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      month: 10
      day: 26
  title: A brief note on the Karhunen-Loève expansion
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- id: AlpayWiener2015
  accessed:
    - year: 2021
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- id: CalatayudGregoriCombining2019
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  title: Stochastic Duality and Orthogonal Polynomials
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- id: GhanemPolynomial1990
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  language: en
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  publisher: Springer International Publishing
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- id: GhanemStochastic2003
  author:
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    - family: Spanos
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  citation-key: GhanemStochastic2003
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  language: en
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- id: GratietMetamodelBased2016
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    - year: 2020
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  container-title: Handbook of Uncertainty Quantification
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  language: en
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  publisher: Springer International Publishing
  publisher-place: Cham
  title: >-
    Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and
    Gaussian Processes
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- id: KimGeneralised2013
  accessed:
    - year: 2020
      month: 5
      day: 20
  author:
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      given: Kwang-Ki K.
    - family: Braatz
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  citation-key: KimGeneralised2013
  container-title: International Journal of Control
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  language: en
  page: 1324-1337
  title: >-
    Generalised polynomial chaos expansion approaches to approximate stochastic
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  URL: >-
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- id: KimWiener2013
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      given: Kwang-Ki K.
    - family: Shen
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    - family: Nagy
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  container-title: IEEE Control Systems Magazine
  DOI: 10.1109/MCS.2013.2270410
  ISSN: 1941-000X
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      month: 10
  page: 58-67
  title: >-
    Wiener's Polynomial Chaos for the Analysis and Control of Nonlinear
    Dynamical Systems with Probabilistic Uncertainties [Historical Perspectives]
  type: article-journal
  URL: http://people.duke.edu/~hpgavin/Risk/Kim-CSM-2013.pdf
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- id: LeiDatadriven2018
  accessed:
    - year: 2020
      month: 12
      day: 17
  author:
    - family: Lei
      given: Huan
    - family: Li
      given: Jing
    - family: Gao
      given: Peiyuan
    - family: Stinis
      given: Panos
    - family: Baker
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  citation-key: LeiDatadriven2018
  DOI: 10.1016/j.cma.2019.03.014
  issued:
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      month: 4
      day: 20
  language: en
  title: >-
    A data-driven framework for sparsity-enhanced surrogates with arbitrary
    mutually dependent randomness
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- id: LevajkovicChaos2011
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    - year: 2020
      month: 5
      day: 20
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  citation-key: LevajkovicChaos2011
  container-title: Publications de l'Institut Mathematique
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  language: en
  page: 65-84
  title: >-
    Chaos expansion methods for stochastic differential equations involving the
    Malliavin derivative, Part I
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  volume: '90'

- id: LuoWiener2006
  accessed:
    - year: 2020
      month: 5
      day: 20
  author:
    - family: Luo
      given: Wuan
  citation-key: LuoWiener2006
  DOI: 10.7907/RPKX-BN02
  genre: phd
  issued:
    - year: 2006
  publisher: California Institute of Technology
  title: >-
    Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial
    Differential Equations
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- id: NoorshamsBelief2013
  author:
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      given: Nima
    - family: Wainwright
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  citation-key: NoorshamsBelief2013
  container-title: The Journal of Machine Learning Research
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  ISSN: 1532-4435
  issue: '1'
  issued:
    - year: 2013
      month: 1
      day: 1
  page: 2799–2835
  title: >-
    Belief propagation for continuous state spaces: stochastic message-passing
    with quantitative guarantees
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- id: NualartChaotic2000
  accessed:
    - year: 2020
      month: 5
      day: 21
  author:
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      given: David
    - family: Schoutens
      given: Wim
  citation-key: NualartChaotic2000
  container-title: Stochastic Processes and their Applications
  container-title-short: Stochastic Processes and their Applications
  DOI: 10.1016/S0304-4149(00)00035-1
  ISSN: 0304-4149
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  issued:
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      month: 11
      day: 1
  language: en
  page: 109-122
  title: Chaotic and predictable representations for Lévy processes
  type: article-journal
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  volume: '90'

