Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Implementing Quantum Finite Automata Algorithms on Noisy Devices Kapitel lesen Erstes Kapitel lesen

2021 | OriginalPaper | Buchkapitel

Second Order Moments of Multivariate Hermite Polynomials in Correlated Random Variables

verfasst von : Laura Lyman, Gianluca Iaccarino

Erschienen in: Computational Science – ICCS 2021

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Polynomial chaos methods can be used to estimate solutions of partial differential equations under uncertainty described by random variables. The stochastic solution is represented by a polynomial expansion, whose deterministic coefficient functions are recovered through Galerkin projections. In the presence of multiple uncertainties, the projection step introduces products (second order moments) of the basis polynomials. When the input random variables are correlated Gaussians, calculating the products of the corresponding multivariate basis polynomials is not straightforward and can become computationally expensive. We present a new expression for the products by introducing multiset notation for the polynomial indexing, which allows for simple and efficient evaluation of the second-order moments of correlated multivariate Hermite polynomials.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Uncertainty Quantification of Coupled 1D Arterial Blood Flow and 3D Tissue Perfusion Models Using the INSIST Framework
Fußnoten
1
Note that the set \(\min \{i \in [n] \mid \ell \le \sum _{r= 1}^n \alpha _r\}\) is nonempty, because \(\ell \le k= \sum _{r = 1}^n \alpha _r,\) so \(i = n\) always satisfies the condition that \(\ell \le \sum _{r = 1}^i \alpha _r.\)
 
2
Note that these multivariate \(H_{\boldsymbol{\alpha }}\) are not normalized to force \({\mathbb {E}}(H_{\boldsymbol{\alpha }}({\boldsymbol{\xi }})^2) = 1\), as is sometimes done in other literature [15].
 
3
Accordingly, we introduce \(s(\boldsymbol{\alpha })\) as multiset notation rather than a literal multiset. An underlying philosophy of multisets is that copies of elements cannot be picked out or distinguished by (say) an indexing convention [8, 10, 16]. For our purposes, however, we want to treat such copies as distinct.
 
4
For instance, in Python3, the combinatorics module in itertools [20] suffices.
 
5
Counting the number of such index matrices, which are often called contingency tables with fixed margins in statistics literature, is well-studied [3, 6] and can be done in poly(n) time [4]. This does not mean that the number of contingency tables is poly(n) but that algorithms can produce the total count of them in poly(n) time.
 
Literatur
1.
Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. 48(2), 385–392 (1947)MathSciNetCrossRef
2.
Constantine, P.: Spectral methods for parametrized matrix equations. Ph.D. thesis, Stanford University (2009)
3.
4.
Dittmer, S.: Counting linear extensions and contingency tables. Ph.D. thesis, University of California, Los Angeles (2019)
5.
Ernst, O.G., Mugler, A., Starkloff, H., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM: M2AN 46(2), 317–339 (2012)
6.
Gail, M., Mantel, N.: Counting the number of \(r \times c\) contingency tables with fixed margins. J. Am. Stat. Assoc. 72(360), 859–862 (1977)MathSciNetCrossRef
7.
Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, Heidelberg (1991)CrossRef
8.
9.
10.
Knuth, D.E.: The Art of Computer Programming: A Draft of Section 7.2.1.1, Generating all \(n\)-Tuples. Addison-Wesley (06 2004)
11.
Lovasz, L., Pelikan, J., Vesztergombi, K.: Discrete Mathematics: Elementary and Beyond. Springer (2003)
12.
Lyman, L., Iaccarino, G.: Extending bluff-and-fix estimates for polynomial chaos expansions. J. Comput. Sci. 50, 101287 (2021)MathSciNetCrossRef
13.
Noreddine, S., Nourdin, I.: On the gaussian approximation of vector-valued multiple integrals. J. Multivariate Anal. 102(6), 1008–1017 (2011)MathSciNetCrossRef
14.
15.
Rahman, S.: Wiener-Hermite polynomial expansion for multivariate Gaussian probability measures. J. Math. Anal. Appl. 454(1), 303–334 (2017)MathSciNetCrossRef
16.
Ruskey, F.: Combinatorial Generation. Preliminary working draft. University of Victoria, Victoria, BC, Canada, pp. 71–73, §4.5.1 (2003)
17.
Slepian, D.: On the symmetrized kronecker power of a matrix and extensions of Mehler’s formula for hermite polynomials. SIAM J. Math. Anal. 3(4), 606–616 (1972)MathSciNetCrossRef
18.
Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (2001)
19.
Szegő, G.: Orthogonal Polynomials. Colloquium publ, American Mathematical Society, American Math. Soc (1975)
20.
Van Rossum, G.: The Python Library Reference, release 3.8.2. Python Software Foundation (2020)
21.
22.
Xiu, D.: Generalized (Wiener-Askey) Polynomial Chaos. Ph.D. thesis, Brown University (2004)
Metadaten
Titel
Second Order Moments of Multivariate Hermite Polynomials in Correlated Random Variables
verfasst von
Laura Lyman
Gianluca Iaccarino
Copyright-Jahr
2021
Buch
Computational Science – ICCS 2021

Print ISBN: 978-3-030-77979-5

Electronic ISBN: 978-3-030-77980-1

Copyright-Jahr: 2021

DOI
https://doi.org/10.1007/978-3-030-77980-1_53