month: 7
      day: 22
  author:
    - family: Zammit-Mangion
      given: Andrew
    - family: Bertolacci
      given: Michael
    - family: Fisher
      given: Jenny
    - family: Stavert
      given: Ann
    - family: Rigby
      given: Matthew L.
    - family: Cao
      given: Yi
    - family: Cressie
      given: Noel
  citation-key: ZammitMangionWOMBAT2021
  container-title: Geoscientific Model Development Discussions
  DOI: 10.5194/gmd-2021-181
  ISSN: 1991-959X
  issued:
    - year: 2021
      month: 7
      day: 12
  language: English
  page: 1-51
  publisher: Copernicus GmbH
  title: 'WOMBAT v1.0: A fully Bayesian global flux-inversion framework'
  type: article-journal
  URL: https://gmd.copernicus.org/preprints/gmd-2021-181/

- id: ZhaoTensorbased2023
  accessed:
    - year: 2023
      month: 2
      day: 21
  author:
    - family: Zhao
      given: Yiran
    - family: Cui
      given: Tiangang
  citation-key: ZhaoTensorbased2023
  DOI: 10.48550/arXiv.2301.09891
  issued:
    - year: 2023
      month: 1
      day: 24
  number: arXiv:2301.09891
  publisher: arXiv
  title: >-
    Tensor-based Methods for Sequential State and Parameter Estimation in State
    Space Models
  type: article
  URL: http://arxiv.org/abs/2301.09891
...