- id: OHaganPolynomial2013
  author:
    - family: O’Hagan
      given: Anthony
  citation-key: OHaganPolynomial2013
  issued:
    - year: 2013
  language: en
  page: '20'
  title: 'Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective'
  type: article-journal

- id: OladyshkinDatadriven2012
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    - year: 2020
      month: 12
      day: 17
  author:
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    - family: Nowak
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  citation-key: OladyshkinDatadriven2012
  container-title: Reliability Engineering & System Safety
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  language: en
  page: 179-190
  title: >-
    Data-driven uncertainty quantification using the arbitrary polynomial chaos
    expansion
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- id: RahmanWienerHermite2017
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    - year: 2021
      month: 1
      day: 29
  author:
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      given: Sharif
  citation-key: RahmanWienerHermite2017
  container-title: arXiv:1704.07912 [math]
  issued:
    - year: 2017
      month: 4
      day: 25
  language: en
  title: >-
    Wiener-Hermite Polynomial Expansion for Multivariate Gaussian Probability
    Measures
  type: article-journal
  URL: http://arxiv.org/abs/1704.07912

- id: SchoutensOrthogonal2001
  accessed:
    - year: 2021
      month: 3
      day: 31
  author:
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      given: Wim
  citation-key: SchoutensOrthogonal2001
  container-title: Journal of Mathematical Analysis and Applications
  container-title-short: Journal of Mathematical Analysis and Applications
  DOI: 10.1006/jmaa.2000.7159
  ISSN: 0022-247X
  issue: '2'
  issued:
    - year: 2001
      month: 1
      day: 15
  language: en
  page: 515-531
  title: Orthogonal Polynomials in Stein's Method
  type: article-journal
  volume: '253'

- id: SchoutensStochastic2000
  accessed:
    - year: 2020
      month: 5
      day: 21
  author:
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      given: Wim
  citation-key: SchoutensStochastic2000
  collection-title: Lecture Notes in Statistics
  DOI: 10.1007/978-1-4612-1170-9
  event-place: New York
  ISBN: 978-0-387-95015-0
  issued:
    - year: 2000
  language: en
  publisher: Springer-Verlag
  publisher-place: New York
  title: Stochastic Processes and Orthogonal Polynomials
  type: book
  URL: https://www.springer.com/gp/book/9780387950150

- id: WanMultiElement2006
  accessed:
    - year: 2020
      month: 12
      day: 17
  author:
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      given: Xiaoliang
    - family: Karniadakis
      given: George Em
  citation-key: WanMultiElement2006
  container-title: SIAM Journal on Scientific Computing
  container-title-short: SIAM J. Sci. Comput.
  DOI: 10.1137/050627630
  ISSN: 1064-8275
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  issued:
    - year: 2006
      month: 1
      day: 1
  page: 901-928
  publisher: Society for Industrial and Applied Mathematics
  title: >-
    Multi-Element Generalized Polynomial Chaos for Arbitrary Probability
    Measures
  type: article-journal
  URL: https://www.math.lsu.edu/~xlwan/papers/journal/megpc2.pdf
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- id: WienerHomogeneous1938
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    - year: 2020
      month: 5
      day: 20
  author:
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      given: Norbert
  citation-key: WienerHomogeneous1938
  container-title: American Journal of Mathematics
  container-title-short: American Journal of Mathematics
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  ISSN: '00029327'
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  language: en
  page: '897'
  publisher: Johns Hopkins University Press
  title: The Homogeneous Chaos
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- id: WitteveenModeling2006
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    - year: 2020
      month: 12
      day: 17
  author:
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    - family: Bijl
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  citation-key: WitteveenModeling2006
  container-title: 44th AIAA Aerospace Sciences Meeting and Exhibit
  DOI: 10.2514/6.2006-896
  issued:
    - year: 2006
  publisher: American Institute of Aeronautics and Astronautics
  title: Modeling Arbitrary Uncertainties Using Gram-Schmidt Polynomial Chaos
  type: chapter
  URL: https://arc.aiaa.org/doi/abs/10.2514/6.2006-896

