Skip to main content

2014 | OriginalPaper | Buchkapitel

Efficient Calculation of Uncertainty Quantification

verfasst von : E. Jan W. ter Maten, Roland Pulch, Wil H. A. Schilders, H. H. J. M. Janssen

Erschienen in: Progress in Industrial Mathematics at ECMI 2012

Verlag: Springer International Publishing

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

search-config
loading …

Abstract

We consider Uncertainty Quantification (UQ) by expanding the solution in so-called generalized Polynomial Chaos expansions. In these expansions the solution is decomposed into a series with orthogonal polynomials in which the parameter dependency becomes an argument of the orthogonal polynomial basis functions. The time and space dependency remains in the coefficients. In UQ two main approaches are in use: Stochastic Collocation (SC) and Stochastic Galerkin (SG). Practice shows that in many cases SC is more efficient for similar accuracy as obtained by SG. In SC the coefficients in the expansion are approximated by quadrature and thus lead to a large series of deterministic simulations for several parameters. We consider strategies to efficiently perform this sequence of deterministic simulations within SC.

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!

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!

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!

Literatur
1.
Zurück zum Zitat Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia (2005)CrossRefMATH Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia (2005)CrossRefMATH
2.
Zurück zum Zitat Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., Wever, U.: Polynomial chaos for the approximation of uncertainties: chances and limits. Eur. J. Appl. Math. 19, 149–190 (2008)CrossRefMATHMathSciNet Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., Wever, U.: Polynomial chaos for the approximation of uncertainties: chances and limits. Eur. J. Appl. Math. 19, 149–190 (2008)CrossRefMATHMathSciNet
3.
Zurück zum Zitat Baur, U., Beattie, C., Benner, P., Gugercin, S.: Interpolatory projection methods for parameterized model reduction. SIAM J. Comput. 33, 2489–2518 (2011)CrossRefMATHMathSciNet Baur, U., Beattie, C., Benner, P., Gugercin, S.: Interpolatory projection methods for parameterized model reduction. SIAM J. Comput. 33, 2489–2518 (2011)CrossRefMATHMathSciNet
4.
Zurück zum Zitat Benner, P., Schneider, A.: Model reduction for linear descriptor systems with many ports. In: Günther, M., Bartel, A., Brunk, M., Schöps, S., Striebel, M. (eds.) Progress in Industrial Mathematics at ECMI 2010. Mathematics in Industry, vol. 17, pp. 137–143. Springer, Berlin (2012)CrossRef Benner, P., Schneider, A.: Model reduction for linear descriptor systems with many ports. In: Günther, M., Bartel, A., Brunk, M., Schöps, S., Striebel, M. (eds.) Progress in Industrial Mathematics at ECMI 2010. Mathematics in Industry, vol. 17, pp. 137–143. Springer, Berlin (2012)CrossRef
5.
Zurück zum Zitat Benner, P., Hinze, M., ter Maten, E.J.W. (eds.): Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74. Springer, Berlin (2011) Benner, P., Hinze, M., ter Maten, E.J.W. (eds.): Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74. Springer, Berlin (2011)
6.
Zurück zum Zitat Bi, Y., van der Kolk, K.-J., Fernández Villena, J., Silveira, L.M., van der Meijs, N.: Fast statistical analysis of RC nets subject to manufacturing variabilities. In: Proceedings of DATE 2011, Grenoble, 14–18 March 2011 Bi, Y., van der Kolk, K.-J., Fernández Villena, J., Silveira, L.M., van der Meijs, N.: Fast statistical analysis of RC nets subject to manufacturing variabilities. In: Proceedings of DATE 2011, Grenoble, 14–18 March 2011
7.
Zurück zum Zitat Doorn, T.S., Croon, J.A., ter Maten, E.J.W., Di Bucchianico, A.: A yield centric statistical design method for optimization of the SRAM active column. In: Proceedings of the IEEE ESSCIRC’09, 35th European Solid-State Circuits Conference, Athens, pp. 352–355 (2009) Doorn, T.S., Croon, J.A., ter Maten, E.J.W., Di Bucchianico, A.: A yield centric statistical design method for optimization of the SRAM active column. In: Proceedings of the IEEE ESSCIRC’09, 35th European Solid-State Circuits Conference, Athens, pp. 352–355 (2009)
8.
Zurück zum Zitat Ernst, O.G., Mugler, A., Starkloff, H.-J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM Math. Model. Numer. Anal. 46, 317–339 (2012)CrossRefMATHMathSciNet Ernst, O.G., Mugler, A., Starkloff, H.-J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM Math. Model. Numer. Anal. 46, 317–339 (2012)CrossRefMATHMathSciNet
9.
Zurück zum Zitat Feng, L., Benner, P.: A robust algorithm for parametric model order reduction. PAMM Proc. Appl. Math. Mech. 7, 1021501–1021502 (2007)CrossRef Feng, L., Benner, P.: A robust algorithm for parametric model order reduction. PAMM Proc. Appl. Math. Mech. 7, 1021501–1021502 (2007)CrossRef
10.
Zurück zum Zitat Fernández Villena, J., Silveira, L.M.: Multi-dimensional automatic sampling schemes for multi-point modeling methodologies. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 30(8), 1141–1151 (2011)CrossRef Fernández Villena, J., Silveira, L.M.: Multi-dimensional automatic sampling schemes for multi-point modeling methodologies. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 30(8), 1141–1151 (2011)CrossRef
11.