- id: XiuHighOrder2005
  accessed:
    - year: 2021
      month: 1
      day: 29
  author:
    - family: Xiu
      given: Dongbin
    - family: Hesthaven
      given: Jan S.
  citation-key: XiuHighOrder2005
  container-title: SIAM Journal on Scientific Computing
  container-title-short: SIAM J. Sci. Comput.
  DOI: 10.1137/040615201
  ISSN: 1064-8275
  issue: '3'
  issued:
    - year: 2005
      month: 1
      day: 1
  page: 1118-1139
  publisher: Society for Industrial and Applied Mathematics
  title: High-Order Collocation Methods for Differential Equations with Random Inputs
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  volume: '27'

- id: XiuNumerical2010
  author:
    - family: Xiu
      given: Dongbin
  citation-key: XiuNumerical2010
  DOI: 10.2307/j.ctv7h0skv
  event-place: USA
  ISBN: 978-1-4008-3534-8
  issued:
    - year: 2010
      month: 7
      day: 1
  language: en
  number-of-pages: '142'
  publisher: Princeton University Press
  publisher-place: USA
  title: 'Numerical Methods for Stochastic Computations: A Spectral Method Approach'
  type: book

- id: XiuWienerAskey2002
  accessed:
    - year: 2021
      month: 2
      day: 12
  author:
    - family: Xiu
      given: Dongbin
    - family: Karniadakis
      given: George Em
  citation-key: XiuWienerAskey2002
  container-title: SIAM Journal on Scientific Computing
  container-title-short: SIAM J. Sci. Comput.
  DOI: 10.1137/S1064827501387826
  ISSN: 1064-8275
  issue: '2'
  issued:
    - year: 2002
      month: 1
      day: 1
  page: 619-644
  publisher: Society for Industrial and Applied Mathematics
  title: The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations
  type: article-journal
  URL: http://www.sci.utah.edu/publications/xiu02/Xiu_SIAM_JSC2002.pdf
  volume: '24'

- id: ZhangQuantifying2019
  accessed:
    - year: 2020
      month: 9
      day: 15
  author:
    - family: Zhang
      given: Dongkun
    - family: Lu
      given: Lu
    - family: Guo
      given: Ling
    - family: Karniadakis
      given: George Em
  citation-key: ZhangQuantifying2019
  container-title: Journal of Computational Physics
  container-title-short: Journal of Computational Physics
  DOI: 10.1016/j.jcp.2019.07.048
  ISSN: 0021-9991
  issued:
    - year: 2019
      month: 11
      day: 15
  language: en
  page: '108850'
  publisher: Academic Press
  title: >-
    Quantifying total uncertainty in physics-informed neural networks for
    solving forward and inverse stochastic problems
  type: article-journal
  URL: >-
    https://deepai.org/publication/quantifying-total-uncertainty-in-physics-informed-neural-networks-for-solving-forward-and-inverse-stochastic-problems
  volume: '397'

- id: ZhengAdaptive2015
  accessed:
    - year: 2020
      month: 12
      day: 17
  author:
    - family: Zheng
      given: Mengdi
    - family: Wan
      given: Xiaoliang
    - family: Karniadakis
      given: George Em
  citation-key: ZhengAdaptive2015
  container-title: Applied Numerical Mathematics
  container-title-short: Applied Numerical Mathematics
  DOI: 10.1016/j.apnum.2014.11.006
  ISSN: 0168-9274
  issued:
    - year: 2015
      month: 4
      day: 1
  language: en
  page: 91-110
  title: >-
    Adaptive multi-element polynomial chaos with discrete measure: Algorithms
    and application to SPDEs
  type: article-journal
  URL: https://www.math.lsu.edu/~xlwan/papers/journal/megpc_discrete.pdf
  volume: '90'
...