Zurück zum Zitat Fijnvandraat, J.G., Houben, S.H.M.J., ter Maten, E.J.W., Peters, J.M.F.: Time domain analog circuit simulation. J. Comput. Appl. Math. 185, 441–459 (2006)CrossRefMATHMathSciNet Fijnvandraat, J.G., Houben, S.H.M.J., ter Maten, E.J.W., Peters, J.M.F.: Time domain analog circuit simulation. J. Comput. Appl. Math. 185, 441–459 (2006)CrossRefMATHMathSciNet
12.
Zurück zum Zitat Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATH Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATH
14.
Zurück zum Zitat Ilievski, Z., Xu, H., Verhoeven, A., ter Maten, E.J.W., Schilders, W.H.A., Mattheij, R.M.M.: Adjoint transient sensitivity analysis in circuit simulation. In: Ciuprina, G., Ioan, D. (eds.) Scientific Computing in Electrical Engineering SCEE 2006. Mathematics in Industry, vol. 11, pp. 183–189. Springer, Berlin (2007) Ilievski, Z., Xu, H., Verhoeven, A., ter Maten, E.J.W., Schilders, W.H.A., Mattheij, R.M.M.: Adjoint transient sensitivity analysis in circuit simulation. In: Ciuprina, G., Ioan, D. (eds.) Scientific Computing in Electrical Engineering SCEE 2006. Mathematics in Industry, vol. 11, pp. 183–189. Springer, Berlin (2007)
16.
Zurück zum Zitat Jansen, L., Tischendorf, C.: Effective numerical computation of parameter dependent problems. In: Michielsen, B.L., Poirier, J.-R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Mathematics in Industry, vol. 16, pp. 49–57. Springer, Berlin (2012)CrossRef Jansen, L., Tischendorf, C.: Effective numerical computation of parameter dependent problems. In: Michielsen, B.L., Poirier, J.-R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Mathematics in Industry, vol. 16, pp. 49–57. Springer, Berlin (2012)CrossRef
17.
Zurück zum Zitat Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification, with Applications to Computational Fluid Dynamics. Springer, Science+Business Media B.V., Dordrecht (2010)CrossRefMATH Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification, with Applications to Computational Fluid Dynamics. Springer, Science+Business Media B.V., Dordrecht (2010)CrossRefMATH
18.
20.
21.
Zurück zum Zitat Pulch, R.: Polynomial chaos for the computation of failure probabilities in periodic problems. In: Roos, J., Costa, L.R.J. (eds.) Scientific Computing in Electrical Engineering SCEE 2008. Mathematics in Industry, vol. 14, pp. 191–198. Springer, Berlin (2010)CrossRef Pulch, R.: Polynomial chaos for the computation of failure probabilities in periodic problems. In: Roos, J., Costa, L.R.J. (eds.) Scientific Computing in Electrical Engineering SCEE 2008. Mathematics in Industry, vol. 14, pp. 191–198. Springer, Berlin (2010)CrossRef
22.
Zurück zum Zitat Pulch, R.: Polynomial chaos for linear differential algebraic equations with random parameters. Int. J. Uncertain. Quantif. 1(3), 223–240 (2011)CrossRefMATHMathSciNet Pulch, R.: Polynomial chaos for linear differential algebraic equations with random parameters. Int. J. Uncertain. Quantif. 1(3), 223–240 (2011)CrossRefMATHMathSciNet
23.
Zurück zum Zitat Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.): Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13. Springer, Berlin (2008) Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.): Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13. Springer, Berlin (2008)
24.
Zurück zum Zitat Stavrakakis, K., Wittig, T., Ackermann, W., Weiland, T.: Parametric model order reduction by neighbouring subspaces. In: Michielsen, B., Poirier, J.-R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Series Mathematics in Industry, vol. 16, pp. 443–451. Springer, Berlin (2012)CrossRef Stavrakakis, K., Wittig, T., Ackermann, W., Weiland, T.: Parametric model order reduction by neighbouring subspaces. In: Michielsen, B., Poirier, J.-R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Series Mathematics in Industry, vol. 16, pp. 443–451. Springer, Berlin (2012)CrossRef
25.
Zurück zum Zitat Tasić, B., Dohmen, J.J., ter Maten, E.J.W., Beelen, T.G.J., Schilders, W.H.A., de Vries, A., van Beurden, M.: Robust DC and efficient time-domain fast fault simulation. COMPEL (2014, accepted) Tasić, B., Dohmen, J.J., ter Maten, E.J.W., Beelen, T.G.J., Schilders, W.H.A., de Vries, A., van Beurden, M.: Robust DC and efficient time-domain fast fault simulation. COMPEL (2014, accepted)
26.
Zurück zum Zitat ter Maten, E.J.W., Wittich, O., Di Bucchianico, A., Doorn, T.S., Beelen, T.G.J.: Importance sampling for determining SRAM yield and optimization with statistical constraint. In: Michielsen, B.L., Poirier, J.-R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Mathematics in Industry, vol. 16, pp. 39–48. Springer, Berlin (2012)CrossRef ter Maten, E.J.W., Wittich, O., Di Bucchianico, A., Doorn, T.S., Beelen, T.G.J.: Importance sampling for determining SRAM yield and optimization with statistical constraint. In: Michielsen, B.L., Poirier, J.-R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Mathematics in Industry, vol. 16, pp. 39–48. Springer, Berlin (2012)CrossRef
28.
Zurück zum Zitat Xiu, D.: Numerical Methods for Stochastic Computations - A Spectral Method Approach. Princeton University Press, Princeton (2010)MATH Xiu, D.: Numerical Methods for Stochastic Computations - A Spectral Method Approach. Princeton University Press, Princeton (2010)MATH
Metadaten
Titel
Efficient Calculation of Uncertainty Quantification
verfasst von
E. Jan W. ter Maten
Roland Pulch
Wil H. A. Schilders
H. H. J. M. Janssen
Copyright-Jahr
2014
DOI
https://doi.org/10.1007/978-3-319-05365-3_50

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.